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In This Chapter

^ Steering clear of the late penalty by enrolling in Part D when you should ^ Understanding what you pay if you don’t sign up on time

He Part D late penalty basically sets a deadline. It’s a device for persuading you to join a Medicare drug plan when you first become eligible —

And not just when you happen to feel like it. If the date of your personal Part D enrollment deadline passes and you enroll in Part D months or years later, prepare to

Pay a penalty in the form of a surcharge that’s added to every monthly Part D premium for as long as you stay in the program.

Face a higher penalty the longer you delay signing up for Part D.

Experience a probable increase in your penalty cost each year.

Pay a lot more for the same drug coverage over time than if you’d signed up for Part D as soon as you were able.

Heavy stuff, huh? People considering Part D often think the late penalty is unfair, though actually it exists for several reasons (see the later sidebar "What were they thinking. . . when they created the late penalty?"). Medicare strictly enforces the late penalty — and takes the view that being confused about Part D is no defense for signing up late. If you have no other prescription drug insurance that’s at least of equal value to basic Medicare drug coverage (or Creditable; See Chapter 6), or if you’re not receiving Extra Help (see Chapter 5), every month that you delay enrolling in Part D makes the penalty bigger. That’s why I’m sharing its implications in the context of deciding whether you need to sign up for Part D right now. Very often, the late penalty is an important part of that decision — or a consequence of it.

So in this chapter, I explain how to avoid the penalty by signing up at the right time — depending on your individual circumstances. Then I describe how the penalty is calculated, how it grows ever larger over time, and whether putting off joining Part D is worth risking it.

What were they thinking. . . when they created the late penalty?

If you’re like many people, you probably feel indignant about the late penalty. How can joining the Medicare drug program be voluntary if you’re going to be penalized for not signing up on time? Why can’t you wait to join Part D until you need it, or when you choose to? Here are the arguments:

Part D gives protection against future risks, just like any other form of insurance. You buy home insurance in case the basement catches fire; you buy car insurance in case you total the station wagon.

If no one joined Part D until she needed prescription drugs, the system couldn’t function. Here comes that insurance concept again: Healthy people must be in the system to spread the financial risk and hold down costs. If only sick people enrolled, drug coverage would be so expensive that most people wouldn’t be able to afford it.

Part D isn’t the only Medicare program that has a late penalty. So does Part B,

Which imposes an extra 10 percent on the premiums for every year you delay signing up, for the same reasons given previously. But because almost everyone signs up for Part B as soon as they’re eligible, the fact that a late penalty exists often goes unnoticed. (Part B helps you pay to see a doctor and use other outpatient services; see Chapter 1 for details.)

Of course, Part D isn’t entirely insurance in the ordinary commercial sense. It’s also a benefit, and that’s why the late penalty seems a bit confusing. The federal government subsidizes your drug coverage in various ways — to a certain degree in the initial coverage period, and to a very large extent if your income is low enough to qualify you for Extra Help (which I explain in Chapter 5) or your drug costs are high enough to take you into the catastrophic phase of coverage. Nonetheless, Part D is built on insurance principles, with the late penalty as a cornerstone.

Avoiding a Late Penalty by Signing Up for Part D at the Right Time

The only way to avoid a late penalty is to meet your own particular deadline. Getting the timing right depends on your situation, which is probably one of the following:

You’re joining Medicare right now (or are about to do so) but don’t have good drug coverage.

I You already have good drug coverage from another source (such as an employer health plan) but are about to lose it or drop it.

I You’re just returning to live in the United States after living abroad.

I You’ve just been released from prison.

In the next few sections, I consider each of these circumstances and also provide a tip for sidestepping an obstacle hidden in the fine print that may trip you up if you don’t know about it.

Ij$jAB££ If you miss your personal deadline for joining a Part D plan, your next chance to join is the annual enrollment period between November 15 and December 31 (with coverage beginning January 1). See the later section "The Price of Missing Your Personal Enrollment Deadline" to find out about the costs you have to pay for letting your deadline slide.

When you join Medicare and don’t have creditable drug coverage

Iijj|kB£* You can get Part D as long as you’re enrolled in Medicare Part A (hospital insurance) and/or Part B (doctor and outpatient services), as explained in Chapter 1. Whether you need to enroll in Part D at the same time as you sign up for Medicare depends on whether you have other drug coverage regarded as creditable under Part D rules. (Coverage is Creditable When it provides at least as much value as minimum Medicare drug coverage. I explain this concept and how it affects decisions about joining Part D in detail in Chapter 6.) If you have creditable drug coverage when you sign up for Medicare, you don’t have to sign up for Part D at the same time.

But what if you Don’t Have such coverage?

If you’re turning 65 soon and Don’t Have creditable drug coverage from another source, you need to enroll in a Part D drug plan at the same time as you enroll in Part A or Part B to avoid a late penalty. That means signing up within the span of your seven-month initial Medicare enrollment period, which starts three months before the month you turn 65 and ends three months after it. For example, if your birthday is in April, your enrollment period runs from the beginning of January to the end of July, and you need to enroll in a Part D plan in July at the latest (and start receiving Medicare drug coverage on August 1) to guarantee avoiding a late penalty.

If you’re joining Medicare at a younger age because of disability and Don’t Have creditable drug coverage from elsewhere, the same rules apply. But in this case, your seven-month initial enrollment period begins three months before your 25th month of receiving disability payments and ends three months later. So, as in the preceding example, if your 25th month on disability falls in April, you need to sign up for a Part D plan in July at the latest to dodge a late penalty.

If you join Medicare at a younger age due to a disability and incur a late penalty because you didn’t sign up for Part D at the same time, you get another chance when you turn 65. You can use the regular seven-month Medicare enrollment period around your 65th birthday to re-enroll in Part D and get rid of the late penalty.

I If your income is limited and you qualify for Extra Help (as explained in Chapter 5), by law you Won’t Be hit with a penalty if you sign up late.

When you lose or drop your current creditable drug coverage

If you have creditable drug coverage from another source, you Don’t Need to enroll in Part D when you sign up for Medicare, as explained in Chapter 6. But what if you lose that coverage? Or your coverage suddenly ceases to be creditable? Or you decide to drop it?

If you lose creditable coverage Involuntarily, Meaning that you lose it through no fault of your own — for example, if your employer’s plan terminates or begins offering benefits that overall are of less value than Medicare drug coverage — you’ll qualify for a special enrollment period in which you can sign up for Part D without incurring a late penalty. That period lasts for 63 days after you’re notified that your creditable coverage will end Or 63 days after the date it actually ends (whichever’s later). See the later section "What the 63-day rule really means" for some crucial details if you’re in this boat.

I If you deliberately drop your creditable coverage, perhaps if it becomes too expensive to maintain, you Won’t Receive a special period to enroll in Part D. Instead, you must sign up during the annual enrollment, which runs from November 15 to December 31 each year. To avoid a late penalty, you can’t go for more than 63 days without creditable coverage. So you’d have to keep your current coverage at least until the end of October, because your Part D coverage wouldn’t begin until January 1. (Ideally, however, you’d keep your own coverage until the end of the year so you continue to be protected by insurance.)

You may find out that your previous sponsored coverage (which you’ve already lost or dropped) Wasn’t Creditable only when you enroll in a Part D plan. Then, almost certainly, you’ll be slapped with a late penalty. At that point, you’ll need to ask the plan for a Reconsideration — a review during which your claim of not realizing that your previous coverage wasn’t creditable will be investigated. I cover this situation, as well as other circumstances in which people feel they’ve received a late penalty unfairly, in Chapter 13. That’s where I also explain in detail how to go about asking for a reconsideration.

When you return to the United States after living abroad

You can’t receive Part D coverage while living abroad, and you Aren’t Expected to sign up for it until you return to live in the United States

Permanently. (This rule is different from the one for Part B enrollment when you’ve been living abroad, which I clarify in Chapter 1.) When you return to the U. S., you can join Part D without risking a penalty in one of two ways, depending on your circumstances:

If you turned 65 while living abroad: You get a special initial enrollment period (IEP) to sign up with a Part D plan on your return. This period lasts seven months — three months before the month of your return to the U. S., the month of your return, and three months after your return. Enrolling in a plan no later than the end of the month before you return ensures you can use your Part D coverage as soon as you arrive back.

I If you were eligible for Part D before moving abroad: You get a special enrollment period (SEP) on your return. You can enroll in a plan without having to apply for an SEP (as I explain in Chapter 17). The SEP begins on the date of your return to the U. S. and ends 63 days later. (See the later section "What the 63-day rule really means" for details.)

When you’re released from prison

You can’t receive Part D coverage while incarcerated in a prison or any other correctional institution, and your stay doesn’t count toward the Part D late penalty. In this situation, the rules are the same as for someone returning from abroad, as I explain in the previous section. If you turned 65 in prison, you get a seven-month initial enrollment period (IEP) to sign up for a Part D plan, lasting from three months before the month of your release to three months after. Otherwise, you get a special enrollment period (SEP) to sign up for a Part D plan, beginning on the day of your release and ending 63 days later. (See the following section for more on the 63-day rule.)

What the 63-day rule really means

The 63-day rule Is usually explained as the time you have during a special enrollment period to enroll in a Part D plan and avoid a late penalty. But this explanation isn’t precisely accurate. Rather, you must be actually Receiving Part D coverage within 63 days to avoid a penalty.

Say you lose your current creditable drug coverage on March 31. Counting 63 days from that date brings you to June 2. If you leave it to the last minute and sign up with a Part D plan on June 1 or 2, you’re still within the 63-day time frame. But you’re not avoiding the late penalty because, under Part D rules, your drug coverage actually begins on the first day of the month After You enroll — in this example, July 1. You’re then penalized for one month without coverage, which may not amount to much money at first but can increase quite a lot with time, as I explain later in this chapter. So think in terms of 60 days instead (or 59 if the time frame includes February), and you’ll be okay.

The Price of Missing Your Personal Enrollment Deadline

The purpose of this section isn’t to scare you half to death, but to give you a practical awareness of what the late penalty means so that — if it figures at all in the decisions you make about joining Part D (and it may not) — you can make an informed choice. Here, I explain how the penalty is calculated and how it can grow over the years. In the end, only you can decide whether ignoring the late penalty is worth the risk.

Looking at how the late penalty is calculated

The basis of the late penalty is something called the National average premium (NAP). Every fall, Medicare works out the average of all the premiums that Part D plans nationwide will charge during the following year. This dollar amount becomes the NAP for that next year. If you incur the late penalty, you’ll pay 1 percent of the NAP for every month you were without creditable coverage and didn’t sign up for Part D. This formula works out at 12 percent a year.

JttNG/ If you miss your personal deadline for joining a Part D plan, your next chance is the annual enrollment period that runs from November 15 to December 31 (with coverage beginning January 1). If you also miss that window, you have to wait another 12 months to sign up, increasing your penalty amount by another 12 percent. Every extra year of delay adds 12 percent to the penalty. Check out the following examples to see the math for yourself:

I Rebecca turned 65 and signed up for Medicare in March 2007. But by the time her personal enrollment period expired at the end of June, she hadn’t signed up for Part D (and had no comparable drug coverage). Her next chance to join Part D was during the open enrollment period between November 15 and December 31, 2007. She decided to do it, and her drug coverage began January 1, 2008. By then she was six months over her deadline (July through December). So her late penalty in 2008 was 6 percent (1 percent x 6 months) of the 2008 NAP, which was $27.93. One percent of that amount is 28 cents. So Rebecca’s monthly late penalty was calculated at 28 cents multiplied by 6 (her months without coverage), which came to $1.68. Medicare rounds the penalty to the nearest 10 cents, so Rebecca actually paid $1.70 a month in 2008, or $20.40 for the whole year, on top of her plan’s premiums.

I Brad was 70 years old and already in Medicare when Part D drug coverage began in 2006. Because the program was just starting, the initial enrollment period for that year was extended into May. But Brad couldn’t figure out what to do about Part D, and he let the deadline pass.

He had another chance to sign up during open enrollment at the end of 2006, but he let that go by, too. Finally, he enrolled in a Part D plan at the end of 2007. By then, he’d been without drug coverage for 19 months (June 2006 to December 2007). So in 2008, he paid a 19 percent penalty — 28 cents multiplied by 19, which came to $5.32. This amount was rounded to the nearest 10 cents, so Brad paid $5.30 a month, or $63.60 over the whole year, on top of his plan’s premiums.

Understanding how the late penalty can add up over time

Maybe the amounts in the preceding section’s examples don’t sound like too big of a deal. But that isn’t the end of it. Rebecca won’t pay the same penalty amount she was first assessed — $1.70 a month in 2008 — for all the years to come. Nor will Brad pay his penalty — $5.30 a month in 2008 — for as long as he’s in the Part D program. They’ll both pay a new penalty amount each successive year, and so will anyone else who has a late penalty. That’s because Medicare recalculates the NAP annually. If the NAP changes, the crucial 1 percent also changes — and so does everyone’s penalty amount.

One part of the calculation doesn’t change — the number of months anyone goes without drug coverage. For example, Brad will continue to pay a penalty of 19 percent of the NAP due to the 19 months he lacked drug coverage. But the dollar amount of that 19 percent will change each year as the dollar amount of the NAP changes. So in 2009, he’ll pay 19 percent of the 2009 NAP, and the next year he’ll pay 19 percent of the 2010 NAP, and so on every year.

I cover the NAP’s yearly variation and provide some estimates of the effect of NAP changes over time in the following sections.

Wondering how much the NAP will change each year

If the NAP rises, everybody with a penalty pays more as time goes by. But the truth is, nobody knows how much the NAP will change annually. Even Medicare doesn’t know until September of each year when it works out the average of all Part D plans’ premiums for the following year. Whether the NAP rises or falls depends entirely on the plans.

Certainly you can find out what the NAP will be for Next Year. That information is included in the Medicare & You Handbook that Medicare sends to all of its beneficiaries every October. The last section in the handbook explains Medicare costs for the following year, including the Part D national average premium (referred to more bureaucratically in the handbook as the National base beneficiary premium) And the 1 percent penalty amount. (You can also read the Medicare & You Handbook online at Www. medicare. gov.)

As this book goes to press, the 2009 NAP is unknown. So to date, only one NAP change, from 2007 — the first year the penalty was imposed — to 2008, has occurred. Between those two years, the NAP rose only 58 cents (from $27.35 in 2007 to $27.93 in 2008), which was far less than anyone expected. Will it always stay so low? Nobody knows. But it’s worth remembering that in these days of Part D’s infancy, the plans have tried to keep their premiums relatively low in order to attract customers. If the market shakes out in the future, with fewer plans competing, you can expect premiums to rise, taking the NAP upward as well.

Estimating how the late penalty may grow long-term

At this point, presenting a chart that shows exactly how much penalty amounts can grow over 5 years, 10 years, 20 years, or more would be useful. Of course, that’s impossible due to the annual reassessment of the NAP. But, what the heck, I’m going to do some educated guesswork so you can at least get an idea of how those amounts can accumulate long-term.

The NAP may creep up very gradually by a dollar or less every year throughout the next decade or so. Or it may jump around all over the place — going up by $5 or $10 one year, or even going down by a few dollars or cents the next. But here I make very conservative assumptions. In fact, in Table 8-1, I make a totally unrealistic assumption — that the NAP doesn’t rise At all, But remains basically the same ($28) as in 2008. Even so, you can see how the penalties mount up and become substantially higher with every year you delay joining Part D. (Note: These figures are raw calculations that haven’t been rounded to the nearest 10 cents.)

Table 8-1 How Penalties Mount Up If the NAP

Remains the Same as in 2008 ($28)

$3.36

Now I’m going a bit further out on a limb, but still very conservatively. In Table 8-2, I assume the NAP will rise by $2 each year in the ten years from 2009 through 2018 — so it becomes $30 in 2009, $36 by 2012, and $48 by 2018. And since these numbers are just raw calculations, I haven’t rounded to the nearest 10 cents.

Table 8-2 How Penalties Grow If the NAP Increases

By $2 a Year from 2009 to 2018

Deciding whether to risk ignoring the late penalty

I know what you’re thinking. You’re looking at that last figure in Table 8-2 — $1,411.20 as the possible accumulated ten-year penalty for someone who delayed signing up for more than three years — and wondering whether it’s less than what you’d have paid in premiums for those 40 months. It may well be. But before deciding to ignore the late penalty and stay out of Part D for a few more years, consider the following:

W Table 8-2′s numbers aren’t real. They’re purely a guesstimate. The actual penalty amounts may be higher.

U Table 8-2′s numbers only account for the next 10 years. Penalties accumulated after 15 years or more — when you may be in your 80s and on a fixed income that doesn’t go as far as it once did — are going to be a lot higher.

U At present the monthly late penalty is calculated on 1 percent of the NAP. But Medicare law allows for this amount to be increased to 2 percent at some unspecified future date. If that happens, the penalties would be doubled.

So delaying Part D enrollment is your choice — but recognize that it’s a gamble. Remember the famous line from Dirty Harry Where Clint Eastwood is facing down a bad guy who doesn’t know whether Clint still has a bullet left in his.44 Magnum? He says, "I know what you’re thinking . . . [and] you’ve got to ask yourself a question: ‘Do I feel lucky?’ Well, do ya, punk?" Well, do you?

Part III

/if This Chapter

► Following guidelines for receiving a massage

► Getting in tune and staying in touch

Torn the day you were born, your body has been hanging around you like

M A shadow. It never leaves you alone. You wake up in the morning, and there your body is, faithful as a puppy, thumping its little tail against your freshly washed bedspread. At first, having a body is a novelty, a fact that you can see reflected in the faces of babies and small children. Even the most mundane details about their bodies fill them with delight. "Oh boy, there’s my hand again!"

As you mature, however, you become more accustomed to having a body, and it begins to bore you. This boredom usually occurs as young people enter their teenage years. "Oh boy, my hand again, big deal." At this point, they begin to pierce their bodies in various locations and cover them with decorative tattoos. By the time people are full-fledged adults, though, most of them have begun to concentrate on other things, leaving their bodies far behind. The only time they really get connected to their bodies is when they’re learning a new skill of some kind, like soccer, or neurosurgery.

The result? Most people take their bodies for granted. One of massage’s main objectives is to get you back "into" your body again. A good massage should rekindle your childlike enthusiasm for life.

In order for massage to help you achieve the lofty goal of getting back in touch with yourself, you need to follow certain guidelines, which I just happen to outline in this chapter. At first, some of these "rules" may seem a little simplistic to you. Others may appear irrelevant. However, I give you my personal guarantee that if you try them out when you’re on the receiving end of a massage, you’re going to get much more out of the experience.

So, approach these guidelines with an open mind, apply them when you feel that doing so is appropriate during your own massage exchanges, and watch your enjoyment of massage soar to levels beyond your expectations.

The rules for receiving massage are, in fact, quite similar to the Ancient Secrets of Life as passed down by Big Important Spiritual Leaders for thousands of years. Yes, it’s true; you can learn every really important thing in life by lying down and getting rubbed.

Honing your skills at receiving massage is more than simply a way to feel better. It’s also a way to improve your life. Read through these rules, practice them, and you’ll see what I mean.

Rule #1: Keep Breathing

When you receive a massage from a professional, she may remind you several times in a soft, soothing voice to breathe. And you may be tempted to say right back to her in a not-so-soothing voice, "I’m already breathing, in case you haven’t noticed."

Don’t be offended. The massage therapist’s comments aren’t meant to imply that she thinks you’re deceased, and she’s not trying to insult you for your poor breathing skills. In fact, many massage therapists start each and every massage with a series of deep breaths, regardless of how obviously alive you are to begin with.

A massage therapist may tell you to take deep breaths during a massage for the following reasons:

W To help you focus on the sensations you’re feeling in your body rather than the internal monologue going on in your mind

To get you to fill your lungs and thus all your cells with fresh oxygen, enlivening your entire body

W To help you become aware of muscles that you’ve been holding tense so you can start to relax them

Most people walk around not actually breathing much. People tend to use only a tiny percentage of their lung capacity, just like they use only a tiny percentage of their brain capacity. Proper breathing changes that.

While receiving a massage, focus your mind as fully as possible upon the very important act of breathing. Focusing your mind on your breath brings your awareness back to your body quicker than anything else.

Going With the diaphragm’s flow

The Diaphragm Is a muscle in your abdomen — it looks like a soft pizza shaped into a double-headed dome — that is responsible for keeping you breathing (see Figure 7-1). Most of the time, your diaphragm is contracting and relaxing without conscious thought from you, but you can teach yourself to control this activity. In the section "Exercising your breathing muscle’s breath," I give you an exercise that helps you use this muscle more consciously, which enables you to exert more control over your breathing, making it fuller and deeper.

Exercising your breathing muscles

The next time you have the chance, spend a few minutes observing a sleeping — or at least relaxed — infant or toddler breathe. Pay close attention to the abdomen, and you can see the entire area gently lift and lower. This movement is the result of an active, uninhibited diaphragm at work.

Then look down at your own abdomen while you breathe for a few minutes. Notice a difference? Where did all the lifting and lowering go? You still have the same breathing mechanisms you always did; they’re not something you grow out of. With each breath you take, you should indeed have a visibly rhythmical, moving body. Somewhere along the line, though, most people stifle themselves into taking shallow, insufficient breaths. This type of breathing is a common reaction to the act of growing older. Don’t worry, you’re still getting enough oxygen to survive. But, are you getting enough to thrive? By practicing deep breathing during massage, you can literally rejuvenate your body, sending extra-oxygenated blood out all the way to your toes.

The key to breathing properly while getting a massage is to take Whole breaths, A term that basically means "breathing like a kid." Go ahead and try a whole breath now. Lie down on your back, placing your palms gently on your abdomen, and then begin this four-step process:

1. Breathe deep and low into your lungs so that your abdomen pushes your hands upward.

Make sure that you’re not just pushing up with your stomach muscles, but that you’re actually expanding the entire abdominal area.

2. Continue the expansion up into your ribs, allowing them to push outward toward each side.

3. When your ribs have expanded Out As far as they’ll go, then expand them Up Toward your head, taking the last bit of breath into the area just beneath your collar bones.

4. Let the whole thing collapse.

You don’t need to try and push the air out; just let it flow. When your lungs feel empty and your abdomen is flat once again, you can restart the process.

Rule #2: Stag Loose

As you probably know, one of the main points of getting a massage is to relax. Logically, you may then think that you can just give your body to a massage therapist who will relax your body for you, like giving your car to a mechanic and expecting him to fix it.

Please release mef let me go…

After you receive several massages, you’ll gradually become accustomed to relaxing your own muscles. Eventually you notice that you can do the same thing even when you’re not receiving massage, like when you’re waiting in line at the grocery store, stuck in traffic, or sitting in a meeting with your boss. "Twang," will go one of

Your muscle fibers, and you’ll feel it beginning to tighten up. Then, silently, without anyone noticing, you send a mental message to the growing knot, telling it to go away, in the same way that your massage therapist helps you do during a massage. You can take this side benefit of massage with you wherever you go.

Expecting a massage therapist to do all your body’s relaxing is called Giving up responsibility for your own relaxation, And it’s a no-no. Staying loose is your responsibility; the massage therapist can help you, but you basically have to do the relaxing yourself. So how do you do that?

You accomplish relaxation by becoming more aware of what you’re feeling in your own body. During the massage, your massage therapist often reminds you to focus on "knots" or tight areas. In those moments, using the power of your own imagination, you can begin to visualize what those knots may look like in your muscles, and to let go of them.

If you’re not staying loose by engaging your mind to relax your own muscles, you’re missing more than half the benefits and effects of the massage.

Rute #3: Let Go

When you receive a massage, especially the first time, you may have a tendency — like just about everyone else in the world — to "help" the person working on you. You may graciously lift your limbs, hold your head up, and twist your body around, all to make things easier for the other person. Although this "helping" may seem like the friendliest thing to do, you’re actually hindering the massage process and making your massage therapist’s job a little more difficult. Relaxing a person who is holding her own arm up in the air as stiff as a flag pole is pretty darn hard.

The technical term for this tendency during massage is Hanging on, And you want to do exactly the opposite, which is letting go. But what, exactly, does "letting go" mean?

The limp-arm experiment

You’re basically hanging onto yourself for dear life, even the parts of your body that are painful, stiff, or tense. This hanging on is a natural tendency, but to get the most out of a massage, you have to let go. The "limp-arm experiment" is an easy way to begin training yourself to let go. All you need is a partner and someplace comfortable to lie down.

1. Lie down on your back and have your partner lift your arm up in the air several inches.

2. After a few seconds, have your partner let go of your arm without any warning to you.

Let your arm drop back down. (Make sure that you’re lying on a soft surface.)

3. Watch to see whether your arm plops back down, limp as a noodle, or whether you hold it right where she left it, stiff as a board.

What do you have to do to let your arm drop back down? What thought process do you

Have to go through? What mental image? What body sensation?

4. Tell your partner to lift your arm a little higher each time.

Instead of dropping your arm all the way back down, tell her to catch it in her other

5. Keep repeating this exercise until your arm completely lets go and your partner can drop it from any height with absolutely no resistance.

This ability may come naturally to you the very first time you try to let go, but normally the exercise takes quite a bit of conscious effort. You may not be able to let go until you make several separate attempts on different days. After you master one arm, you can try the other arm, a leg, or your head.

Use this newly formed skill to let go the next time you receive a massage.

If you were to take a microscope and look deep within your muscles and joints while you’re getting a massage, you’d discover some specialized nerve cells that monitor the position and relative movement of your body. These cells are called Proprioceptors (see Chapter 4).

These cells constantly tell you where you are in space, something everybody likes to have a pretty firm control over all of the time, even while asleep. These cells keep you from just rolling right out of bed every night.

About the only time you completely let go of all your holding patterns, tensions, and proprioceptive rigidities is when you’re under deep anesthesia. Under anesthesia, people sometimes release the tension they normally think of as "built-in" through age or heredity, including stooped shoulders, stiff hips, ugly grimaces, and more. When they come out of anesthesia, they reclaim these habitual patterns almost instantaneously. They can’t sustain the relaxation because it’s unconscious. Massage, conversely, allows people to achieve a conscious relaxation, which can last indefinitely.

One of my clients suffered for years from debilitating pain due to whiplash. Then one day she received a massage from a woman at Gurney’s Inn, a spa in Montauk, New York. After that massage, the pain was almost entirely gone, and it continued to gradually fade away. My client was able to make such a drastic change by letting herself go fully into the healing hands of the massage therapist. When she did, she stopped holding onto the same painful, habitual patterns that had formed in her body since the accident.

Rule #4: Stop Thinking, Start Being

The problem with your mind is that it just works too darn well, thinking and thinking and thinking without stopping all day long from the first moment you wake up until well after your head hits the pillow. This feature is fine during most of your daily activities, but when it comes to getting a massage, too much thinking is definitely a drawback.

Many people get a massage and then ten minutes later can barely remember it because they weren’t really paying attention to it while it was happening. Instead, an ongoing stream of thoughts kept them from fully experiencing the massage.

When you’re getting a massage, don’t think about what you should have done the day before or what you plan to do an hour later. A massage is time to Be Here Now. The sensations you’re feeling offer a great opportunity to quiet your mind, focus, and think of nothing else for a little while. In this way, every massage is a potential meditation. Don’t get me wrong: relaxing and joking around during a massage is perfectly okay, too, but most people, at least once in a while, can benefit from a Massage meditation. (See the sidebar, "A massage meditation," later in this chapter.)

Rule #5: No Pain, No Gain) No Wag!

You may have heard of the massage-masochists who don’t believe they’re receiving a real massage unless they have to grit their teeth to keep from screaming through the whole thing. They’re the ones you can hear yelling from behind massage room doors, "More pressure! More pressure!"

This green-beret school of massage is an unfortunate result of the "no-pain, no-gain" mentality that military academies, full-contact sports enthusiasts, and certain daytime-television talk show hosts foster. You don’t need to buy into this way of thinking, and you shouldn’t let this attitude scare you away from getting a massage.

A massage meditation

Meditation, in a nutshell, is the act of focusing your entire attention on just one thing, thus stopping the constant chatter inside your head and experiencing a state of timelessness, contentment, and wholeness. People achieve this state in many ways — through sports, or silence, or prayer, for example — and massage is yet another activity that you can use to effectively shut out the rest of the world and tune into your own inner peace. The next time you receive a massage, try this meditation…

1. Close your eyes and begin to get in touch with your breath, as I describe in the section "Rule #1: Keep Breathing."

Before you receive the first touch of the massage, spend several minutes trying to clear your mind of any other thoughts. Concentrate only on your breathing.

2. When your massage therapist first makes contact, imagine yourself breathing in through that very spot.

For example, if she starts by massaging your neck, imagine a stream of fresh oxygen and energy entering through your neck, exactly where her fingers are.

3. On the exhalation, imagine your muscles in that same area becoming softer, warmer, and looser.

4. Continue with this awareness — breathing relaxation into each successive point that the massage therapist is touching.

Eventually, you become aware that the massage therapist is tuning in to your breath as well, and the massage becomes a shared meditation.

5. Communicate with your massage therapist, both verbally and nonverbaliy. Together, you can create a special massage mood that will help you focus on your experience, making the massage more like a meditation (see Chapter 9).

6. Keep bringing your mind back to the massage.

You may realize, at various points during the massage, that your mind has wandered off on some train of thought. This is completely natural and happens even to advanced meditation practitioners. Simply bring your mind gently back to the breath and the relaxation. Don’t worry about how "good" you are at meditation. See Meditation For Dummies (IDG Books Worldwide) for more guidance and tips about meditation.

So, how much pain should you experience during a massage? In my opinion, none. Zero. However, the line is indeed thin between the pleasure you receive during massage and a certain kind of therapeutic pain. Some people like to walk that line while they’re getting a massage. If you want to experiment walking this line yourself, make sure to do so with an experienced professional.

Although certain muscle knots and patterns of tension do respond well to firm, well-focused pressure, you don’t necessarily need to experience it for yourself. Harder massage is not always better massage, and at times the lightest touch can achieve the most profound benefits.

#6: Listen to \lour Emotions

Don’t be surprised if during a massage one day you suddenly, for no reason at all, feel like crying your eyes out, or laughing hysterically. Massage sometimes has that effect on people. Some of the reasons for this emotional response include:

V0 Certain emotional memories — usually the result of powerful experiences — can resurface when your body is massaged.

No one has touched you with care, compassion, and gentleness for a very long time. In that case, the experience suddenly overwhelms you with gratitude, bringing forth tears.

You’re a very ticklish person.

As esoteric as the first two explanations may sound, they’re entirely plausible. In fact, certain types of massage are famous for stirring up emotions. Rolfing, for instance, often triggers this type of experience. The explanation for this emotional component of massage is straightforward — your body and mind have faithfully recorded your every experience, but some of these experiences were so unpleasant that you filed them away in your unconscious and shut down certain feelings in the corresponding part of your body. Massaging the affected areas can bring your awareness back to your body, thus unlocking the memories.

If you encounter one of these emotional peaks yourself during a massage, relax, breathe, and allow it to happen. Remembering that you are safe in your present environment, let your mind drift to whatever images or memories seem to be surfacing. You may find yourself remembering all sorts of things that you hadn’t thought of for years, and you can benefit from letting the attendant emotions flow, freely through your body, without trying to stifle them. Professional massage therapists are accustomed to this type of emotional release and know how to make you feel comfortable while it’s happening. There’s no need to feel embarrassed by the experience.

If, as occasionally happens, one of these resurfacing memories is particularly traumatic, as in the case of abuse, do whatever is necessary to comfort yourself. Communicate with the person massaging you, letting her know that you need to sit up again, or get wrapped in a blanket for a feeling of safety. Have some tissues nearby to dry away tears. Later you can decide whether you want to pursue these memories further with the guidance of a psychologist or other counselor.

Kufe #7: Btissinq Out Is Okay

Sometimes, massage doesn’t just make you feel great; it makes you feel ecstatic, rapturous, and filled with bliss. The feeling is visceral. You’re lying there one minute relaxing, hopefully concentrating on your breathing, but perhaps just going over your grocery list in your head, when KABOOM!, it hits you, and suddenly you’re just floating there in a syrupy sea of endorphins, not knowing what to do with yourself.

I can tell you what to do: enjoy this feeling while it lasts, because, like every other human experience, it passes.

These experiences are different for everyone, and nobody knows exactly what causes them. They’ve been responsible for many people changing their entire lives and heading into a career as a massage therapist. And people with spiritual inclinations, once touched in this fashion, have created entire ministries devoted to the "laying on of hands."

A minister named Zach Thomas from North Carolina once had such a powerful experience receiving a massage that he went on to become a massage therapist himself. At first, his church was opposed to his hands-on work, and Zach had to practice massage privately. Eventually, though, he took his skills and his compassionate touch out to the public, performing massage for dying people in hospices and hospitals. He helped form a group called the National Association of Bodywork in Religious Services (NABRS), which is active today with hundreds of members. Much of the work the members of this association do is for those people who wouldn’t otherwise be able to afford it. The nuns and priests and other clergy involved practice the actual "laying on of hands" as written about in the Bible. For more information, you can write to the organization at 337 Tranquil Avenue, Charlotte, NC 28209.

The spiritual secret behind massage? Simple. What massage really boils down to is two people just being together fully in the present moment, which has been the essence of spiritual traditions forever, especially in the East. The mystical traditions of the West have expressed similar sentiments, as noted in the phrase, "Be still and know that I am God." These understandings are mystical in nature, not reserved for any one particular religion.

Think of it this way — massage is one sure-fire way to follow the Golden Rule that exists in almost all cultures and every religion, from the Good Samaritan to the compassionate Buddha. Do unto others as you would have them do unto you. Well, who doesn’t want to be touched with care and compassion? Who doesn’t like others to help them feel better and lighten their load?

Massage is compassion turned into action.

Me #8: It’s Coot to Be Nude (Or Not)

You are, whether you like it or not, naked all the time beneath your clothes. You were born nude, just like every other human on the planet. Nudity is natural. However, each culture develops its own peculiar attitudes about nudity, ranging from those who consider it extremely awkward, embarrassing, and inappropriate at all times to those who don’t think twice about it, anytime, anywhere, for any reason.

Neither attitude is healthier than the other, they’re just different. The key for massage situations is to respect the attitudes of both people at all times. If either the person receiving or giving the massage is uncomfortable with any kind of skin exposure whatsoever, you’re much better off to cover that area up and keep it covered than to cause discomfort. This applies to the entire body, even the legs and arms, which most people are comfortable exposing. Although gliding an oiled palm is definitely easier over bare skin than covered skin, massage has other moves besides gliding, and you can give a very good massage to a fully clothed person (I show you how in Chapter 11).

Remember this message: When you receive a massage, you’re okay the way you are — nude or totally covered up. Just be comfortable.

Rule #9: i/ou’re the Boss

Even though you’re lying down with your eyes closed during most massages, you’re still in charge. With the slightest word or gesture, you can change the course of the proceedings. Deeper pressure? It’s up to you. Slower pace? That’s your call, too. Less chit-chat? Your decision.

You have complete authority to change anything that may be making you uncomfortable. Requesting a change of music, for example, is perfectly permissible, as is turning the music off altogether. If you want to be covered more modestly, just ask. Whatever you say goes. You can say exactly what you’re feeling, even ending the massage at any time, for any reason you want. Period. You always have the option of standing up and saying, "Enough!"

Of course, when you’re receiving a massage from a professional massage therapist, it makes sense to listen to her suggestions. If she thinks you should quiet down and focus on the massage rather than conversation, for example, it’s probably best to follow her advice. However, don’t mistakenly place yourself in a submissive role just because you’re lying down. Even if the other person knows more about massage than you, is older than you, or has a louder personality than you, the bottom line is, when you’re receiving a massage, you’re the boss.

Rule #10: Be Grateful

During the massage itself, spend some time being grateful for what you’re experiencing in the moment. This course is by far the best one to take, instead of the alternatives, which consist of

Wondering when the massage is going to end Plotting the next time you can get a massage Planning your next business trip Worrying about the world economy

Also, be sure to share your feelings of gratitude with the person who just gave you the massage, being especially vocal about her fantastic skills and techniques. That way, she’ll look forward to giving you your next massage as much as youTl look forward to getting it.

Chapter 8

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In This Chapter

^ Discovering the origins of Indian medicine ^ Analysing your type the Ayurvedic way ^ Exploring Ayurvedic therapies ^ Researching evidence that Ayurveda works ^ Finding a practitioner

Yurveda, the ancient medical tradition of India, has become increasingly popular in the West in recent years. It is regularly featured in the media and even seems to be the therapy of choice for some celebrities.

Some find Ayurveda’s language a bit confusing, so in this chapter I show you how to tell your Vata (wind) from your Kapha (phlegm) and how to identify your predominant constitutional ‘type’. This information gives you a chance to explore Ayurvedic recommendations for appropriate lifestyle and diet for each constitutional type. These recommendations are designed to balance your body and promote your health.

In this chapter, I also take you on a tour of the way in which an Ayurvedic practitioner diagnoses health problems and introduce you to the wonderful world of Ayurvedic therapy and its treatments for cleansing the body and purifying the mind.

Along the way I share with you a few stories of my experiences at Ayurvedic clinics in India illustrating how this amazing system of medicine works in practice.

A (Very) Brief History of Ayurvedic Medicine

The word ‘Ayurveda’ is derived from Ayus Meaning ‘life’ and Veda Meaning ‘knowledge’ and is often translated as the Science of Life. Ayurvedic medicine’s roots go back thousands of years. Legend has it that this medical art and science originally came as a divine gift from the Hindu god Brahma and that its wisdom is contained in the Vedas, The ancient Hindu sacred texts. However, little real evidence exists for this idea as these texts actually include only a few mentions of medicine or health practices. Instead, nowadays Ayurveda is believed to have developed over time, through trial and error, absorbing information from various sources along the way.

Even so, the earliest medical texts for Ayurveda are around 2,000 years old. Two great compendiums written by the early physicians, Charaka and Sushruta, known as the Charaka Samhita And the Sushruta Samhita, And a later one by the physician Bhela, contain a vast array of information ranging from diagnosis, therapy, and surgery to health tips for daily life. These books are so detailed and useful that they’re still used today in the training of Ayurvedic physicians.

According to legend, the king of the Hindu gods, Indra, received medical teachings direct from Lord Brahma and then passed them down to the sage Atreya, who in turn passed them on to his student, Agnivesa. Charaka, a famous physician who lived around the first or second century AD, rewrote Agnivesa’s work and, over time, others also contributed to his new medical compendium, the Charaka Samhita.

The compendium has 120 chapters divided into eight sections: Sutra (pharmacology, diet, and philosophy), Nidana (eight main causes of disease), Vimana (nutrition and pathology), Sarira (anatomy and embryology), /ndriya(diag-nosis and prognosis), Cikitsa (therapy), Kalpa (pharmacology), and Siddhi(general therapy). Charaka Samhita Also discusses the idea of rebirth.

Sushruta is said to have received the knowledge for his compendium, the Sushruta Samhita,

Direct from the god Dhanvantari, who also received them from Indra. However, historical analysis suggests that the text was in fact a compilation from various authors, including the sage, Nagarjuna. His compendium eventually had six sections: Sutra (the origins of medicine and medical training), Nidana (pathology, prognosis, and surgery), Sarira (embryology and anatomy), Cikitsa (therapy), Kalpa (dealing with poisons), and Uttara (children’s diseases, eye disease, dentistry, and demonic attack!).

Two later texts, The Heart of Medicine, or Astangahrdaya, And the Tome on Medicine, Or Astangasamgraha, Probably compiled by Vagbhata around AD 600, brought all the strands together and first described Ayurveda as a complete system of medicine. These texts were translated into many languages, influencing Tibetan, Chinese, and Arabic medicine, and are still important today.

In Hinduism, the three gods that make up the the gods, he rose out of the depths carrying a

Holy trinity (trimurti) Are known as Brahma, The vase filled with the nectar of immortality that

Creator; Vishnu, The Preserver; and Shiva, Could cure all diseases and began to teach the

The Destroyer. These gods take on different art and science of Ayurveda for health and heal -

Incarnations for different purposes and one of ing. Statues of him can be found in many

Vishnu’s incarnations is said to be Dhanvantari, Ayurvedic hospitals and clinics and he is wor -

The god of Ayurveda. Legend has it that when shipped by Hindus for his healing powers. the oceans of the world were being churned by

Early Ayurvedic medicine concentrated on herbal remedies and dietary and lifestyle recommendations but also developed pioneering forms of surgery.

Ayurveda was not encouraged during British rule in India in the 19th century but was revived with that nation’s independence movement. Nowadays it enjoys the support of the Indian government and is a dominant form of medicine in India, with Ayurvedic colleges, clinics, and pharmacies across the country. Its popularity has also spread to the West where you can find many Ayurvedic practitioners and various Ayurvedic training courses. Ayurvedic beauty and rejuvenation treatments have also become popular in beauty salons.

Deciphering Disease in Ayurvedic Medicine

Ay—

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In Ayurveda, the body is seen as a mini-universe where the five great elements of the cosmos – ether (akasha), Air (vayu), fire (agni), Water (jala), And earth (prithvi) - known as the Panchamahabhutas Combine to form three humours.

The humours, known as Doshas, Each have their own special qualities and manifest in the body in different ways to determine your individual constitution (prakrti) Or type.

Your type is also affected by the strength of your ‘digestive fire’, known as Agni, Which is the ability of your body to digest food and absorb nutrients efficiently, and Kostha, Which is the regularity and efficiency of your bowel function and indicates how well your body can expel waste materials.

Hindu beliefs include the concepts of Reincarnation, Of being reborn in a new form after death, and the law of Karma, The law of cause and effect. These beliefs have also crept into Ayurvedic medicine and so karma, resulting from inappropriate actions in a past life, is also considered as a possible cause of disease

And is taken into account in therapy. When I worked in Ayurvedic clinics in India, I encountered many patients who cheerfully accepted this to be the case because they also believe that good deeds and prayers in this life can help to dissolve the karma and ease their condition.

Ayurveda also pays a lot of attention to your mental and emotional state and the strength of your spirit. Three tendencies, or Gunas, Are said to interplay to determine mental state:

Sattva Represents purity and balance and predominates when you feel calm and peaceful and have a clear and ordered mind.

Rajas Is behind activity and dynamism and triggers desires for material things or for meeting physical needs and the fear of losing them. For example, Rajas Is the force behind the mechanism of feeling hungry, thinking about food, and then obtaining it and eating it or fearing you won’t have enough or someone will take it away.

Tamas Relates to solidity but is also associated with slowness and mental ignorance. This can relate to mental dullness or confusion, lethargy, or self-destructive tendencies such as depression.

Imbalances in the humours and the Gunas Caused by faulty lifestyle, poor dietary habits, lack of exercise, and negative thoughts are believed to be the root cause of disease.

The three Gunas Can also be applied to foods and their effects on the mind and body. For example, Sattvic Foods such as fresh, organic fruit and vegetables are light and easily digestible and said to have a calming effect on the mind; Rajasic Foods are spicy and hot and heat the body and can disturb the mind, triggering anger and irritation; and Tamasic Foods are cold and heavy (such as ice cream) and can slow the mind and trigger lethargy and dullness.

Understanding Your Health the Ayurvedic Way

In Ayurveda establishing whether your basic type is Vata (wind), Pitta (choler or bile), or Kapha (phlegm) is important. You can then make simple changes to your diet, lifestyle, and daily habits to help restore balance in your body and improve your health. This section describes how to figure out your personal type.

Go through the checklists in Table 5-1, answering the questions. Then tally your totals for each checklist. The one that has the most ‘yes’ answers is your current type according to Ayurvedic medicine.

Table 5-1 Figuring Out Your Type

CHECKLIST A Yes No

Are you slim and do you find it hard to gain weight?

Does your skin tend to be dry and rough?

Are your nails dry, brittle, and easily breakable?

Is your hair dry, thin, and brittle?

Are your eyes small and deep-set?

Is your appetite irregular and do you suffer from digestive problems?

Are you talkative, active, prone to anxiety, imaginative, and creative?

Are you often forgetful?

Do you often feel tired when you wake up?

Do you lack regular routines and often feel stressed?

TOTAL ‘YES’ _/10

(continued)

Here’s a guide to your Ayurvedic type:

Most ticks for Checklist A in Table 5-1 means that you’re predominantly a Vata Type. This means that you are always doing something and tend to suffer from stress, anxiety, and digestive problems like IBS (irritable bowel syndrome).

Most ticks for Checklist B means that you’re predominantly a Pitta Type. This means you’re quite a dynamic and fiery person with a quick intelligence but you don’t suffer fools gladly and may be domineering. You may suffer from digestive problems and headaches.

If you have most ticks for Checklist C, then you’re predominantly a Kapha Type. This means that you’re practical, down-to-earth, and resilient but also tend to be a bit lethargic and may suffer from fatigue, bloating, sugar-cravings, and water retention.

For details on how to balance your type, look for the tips in the later section ‘Restoring Balance with Ayurvedic Therapies’.

Making Diagnoses in Ayurvedic Medicine

Ayurvedic practitioners use an eight-fold examination (astavidha pariksha) To investigate your health:

The pulse (nadi): Your wrist pulse is used to determine pulse qualities. A Vata Pulse is fast and slippery, a Pitta Pulse is jumpy and irregular, and a Kapha Pulse is slow and steady.

The tongue (jihva): This looks at the colour and coating of your tongue. A Vata Tongue is dry, rough, and cracked with no coating; a Pitta Tongue is red with an oily, yellow coating; and a Kapha Tongue is swollen and moist with a greasy, white coating.

The voice (sabda): Vata Types have rough, throaty voices; Pitta Voices are erratic and may break during speech; and Kapha Types have deep voices and often have to clear their throat when speaking.

The skin (sparsa): The physician looks at the colour and texture of your skin and may palpate certain points to check for tenderness or redness. Vata Types have dry, rough, sensitive skin that is cool to touch; Pitta Types have reddened areas of skin that feel hot to the touch; and Kapha Types have clammy, moist, cold skin.

Vision (drka): The Ayurvedic practitioner checks the whites of your eyes and registers any other visual abnormalities. Vata Types have dry, sensitive eyes with dull eye whites; Pitta Types have a burning sensation in their eyes and yellowed eye whites; and Kapha Types have heavy, drooping eyelids and frequent watery eyes.

General appearance (akrti): This covers your posture and gait. Vata Types are often thin and wiry and move rapidly; Pitta Types are often strongly built and restless; and Kapha Types are often heavily built and rather slow moving.

Urine (mutra): The Ayurvedic practitioner examines the colour and odour of your urine and you’re asked how frequently you urinate. Vata Types have frequent, clear, odourless urine; Pitta Types have small amounts of brown or deep yellow urine with a burnt odour; and Kapha Types have copious turbid, whitish urine with a stale smell.

Stools (mala): The colour, consistency, and odour of your stool is considered. A normal healthy stool is firm and light brown in colour. Vata Types have hard, dry stools that are grey in colour; Pitta Types have loose stools that are yellow, dark brown, or green in colour; and Kapha Types have slimy, pale stools containing mucus and bits of undigested food.

These eight examinations are combined with questions to enable the Ayurvedic practitioner to decide on the relative balance of the Doshas For the person who may not be just one type but has more than one fighting for predominance.

The practitioner also takes into account other factors such as your age, vigour, mental state, and physical condition, as well as possible genetic and environmental factors, diet, and lifestyle. Analysis of dreams and astrological birth charts are also sometimes used to determine possible spirit, or karmic, influences.

After the Ayurvedic practitioner has determined the cause and stage of the disease and the nature of imbalance of the Doshas, Then an appropriate therapy can be selected.

Restoring Balance with Ayurvedic Therapies

Ayurveda aims to treat the whole person, not just the symptoms or disease. The key aim is to re-balance the Doshas In the body, but secondary aims are to improve general health and vitality, strengthen digestive fire (agni), And promote longevity. The ultimate aim is to assist the person in following the three goals of life outlined in the Vedas: dharma (virtuous living), Artha (prosperity), and Kama (pleasure), which together may lead to the ultimate goal of Moksha (spiritual liberation).

Ayurvedic healing includes the following:

IU Medicinal treatment, or Ausadha (using herbs and such)

IU Cleansing and purifying techniques, or Panchakarma

IU Dietary therapy

IU Lifestyle modification, or Pathya

IU Exercise (yoga) and massage therapy

I Meditation and other spiritual remedies

You might also use self-help techniques, such as personal hygiene, methods for enhancing fertility and virility, and rejuvenation techniques.

The physician, medical assistants, patients, and even the remedies themselves must all combine together effectively to obtain a cure.

These treatments are divided into three main types according to whether they’re based on reasoning, the sacred, or character. Those based on reasoning are the medicinal treatments, the Panchakarma Cleansing and purification techniques, and the dietary and lifestyle advice. Those based on the sacred are the spiritual remedies such as reciting Mantras (repeated words or phrases) or prayers, carrying out fasts, going on pilgrimages, or wearing sacred gemstones. Those based on character involve cultivating moral qualities and integrity and avoiding harmful influences such as alcohol, gambling, lying, cheating, and, in modern-day speak, ‘sex, drugs, and rock and roll’!

The rest of this section describes these Ayurvedic therapies in more detail.

The ancient Ayurvedic physician Charaka wrote that four pillars determine successful therapy. The first is the physician, who must be knowledgeable and experienced and should also demonstrate physical cleanliness and purity of mind.

The second is the medicine, which should be grown, harvested, and prepared correctly to be effective – and, of course, prescribed correctly too.

The third is the medical attendant, who assists the physician in carrying out the therapies. They too should be knowledgeable, experienced, clean, and pure of mind, as well as compassionate and empathic with patients.

The fourth and final pillar is the patient, who has a duty to describe symptoms accurately, to follow the physician’s instructions, and to be strongly motivated towards healing.

Ayurvedic medicines

Ayurvedic medicines are mainly made from plant material such as leaves, flowers, fruits, and spices but can also contain ground minerals and gem-stones and animal products such as ghee (clarified butter), animal fats, beeswax, or honey.

The different plants and medicinal materials are classified according to their tastes, potencies, and ‘ripening’ effects (vipaka):

IU Sweet, sour, salt, bitter, pungent, or astringent qualities IU Warming or cooling properties

IU Post-digestive effects (the physical effects of their components after they’ve been digested in the body)

IU Specific healing effects on the body (prabhava)

Herbs are cultivated and harvested carefully and then prescribed singly or combined together according to both ancient and modern formulae for maximum efficiency. They may be used dried or fresh and go through various processes, such as being ground, boiled, soaked, juiced, heated, and so on, to be made into medicines.

When I’m in India, I often visit the Ayurvedic pharmacies and am amazed at the range of products on offer. Few are in their raw state, as in the old-style traditional Chinese medicine pharmacies; most are carefully packaged, with extensive medical claims. Some aren’t labelled in English and great care is needed here. Not all have gone through rigorous quality control and some contain contaminants and heavy metals such as lead, mercury, or arsenic.

A recent tragic case involved an Anglo-Indian woman who bought Ayurvedic remedies for weight loss while in India and took them extensively on returning to Britain. These medicines

Contained contaminants and she unfortunately died of liver failure before it was realised that it was these remedies that were making her ill.

Ayurvedic remedies sold in this country by responsible companies are carefully batch tested for correct ingredients and purity so that they’re safe to take. However, not all distributors act this responsibly. Therefore, only take remedies from a reputable practitioner and supplier that specifies that they implement these checks and only take products clearly labelled in English showing that they’ve been batch tested.

The medicines may then be taken as pills (vati), Powders (churna), Pastes (kalka), Juices (svarasa), Decoctions (kvatha), Or medicated oils (tailas). Herbal teas and rejuvenation elixirs are also popular over-the-counter remedies.

In general Pitta Remedies are cooling, Kapha Remedies are warming and help clear phlegm, and Vatta Remedies are calming. However, different combinations of tastes and properties are used according to individual balance.

The Panchakarma purification techniques

The Panchakarma Are five Ayurvedic techniques for cleansing and purifying the body and rebalancing the Doshas That are used in both treatment and prevention of disease.

The body must be carefully prepared for these techniques. First, oil therapy (snehana) Is used, whereby vegetable oils are taken orally, given as enemas, and/or massaged into the skin. This therapy helps to soften the skin and get rid of old waste matter that may have accumulated in your intestines. Second is sweating therapy (svedana) Whereby the body is exposed to external heat from sweat baths or hot packs, or made to generate heat internally by means of exercises or swaddling in blankets. This sweating also helps to remove impurities from the body.

After the preparations are complete, the appropriate Panchakarma Can be selected:

IU Vamana: The use of herbs, rock salt, and honey to induce vomiting in order to clear the digestive system and relieve mucus. It is often used for Kapha Conditions such as chest and nasal congestion as in bronchitis and rhinitis.

IU Virechana: Involves drinking laxative and purgative herbs to cleanse the body. It is often used for Pitta Conditions such as fevers, intestinal worms, and skin diseases.

Virechana Is quite a rigorous therapy and isn’t suitable for young children, the frail, or the elderly.

I Vasti: The use of enemas to cleanse the bowels. These are often used for Vata Conditions such as dry skin, irritable bowel, nervousness, and fatigue.

IU Nasya: Nostril cleansing that involves inhaling medicinal oils, powders, or steam in order to relieve blocked noses, sinusitis, headaches, and nasal congestion, often signs of Kapha Imbalance.

IU Raktamoskshana: Blood-letting that uses leeches or surgical instruments to extract small amounts of blood, generate new blood cell production, and increase circulation. It is used for Rakta (blood) disorders and Pitta Conditions such as boils, abscesses, and skin diseases.

Don’t partake in the Raktamoskshana Therapy during pregnancy or if you suffer from anaemia.

Always follow Panchakarma Treatment with rest, appropriate dietary therapy, and any other relevant therapies. Such treatments must be done carefully and correctly by qualified practitioners; in India, they’re usually only carried out in hospitals or clinical settings.

Ayurvedic dietary therapy

In Ayurveda, foods are classified in the same way as herbs, that is, according to their six Savours Or tastes, their energetic potencies of warming or cooling, and their post-digestive effects. Table 5-2 shows recommended foods and cooking methods for the three different Dosha Types.

Pitta

Hot, spicy, fried, pungent, Cool and raw or lightly steamed foods;

Sour, and salty foods, sweet, bitter, and astringent foods;

Such as curried foods, sweet and bitter vegetables; and items

Pickles, and salted nuts. such as mint tea, salads, cucumber,

Oats, figs, rice, tofu, mushrooms, lentils, and ghee (clarified butter).

Kapha Sweet, sour, phlegm -

Producing and salty foods including sugar and confectionary, dairy produce, and salt and salted foods such as crisps.

Light, dry, spicy, warm, and easily digestible foods with pungent, bitter, and astringent taste such as cranberries, fennel, aubergine, apples, lentils, chickpeas and other pulses, millet, couscous, and natural diuretics such as watermelon.

Ayurveda also advises that you eat food in moderation, chew slowly, and eat in a calm setting. This is especially important for Vatta And Pitta Types. Kapha Types need to eat little and often.

Lifestyle modification (pathya)

Ayurvedic practitioners recommend living according to the seasons (rit-ucharya). Doing so means wearing warm clothes and eating warm foods in cool or cold seasons, wearing cool clothes and eating cooling foods in warm or hot seasons, and drinking plenty of fluids in dry seasons. Cleanliness, regular habits, and sleep are regarded as important, and daily exercise, meditation, and offerings to Hindu gods are all encouraged.

Also recommended is that natural urges such as hunger, thirst, sneezing, yawning, and urination shouldn’t be unnaturally suppressed and that good habits and virtues is cultivated. These good behaviours include telling the truth, being kind and generous, respecting your parents and teachers, and keeping good company. Harmful behaviours such as lying, cheating, envying, anger, and greed need to be overcome.

Vatta Types need to try to slow down and keep to regular routines; Pitta Types need to try to keep cool, avoid stuffy environments, and wear natural fibres; and Kapha Types should vary their routine and incorporate new forms of stimulation and mental challenge into their lives.

Exercise and massage therapy

Ayurvedic medicine advises yoga exercises for keeping the body supple, the muscles toned, and for promoting mental concentration and calm. Vata Types need to concentrate on gentle exercise and stationary yoga poses. Pitta Types can take moderate exercise combining yoga poses with breathing exercises. Kapha Types should take vigorous exercise combining aerobic exercise with yoga or doing energetic forms such as Ashtanga yoga.

Massage is done with the hands or feet or using heated linen pads. Different oils may be used to warm or cool the body according to Dosha Type. In Marma Therapy, different points on the body, rather like the acupoints of acupuncture, are stimulated.

Meditation and other spiritual remedies

Yoga breathing exercises (pranayama), Recitation of mantras, meditation and prayer are all encouraged to calm the mind, purify karma, and connect with

The divine. Vatta Types find sitting still hard, so can try walking meditation; Pitta Types can benefit from developing a regular seated meditation practice incorporating breathing techniques to calm the senses; and Kapha Types can benefit from active mantra chanting.

Deciding when to use Ayurvedic therapies

In India and Sri Lanka, Ayurvedic treatment is widely used for chronic conditions such as arthritis, joint pain, digestive problems, skin problems, and respiratory problems as well as for post-operative recovery, such as in the case of cardiac surgery and in the treatment of cancer.

If you’re taking any form of Ayurvedic medicine and also Western medicine, always ensure that you advise both your Ayurvedic and orthodox medical practitioner in case of any potential interaction between the medicines.

Finding Out Whether Ayurvedic Medicine Works

A growing database of research into the therapeutic effect of Ayurvedic herbs and herbal formulae now exists, as well as some research on the therapeutic effects of yoga exercise. Unfortunately, however, even in India, only very limited good quality research has been carried out on Ayurvedic therapies in general. Therefore, its efficacy in treating various diseases remains largely clinically unproven, although of course it has been extensively tried and tested throughout centuries of use.

Some work on the effect of individual herbal remedies has been encouraging, such as the use of neem leaves to suppress head lice and other infections, Shatavari root to treat menopausal symptoms, Triphala (a mixture of three fruits) to treat constipation, Guggul for reducing cholesterol, Ashwagandha for treating stress and fatigue, and so on, but much of the evidence is anecdotal and more well-designed clinical trials are needed.

Finding a Practitioner of Ayurvedic Medicine

The practice of Ayurveda is not yet regulated in the UK, so currently anyone can call themselves an Ayurvedic practitioner. The regulation process is now

Beginning and university accredited training courses for practitioners have already been established. However, until proper regulation is in place, and given that some of the remedies used may potentially be toxic, take great care to check that your practitioner is well-trained and experienced and able to practice safely.

You can find practitioners in the UK via:

The Ayurvedic Practitioners Association (Www. apa. uk. com or phone 07983-124950). This association has established a new three-year full-time BSc course in Ayurveda (BSc Honours Complementary Health Sciences (Ayurveda)) with the University of Middlesex, plus a year of post-graduate clinical training and an optional MSc training.

IU The Ayurvedic Medical Association (AMA UK). For details ring 0208-6576147 or 0208-6823876. Some of their members are Ayurvedic physicians who have completed a five-year course in India or Sri Lanka and have the initials BAMS: Bachelor of Ayurvedic Medicine and Surgery (India); or DAMS: Doctor of Ayurvedic Medicine and Surgery (Sri Lanka) after their names.

IU The Ayurvedic Company of Great Britain (Www. ayurvedagb. com or

0207-2246070). Their members may hold a Diploma in Ayurveda (Dipl. Ayu.) or a BA Hons Ayu from the Manipal Ayurvedic University of Europe (a joint venture between the Ayurvedic Company of Great Britain and Manipal University, India and formerly with Thames Valley University). This company has also established a British Ayurvedic Medical Council (BAMC) and a British Association of Accredited Ayurvedic Practitioners (BAAAP).

IU Maharishi Ayurveda (Www. maharishiayurveda. co. uk/

Practitioners. htm or via the Maharishi Ayurveda health centre on 01695-51008). Members are medical doctors who have also trained in the Maharishi Mahesh Yogi’s Vedic Approach to Health.

In the US, practitioner details can be obtained from the National Ayurvedic Medical Association on Www. ayurveda-nama. org

In This Chapter

^ Switching from fractions to decimals to percents and back again ^ Investigating both the practical and impractical with percents ^ Using percentages to your advantage — in your best interest

A Decimals and percents are really just fractions — in a more manageable

Format. Doing problems that involve percents of things involves changing the percents to decimals and then doing the indicated operations. That’s not a big deal, if you handle the decimals correctly. And the computations are much easier than with fractions, which can have very uncooperative denominators.

In this chapter, you see how to figure percent increase and percent decrease and determine whether what you see advertised is a good deal. Everyone is affected by interest on money — whether you’re borrowing or saving — so I include problems dealing with computing interest as well.

Relating Fractions, Decimals, and Percents

The usual move from fractions to percents is through decimals — the decimal format is the middleman in the process. You probably already know some of the more common equivalences of percents and fractions. You know that 50 percent is equivalent to >2 and 25 percent is equivalent to!4. Well, I’m assuming that you know this, but, just in case, here are some properties and techniques that you can use to make the transitions easier. You don’t have to memorize these properties, but having some of them in mind as you’re working on percentage problems is helpful.

Changing from fractions to decimals to percents

A decimal is a fraction and vice versa. To change a fraction to a percent, you first determine the decimal value and then fiddle with the decimal point. That’s all there’s to it — really.

To write a fraction as a decimal, you divide the numerator by the denominator, inserting the decimal point where needed. To change a decimal to a percent, you move the decimal point two places to the right and use a percent sign (%).

For instance, changing the fraction j-^j to a percent, you first find the decimal

By dividing the numerator, 7, by the denominator, 16. The decimal that you get is 0.4375. Here’s what the division looks like:

0.4375

16 )7.0000

64 60 48 120 112 80

80

To change the decimal to a percent, you move the decimal point two places to the right to get 43.75 percent. This makes sense, because 7 is not quite half of 16, and 43.75 percent is just short of half of 100 percent.

Finding terminating decimal values

Fractions all have decimal values, but some of these decimals Terminate (come to an end) and some Repeat (never end). As long as the denominator of the fraction is the product of 2s and 5s and nothing else, then the decimal equivalent of the fraction will terminate. To find this terminating decimal, you divide the denominator (bottom) of the fraction into the numerator (top) and keep dividing until there’s no remainder. You may have to keep adding 0s in the divisor for a while, but the division will end.

For example, to find the decimal equivalent for the fraction -25, you divide 25

Into 14. To begin the division, you put a decimal point to the right of the 4 and add a 0 to the right of the decimal point. You keep adding 0s as needed in the division. This is what I get when I do the division.

0.56 25 YT41)0 125 150 150 0

So 25 = 0.56. As you see, this decimal Terminates. The division stopped

Because I eventually ended up with no remainder. You can predict that this will happen (the termination of the decimal), because the denominator, the 25, is equal to 5 times 5. The only factors of 25 are 5s.

Next, you see the division required to find the decimal equivalent of the

Fraction 119g. This decimal will terminate, too, because the denominator is

Equal to 2 X 2 X 2 X 2 X 2 X 5. Only 2s and 5s are factors of the denominator. It doesn’t matter what the numerator is; the decimal will terminate. Finding the decimal equivalent:

0.050625 160 G 9.000000 800 100

_0

1000 960 400 320 800 800 0

It took a while, but you can now say that -MTt = 0.050625.

160

Computing decimals that keep repeating themselves

Terminating decimals are just dandy, but they’re in no way the only type of decimal value out there. Repeating decimals occur when you change a fraction to a decimal and the denominator of the fraction has some factor other than 2 or 5. It only takes one such factor to create the repeating situation. For

Example, the fraction 12 repeats when you divide the numerator by the denominator, because the denominator has a factor of 3.

0.4166… 12 G 5.0000

48 20

12

80

72

80

72

8

As you can see, the remainder will now forever be 8, and the corresponding number in the quotient (answer of a division problem) will be 6. The three dots following the last 6 shown indicates that the 6 keeps repeating forever and ever.

To indicate a repeating decimal, you can either write three dots (ellipsis) after some repeated digits, or you can draw a horizontal bar across the digits that repeat. For instance, 0.41666 . . . can also be written 0.416. Note that the bar is over the 6 only. The first two digits don’t repeat.

When you’re using repeating decimals in a problem, you decide how many decimal places you want and then round the number to that Approximate Value. Rounded decimals aren’t exactly the same value as the repeating decimals, but you can make them pretty accurate in an application by using enough digits in the decimal equivalent.

Making the switch from fractions to percents

The middle step in changing a fraction to a percent is finding the decimal equivalent (or, in the case of a repeating decimal, the approximate). Table 6-1 shows you some fractions, their decimal value, and then the percent that you get by moving the decimal point two places to the right.

Changing from percents back to fractions

Percents are very descriptive — they tell you how many you have out of 100 things. The only problem with percents is that you can’t use the percent format when doing computations. You need to change a percent back to a decimal or a fraction, if you want to multiply or divide using the percent.

To change a percent to a decimal number, move the decimal place in the percent two places to the left, adding 0s if necessary. To change that decimal number to a fraction, write the digits in the decimal over a power of 10 that has as many zeros in it as there are digits in the decimal — then reduce the fraction if you can.

For example, 45 percent has a decimal value of 0.45, and the fraction equiva -

Lent is

45 _ 9

. Another example is 0.032 percent, which has a decimal 32 1

Value of 0.00032 and a fraction of, AA AAA = 010,.

100,000 3125

Tackling Basic Percentage Problems

The nice thing about percents is that their values are easy to relate to. If you’re 75 percent finished with a project, then you know that you’re well on your way. You compare the percentage to 100 — a nice, round number — and have a good idea of what the value is in the comparison. To be more exact with an answer, though, you need to convert percentages to decimals and create a more exact value to use in computations. Using the decimal equivalents, you can solve for the percent of a value and get the answer in items, and you can also solve for how many items are needed to reach a certain percent.

Finding the percent amount

When you’re told that you have 60 percent of the work done or 85 percent of the problems correct, you multiply the total number of hours needed to do the work or the total number of problems on the test by the percent to get the numerical value of what you’re discussing. Percents are convenient amounts for comparison. You convert percents to decimals to use them in problems.

The Problem: You sign up for Weight Watchers, and you’re told that you need to lose 10 percent of your current weight. If you weigh 160 pounds, how much do you need to lose?

Changing 10 percent to a decimal, you get 0.10. Multiply 160 X 0.10 and you get 16 pounds. That should be a piece of cake. Oops! Not on Weight Watchers — make that a carrot stick.

The Problem: You’re told that 95 out of 100 of the people who buy a Honda motorcycle will buy another Honda when they need to buy another motorcycle. Last year, the number of Honda motorcycles sold in North America was 570,000. How many of these owners will buy a Honda when they make their next motorcycle purchase?

^VLA/jf First, change the fraction to a decimal. The value of 95 out of 100 is 95

Percent, but you can’t use the percentage in a computation — you want the decimal value. This fraction is equal to 0.95, so 570,000 X 0.95 = 541,500 people who will buy a Honda motorcycle the next time they make a purchase.

The Problem: To get an A in your math class, you need to have an average of 92 percent of all the points available. You currently have 540 out of a possible 600 points, and you still have the 200-point final to take. What do you need to score on the final to get an A?

^VLA/V This problem involves several operations: addition, multiplication, and subtraction. First, determine the total number of points needed. Add 600 + 200 for a total of 800 points. You need 92 percent of 800 points to get an A, so multiply 0.92 X 800 = 736 points. You currently have 540 points. Subtract 736 – 540 = 196. You need a score of 196 out of 200 points, which

Is 196 = Tkr = 0.98 = 98 percent. You have your work cut out for you. 200 100

Finding the whole when given the percent

Working backward to find out where you get a certain percent amount requires division instead of multiplication. This process of using division makes sense, because multiplication and division are inverse operations — one Undoes The other.

The Problem: Forty-five percent of the class is boys. And you’re told that the class has 18 boys in it. How many students are in the class?

Falling golf balls

Two large barrels contain the same amount of height above the two barrels, are dropped at water. The water temperature in the first barrel exactly the same time. Which ball will touch the is 49°F, and the temperature in the other is 29°F. bottom of the barrel first? Two identical golf balls, positioned at the same

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VLAiV Divide 18 by 45 percent — 18 0.45 = 40. There are 40 students in the class. You don’t believe this works so easily? Then check the work by finding 45 percent of 40 to see if you get 18 boys. Multiplying 0.45 x 40 = 18 boys. By golly, it works!

The Problem: In a large bag of Skittles, 20 percent of the candy is colored yellow and 18 percent is colored red. If you counted 100 yellow candies, then how many are red?

^VIA* First, determine how many pieces of candy there are altogether, and then determine what 18 percent of that number is to find out how many are red. You know that 20 percent of the candy corresponds to 100 pieces, so divide 100 by 20 percent — 100 ■ 0.20 = 500 pieces of candy. Now multiply 500 times 18 percent, or 500 x 0.18 = 90 pieces of candy that are red in color.

Looking At Percent Increase and Percent Decrease

You’ve been drawn to a store when it advertises "All Prices Slashed 20 Percent" or "Take 15 Percent Off the Reduced Price." Who wouldn’t be tempted when you’re offered such deals? And what about that meeting with the boss when she says that you need to increase productivity by 25 percent? Where does that put you as far as output? Can you do it? Percent decrease and percent increase are both based on changing the amount from 100 percent, or the full amount. You use the difference from 100 percent to help you when doing the problems.

Decreasing by percents

Figuring a percent decrease — or the new value of an item after the decrease is applied — just takes a deep breath and a little common sense. You can get messed up with the arithmetic if you don’t think about what the answer should be ahead of time. In general, you multiply the total amount by the percent to get the decrease in the amount. You can then subtract that decrease from the original amount to get the new result. Another way of finding the resulting amount is to subtract the percent decrease from 100 percent and multiply this difference times the original amount.

To determine the net result or amount after applying a percent decrease, you use one of the following methods (either one works):

Total amount X Percent decrease = decrease in amount Total amount – decrease in amount = net result 100 percent – percent decrease = decreased percent Total amount X Decreased percent = net result

The Problem: A local store is going out of business and has advertised that all items are 60 percent off the original price. You buy a toaster oven that’s currently marked $49.95. What will you pay for the toaster oven after the store applies the discount?

Using the first method, earlier, you first multiply $49.95 X 0.60 = $29.97. This is the amount of the decrease. Next, subtract $49.95 – $29.97 = $19.98. Wow! Such a deal!

Now, using the second method, you subtract 100 percent – 60 percent = 40 percent. The item will cost 40 percent of the original cost. Taking $49.95 X 0.40 = $19.98. You get the same answer, of course.

You’re pretty happy with your purchase of the toaster oven until, the next day, you see that the same store is offering to reduce the Previously reduced price By another 20 percent. Does that mean that the new reduction is 80 percent, or is the reduction 20 percent of the previous 60 percent? Is there even a difference? Oh, yes, there is.

The Problem: What is the difference between an 80 percent decrease and a 60 percent decrease followed by a 20 percent decrease? For the sake of comparison, use an item that has an original cost of $100.

To see if there is a difference at all, do the computations two different ways: Find the price after an 80 percent decrease. Then go back to the original price and figure a 60 percent decrease followed by a 20 percent decrease.

Determining the cost of the item if there’s an 80 percent decrease in the price, you subtract 100 percent – 80 percent, giving you a final cost of 20 percent of $100. Multiplying 100 x 0.20 = $20. That’s the cost of the item with a straight 80 percent decrease.

Now, find the cost when there’s first a 60 percent decrease in cost followed by a 20 percent decrease in that result — 100 percent – 60 percent = 40 percent. Multiplying 100 x 0.40 = $40. A 20 percent decrease becomes 100 percent – 20 percent = 80 percent. And, multiplying, $40 x 0.80 = $32. That’s quite a bit different from the $20 when figured the other way.

Not that I’m beating up on this poor store that’s going out of business, but here’s another scenario to consider when dealing with percent decreases. Consider a less-than-scrupulous manager who advertises that prices are going to be decreased by 60 percent off the original amount. What you don’t know is that he changes all the prices the night before the big sale so that the 60 percent decreases result in the cost of the items all being the same as they were the previous day. How does he do the necessary math?

The Problem: A store advertises a 60 percent decrease in cost on all items. What does the price tag have to read so that an item costing $100 originally will still cost $100 after the 60 percent decrease?

Go back to the same process of subtracting the 60 percent from 100 percent to get a net cost of 40 percent. Then divide instead of multiplying to get the required price. Dividing $100 ■ 0.40 = $250. The price needed on the item is $250. When you apply a 60 percent decrease to $250, you get back to the original price of $100. Somehow, I think that the shoppers just may notice the price hike.

Matching socks

You have 40 socks in a drawer in your bedroom. are blue. How many socks do you have to take The power is out, though, and you can’t see what out of the drawer to be absolutely sure that you color the socks are. You know that 35 percent of have a matching pair of socks? You don’t care the socks are black and 65 percent of the socks which color, as long as they match.

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Making the discount count

You’re going to take advantage of the end-of-year discounts being offered at the local hardware store. You get a 25 percent discount on all items, but you have to figure in the 8 percent sales tax.

The Problem: You have purchases totaling $75.45 and get to the checkout counter. Your coupon says 25 percent off, but there’s an 8 percent sales tax, so the clerk tells you that she’ll just subtract the 8 percent from the 25 percent, leaving a 17 percent discount, which means that you pay 83 percent of the total before all this magic arithmetic. Is this right?

You sense a shell game going on. The numbers are flashing through your head like those visions of sugar plums. Take a deep breath, and do the computations yourself. First, find the reduced cost, and then add the sales tax. The discount is 25 percent off, so you’ll only be paying 75 percent of the original price. (Refer to "Decreasing by Percents," earlier in this chapter, for more on figuring the discounted price.) Computing 75 percent of $75.45, you get 0.75 X $75.45 = $56.5875 or $56.59. Now add on the 8 percent tax by multiplying $56.59 times 108 percent. (This is the same as finding 8 percent and adding that amount on to the price. See "Determining an increase with percents," later in this chapter, for more information on this process.) The computation is 56.59 X 1.08 = $61.12. The tax was 8 percent of $56.59 or $4.53.

How does this compare to the clerk’s suggestion of just giving you a 17 percent discount? Computing 83 percent of $75.45, you multiply 0.83 X $75.45 = $62.6235. You would pay $62.62, which is $1.50 more than you should.

Determining an increase with percents

A percent increase may involve your goal in productivity or the amount of rainfall one summer or the amount that a price is increased due to sales tax. In general, to find the new amount after a percent increase, you either determine the increase in the amount that results from the percent increase and add it to the original, or you add the percent increase to 100 percent and multiply by the new percentage.

To determine the net result or amount after applying a percent increase, you use one of the following methods:

Total amount X Percent increase = increase in amount Total amount + increase in amount = net result 100 percent + percent increase = increased percent Total amount X Increased percent = net result

The Problem: A sheep shearer figures that he can improve upon his average per day shearing of 100 sheep. He’s set a goal of increasing the number of sheep by 15 percent. How many sheep per day will he have to shear to make that goal?

Using the first method, he first determines that 15 percent of 100 is 0.15 X 100 = 15. Add 15 to the usual 100 sheep, and he sees that he needs to shear 115 sheep per day.

Using the second method, if the shearer adds 15 percent + 100 percent, he gets 115 percent. Multiply 100 X 1.15, and he gets 115 sheep.

Percentages of more than 100 percent have decimal equivalents that are greater than 1. Be careful when moving the decimal point. For instance, 250 percent = 2.5, 800 percent = 8 and 1,000 percent = 10.

Problems involving sales tax are pretty much just percent increase problems. They get even more interesting when you figure in both a percent decrease because of a sale price and also sales tax.

The Problem: You purchase new shoes that were advertised as being 25 percent off the original price. You look at the sales receipt and don’t agree with the total price. You suspect that the sales tax was computed on the original price of the shoes, not the sale price. The shoes were originally $120, and the sales tax is 8>4 percent. The amount you’re being asked to pay is $97.43. Is this the correct amount?

^ylAJV First, figure out the new price due to the decrease. Then figure the increase in ‘ |SK\ price due to the sales tax. For the decrease in price, multiply the original cost H> \f ) times 75 percent. If you need help in determining where the 75 percent came ^ttmjw From, go back to "Decreasing By Percents," earlier in this chapter. After you

Get the new, lower cost of the shoes, multiply that amount by 108.25 percent.

This percentage is the result of applying a percent increase and adding the

Sales tax percentage to 100 percent.

Doubling up on amoebas

A jar contains seven amoebas. These particular amoebas multiply so fast (split into two) that they double in volume every minute. If it takes

40 minutes for the amoebas to fill the jar (for it to be 100 percent full), how long did it take to fill half the jar (50 percent full)?

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The new price of the shoes is $120 X 0.75 = $90. Now, to figure the total cost with tax, take $90 X 1.0825 = 97.425. The amount on the bill is correct, when you round up to the higher penny. No error was made here. If the tax had been figured on the original amount, the total would have been $99.90. I get this amount by figuring the tax on $120 to be $120 X 0.0825 = $9.90 and adding that tax to the reduced price of the shoes. It doesn’t hurt to check — errors easily can be made when figuring discounted prices and taxes.

Tipping the Waitress without Tipping Your Hand

Did you know that the word Tip Is an acronym for To Insure Promptness? The word Tip May not have started out to stand for those words, but it does seem to fit the situation. Waiters and waitresses are at your mercy when it comes to tipping them properly, so you want to be able to compute their payment with a minimum of hassle and struggle — and do it accurately. Sometimes the restaurant makes it easy for you and adds on a 15 percent tip. Also, if you charge the meal, you’re given a nice slip of paper to do your addition on — if you know how much you want to tip. A bit of a problem arises when you use a discount coupon. All these things have to be taken into consideration — and I walk you through them all in this section.

Figuring the tip on your bill

Even when the service is questionable, most people leave a tip. The amount can serve to indicate to the waiter just what you think of the service. The most commonly used tip percentages are 10 percent, 15 percent, and 20 percent. The first and last are fairly easy to compute. But 15 percent, is a bit of a challenge.

Adding on 10 percent or 20 percent

To figure a tip of 10 percent — which means that you aren’t particularly impressed with the waiter’s work — all you need do is take the total and move the decimal point one place.

Multiplying by 10 percent is accomplished by moving the decimal point one place to the left. If you multiply 48 x 10 percent this is 48 x 0.10 = 4.80 or 4.8.

Figuring the tip in your head

When you’re in a situation where you’re going to give only a 10 percent tip (and I hope that the situation doesn’t come up very often), you can do a quick, neat calculation in your head to find the total amount — the tip plus the charge for the meal. This little method works because it’s a quick computation used to multiply any number by 11.

The method you use is to picture a 0 in front of the meal cost and add the adjacent digits together. If any of the pairs add up to more than

9, you’ll have to carry 1 to the next sum. For example, if your bill is $45.32, then picture in your mind 045.32. Start on the right end and add 2 + 3 = 5, 3 + 5 = 8, 5 + 4 = 9 and 4 + 0 = 4. Write these sums in reverse order, with the decimal point in the correct place, and you get 49.85 — the sum of the meal plus a 10 percent tip. This calculation is going to be off by a penny if the last digit in the meal cost is bigger than 5 — because of the rounding. But, for a quick computation, it works pretty well.

The Problem: How much tip do you leave if you’re going to pay 10 percent on a bill of $18.80? And what is the total payment after adding the tip?

Move the decimal in $18.80 one place to the left to get $1.88. Add $18.80 + $1.88 to get a total of $20.68.

To figure a 20 percent tip, all you do is figure the 10 percent tip and double it. You can do this by either doubling the cost and then moving the decimal point, or you can figure the 10 percent amount by moving the decimal and then doubling that amount for the tip.

The Problem: How much tip will you give the hairdresser if the charge for a haircut and perm is $85 and you want to give a 20 percent tip?

JcVLAilf When you’re moving the decimal point in 85 one place to the left, you have to remember that there Is A decimal point in the number. The decimal point is always assumed to be at the far right of the number, if it isn’t showing. So one place to the left gives you 8.5 — which is $8.50. Double that to get $17. Add that tip to the bill for a total of $85 + $17 = $102.

Computing a 15 percent tip

Multiplying a number by 15 isn’t the hardest thing to do, but it isn’t quite as sweet as multiplying by 10 or 20. So it won’t come as a shock to find that figuring a 15 percent tip is a bit more involved than figuring a 10 percent or 20 percent tip.

The simplest approach is just to multiply the amount by 15 percent and be done with it.

VLAiV

The Problem: How much do you tip the waiter if the bill is $164, and you’re going to tip 15 percent? What is the total bill after the tip is added?

Multiply 164 X 15 percent = 164 X 0.15 = $24.60. So, adding $164 + $24.60, you get a total of $188.60 for the bill.

If you like to figure out the tip in her head, there’s a trick to computing the amount of the tip when you want to leave a tip of 15 percent. First figure the 10 percent tip by moving the decimal point one place to the left. Then take half of that tip and add it on to the whole tip. There’s your 15 percent tip — 10 percent and half of 10 percent.

The Problem: You’ve gone to lunch with nine of your friends and you’ve all agreed to just split the whole bill equally — ten ways — and leave a 15 percent tip. You’re in charge of collecting everyone’s money and then settling up at the cash register. The charge for the ten lunches is $188. How much will each person give you for her share?

You first have to figure the tip, add it on to the cost of the meal, and then divide the total by 10. You’re going to do this in your head — and hope you can do it correctly, or you’re going to get stuck with any shortfall. A 15 percent tip is 10 percent plus half of 10 percent, so it’s $18.80 + $9.40 = $28.20. Add the tip to the bill to get $188 + $28.20 = $216.20. To divide that by 10, you just have to move the decimal point again and get that everyone’s share is $21.62. Of course, what are the chances that everyone will have the correct change? Oh, sure.

Taking into account the discount

Everyone just loves those buy-one-get-one-free coupons or the percentage off the total discounts. These promotions get you in the door and are good for everyone involved. Sometimes you have to pay tax on the amount before the discount, and sometimes you pay tax on the lesser amount. It depends on what the product is — and if the merchant knows how to figure it correctly. That’s why you need to be aware of what’s going on so you can check the computations.

The Problem: Your favorite restaurant is offering a free second entree, as long as it costs less than the entree you’re paying for. This is how most of the buy-one-get-one-free promotions work. You’ll pay tax on the reduced price, but you need to tip the waitress based on the total cost before the discount. Your entree costs $19.95, and your friend’s entree costs $21.95. You’ve ordered beverages totaling $16 and shared an appetizer that costs $6.95. The tax (sales plus restaurant tax) comes to 10.5 percent, and you want to give a 20 percent tip. You and your friend will split the bill. How much will each of you pay?

You’re going to pay tax on all the items except the less-pricey meal, so add up all the items — food and beverages — except the $19.95 and compute the tax on that. You’re going to figure the tip on all the items except the tax, so you’ll need a different sum to do that computation. Last, you’ll add up the cost of the items you’re paying for, the tax, and the tip. Divide that total by 2, and you’ll have the amount that each of you owes.

First, computing the tax, add $21.95 + $16 + $6.95 = $44.90. The tax on that is $44.90 X 10.5 percent = 44.90 X 0.105 = $4.7145, making the tax $4.71.

Next, to compute the tip, add up the cost of all the items, $19.95 + $21.95 + $16 + $6.95 = $64.85. Multiplying $64.85 X 20 percent = 64.85 X 0.20 = $12.97.

Your total cost is the cost of the meals plus the tax plus the tip. Add $44.90 + $4.71 + $12.97 = $62.58. Divide that by 2, and each of you owes $31.29. Now, I know that in practice, most people would round the tip up to $13, but that doesn’t really change the amount by much.

KlSS: Keeping It Simple, Silly — with Simple Interest

Simple interest Is the interest computed when Compounding Doesn’t occur. The interest in a savings account Compounds, Because, if you don’t withdraw any of the money you’ve invested, your interest earns interest. With compound interest, the amount of interest earned is added to the account total, and then the new interest is figured on the new total. Simple interest is computed only on the beginning amount.

The formula for Simple interest Is I= prt, Where IIs the amount of interest earned, P Is the principal or amount of money involved, R Is the interest rate (a percent changed to a decimal for the computation), and T Is the amount of time involved — usually a number of years.

Determining how much interest you’ve earned

Problems involving Interest Are two types: interest earned, and interest you have to pay. The interest earned is the more-fun type. You get to add money to your savings account without even working at it.

The Problem: How much simple interest is earned on $10,000 if this money is deposited for 6 years in an account that earns 4>4 percent interest?

Using the formula for simple interest and replacing the letters with their corresponding values, you get I= $10,000 X 0.0425 X 6 = 2,550. You earn $2,550 in interest, so now the total in the account is $12,550.

The Problem: How much simple interest is earned on $10,000 if you have it in one account for 6K years at 4 percent interest and then move that money and the interest it’s earned to another account earning 6 percent interest for another 332 years?

To determine the total amount of interest earned, you have to add the two different interest amounts. One amount comes from the interest earned at 4 percent, and the other amount comes from the interest earned at 6 percent.

Apply the simple-interest formula on $10,000 at 4 percent for 634 years with I= 10,000 X 0.04 X 6.5 = $2,600.

Next, apply the simple-interest formula on $12,600 at 6 percent for 334 years. Why is the amount of money different? Because the total amount from the first 634 years, the principal plus interest, is all deposited in the new account. You get I= $12,600 X 0.06 X 3.5 = $2,646.

Now, to answer the question, "How much interest is earned?", you add the two interest values together — $2,600 + $2,646 = $5,246.

Figuring out how much you need to invest

The simple-interest formula, I= prt, Gives you the amount that your money has earned for you over a particular period of time. You have a set number of dollars and invest it for a chosen number of years. Of course, the longer you invest the money, the more interest you’ll earn. Another situation is that you may have a target amount of interest or a target total of money, and you need to know how much to invest right now to reach that target. For example, you may want to set up an account (usually called an Endowment) Where only the interest is spent each year while the amount in the account stays the same and is never withdrawn. Or you may want to have a particular total amount of money to buy a boat in ten years and need to make a deposit today so that the money in the account will grow to that target amount over the years.

Spending only the interest

A benefactor wants to donate money to a local charity but doesn’t want the charity to spend it. How does this help the charity? The arrangement is that the charity gets to spend only the interest, every year, and the amount in the account stays there to earn interest the next year and the next and so on.

Inheriting a fortune

A sheikh had two sons who were equally likely the first to the city, because each wanted his

To inherit his position and his fortune. He camel to be the slower one. They happened

Devised a contest to see which son would be his upon a wise man and asked him for advice on

Successor. In this contest, the two sons would how they could finish this contest. The wise

Race their camels to a distant city. The winner man gave it his best. After hearing what he had

Would be the one whose camel was Slower! To say, the two sons jumped on the camels and

The sons started the race and wandered about aimlessly for several days; neither wanted to be

Raced as fast as they could to the city. What did the wise man tell them?

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The Problem: How much money must be invested in an account that earns 5 percent simple interest per year if the interest must come out to be $6,000?

.VLAiV Use the simple-interest formula, I = Prt, Replacing the letters with the corresponding values in the problem. You’ll be solving for P, The principal. The equation becomes: $6,000 = P x 0.05 x 1 or $6,000 = 0.05p. Divide each side of the equation by 0.05 to get P = $120,000. The person donating the money needs to put $120,000 in the account for there to be $6,000 of spendable money (or interest) each year.

Aiming at a future purchase

You have your eye on a new powerboat — one that seats up to 12 people and moves fast enough to pull a skier. You figure that the boat you want will cost close to $100,000 when you’re ready to buy it. You’re going to put away a lump sum of money today, and let it grow in value for ten years — at which time you’ll take all the money and the interest to buy the boat.

The Problem: How much money do you have to deposit in an account earning 8 percent simple interest if you want to have a total of $100,000 in principal and interest in ten years?

^jVLAiV If you use the simple-interest formula, I = prt, you’ll get varying amounts of interest, depending on what the principal is. What you want is for the principal, P, And the interest, Prt, To have a total of $100,000. Your equation would look like: P + Prt = $100,000. You can solve for the principal by dividing the $100,000 by 1 + Rt. Just add 1 to the product of the rate times the time and divide that into the $100,000. You’ll be dividing $100,000 by 1 + (0.08 x 10), which is: $100,000 1.8 = $55,555.56. Actually, the money grows even faster if you use compound interest, but this is pretty impressive just as it is here.

Working out the payments

Many people use credit cards to pay for large purchases, but some stores still offer convenient short-term payment plans on their merchandise. If you want to buy an item, the store figures out the total cost of the item plus the interest over a period of time, divides the total into equal payments, and then lets you purchase the item paying back the same amount for a certain number of months or years.

The Problem: You want to buy an all-leather sectional sofa that costs $3,500. The store will let you pay for it over the next 36 months at 9 percent interest. How much will your monthly payment be if you’re going to be paying for the sofa and three years’ simple interest in equal monthly payments?

Figure out the interest on $3,500 at 9 percent for 3 years. Add the interest to the cost of the sofa and divide the total by 36. The interest is $3,500 X 0.09 X 3 = $945. Add $3,500 + $945 to get $4,445. Divide $4,445 ■ 36 = $123.47222. . . . You can’t just lop off the remainder. The 0.00222 . . . represents a remainder of 8 cents in the division, so you’ll pay $123.48 for 8 months and $123.47 for the other 28 months.

How do you get the 8<t remainder? You multiply $123.47 by 36, and you get $4444.92, which is 8<t short of the total. Long division gives you a remainder of 8<t. Calculators give you the decimal. And, if you’re so lucky, you have a graphing calculator that changes decimals into fractions automatically.

Chapter 7

In This Chapter

^ Using proportions to figure fair shares ^ Working with proportions effectively ^ Weighing all the choices

R

Roportions are nothing more than two ratios or fractions set equal to one another. Proportions have several very handy properties that make working with them much easier to manage. In this chapter, you see how to set up the proportions correctly and how to solve the problems you’ve created with the proportions. You see how to apply the properties of proportions to make the solutions easier.

Working with the Math of Proportions

A proportion is an equation involving two fractions. The ratio of the numerators and denominators of the fractions must be equal. A proportion is a statement saying that two fractions are equivalent or equal in value. The fractions can be reduced in the normal way — the way you’ve see since third grade — and they can be reduced in some rather unique ways, too. You use this prop-^ABEft Erty and several others to solve for unknown parts of a proportion.

Given the proportion % = c, the following also are true:

A x d = B X C The cross products are equal.

% = D The reciprocals are equal.

You can reduce (eliminate common divisors):

% = e # g, % = e——, % = 7T You can reduce the fractions

B e #g b e # g b g

C

E X F e XG> EX F e#G’ F g

Vertically.

You can reduce the fractions horizontally.

%

Solving proportions by multiplying or flipping

Equations involving proportions are solved using the properties of proportions. When you cross-multiply, you get rid of the fraction format, which gives you an equation that is usually simpler to deal with. Also, if you flip the proportion, you make the problem more to your liking — easier to solve.

Cross-multiplying in a proportion

You can solve for the value of X In the proportion <2x +i3 = 5 by cross-multi -

1 • 1 • 1 C TI c t – iri. X — 3 7 J

Plying and getting rid of the fractional format.

2x + 3 = 5 X — 3 7 (2x + 3) X 7 = (X — 3) X 5

14x + 21 = 5x — 15

9x =—36 X=—4

Flipping your lid over a proportion

Even though cross-multiplying is a great tool to use when solving proportions, you can often take an easier route: Flip the fractions (set the reciprocals equal to one another) and then multiply each side by the same number to solve the equation. For example, in the following equation, I flip the proportion and then just have to multiply each side by the number under the X, Reduce, and get the answer.

70 35

— = —

X 21 X 21

= ——

70 35 X=21 X 2 =42

Going every which way with reducing

Reducing fractions in proportions is a blast! You can reduce across the tops, across the bottoms, up and down on the left, or up and down on the right. You just can’t reduce going diagonally — the crisscross motion is for multiplying only. By reducing the proportion, first, before cross-multiplying, you get to work with smaller numbers. That’s a good idea not only for the ease of the problem, but also because it helps prevent errors.

Reducing vertically

Proportions are created from fractions, and the traditional way of reducing fractions is to find a number that divides the numerator and denominator evenly and divide each part of the fraction by that number. Reducing fractions is sometimes referred to as Cancelling. In the following proportion, the numbers in the fraction on the left are each divisible by 5. After reducing the fraction on the left, you can cross-multiply and solve the equation for X.

25 X

— = —

40 16 25 X

- = -

5X

— =

8 16 5 X 16 = 8x

80 = 8X

10 = X

You could also have reduced horizontally in this equation, because 8 and 16 are both divisible by 8. The next section shows you how that works.

Reducing horizontally

The property of proportions that allows you to cross-multiply and have a true statement is the same property that allows for reducing across the proportion horizontally. Both of these clever tools are due to the commutative property of multiplication — the fact that reversing the order of the numbers in a multiplication property doesn’t change the answer. It’s just neat that this property comes in so handy when working with proportions. In the following proportion, the two numerators are each divisible by 11. Then you see that the two numbers in the right fraction are divisible by 7. Reduce that fraction so that, when you cross-multiply, you don’t have to multiply by 63.

X 63 27

— zz -

X 63

71

63

2X9 XX1

18 X

Dividing $17

Sneaky Pete thought that he could pull one over on his buddies and keep all the money that was paid to all of them for taking care of the neighbor’s dog. The neighbor gave Pete $17 and told him to divvy it up according to how much time each friend spent taking care of the dog. Pete took the money and posed this problem: "In my hand, I have $17. Tom can have one-ninth of the money, because he worked one-ninth of the

Time. Tim can have one-third of the money, because he worked one-third of the time. And Ted can have one-half of the money, because he worked half of the time. You aren’t allowed to tear any of the dollar bills, and there’s no change. If you can’t figure out how to divide up the $17, then I’ll just keep it all." Pete’s buddies were just too smart for him and divided up the money equitably. How did they do it?

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Dividing Things Up Equitably

Everyone should play fair. Whether it’s in basketball or a card game, you expect to be treated fairly. The same can be said for sharing candy or money or time. If everyone gets an equal share (all the same amount), then the computation is easy — you just divide by the number of people who are involved. It gets a little more complicated when the shares are to be unequally divided, like someone getting twice as much as another because she did twice the work.

Splitting things between two people unevenly

If two children are to share equally in an inheritance, you just divide the total amount by two. It gets a bit stickier when one person gets more than the other. It strains the family relationship %nd The mathematics.

The Problem: Henry and Hilda are to share their father’s $2.5 million estate. Their father said that Henry is to get 65 percent of the estate and Hilda is to get the other 35 percent. How much does Henry get?

Using proportions, think of Henry’s 65 percent as being 65 out of 100, where 100 is the total amount. Write the proportion with X Being the unknown

Amount out of $2.5 million. The proportion is = 2 500 000. The proportion says that 65 out of 100 is equal to some unknown value out of 2,500,000 (2.5 million).

When writing proportions, put Related Amounts either horizontally or vertically from one another. Units that are alike should be either across from one another or above and below one another.

In the problem, the units that are Related Are the $2,500,000, which is All Of the inheritance and the 100 which represents 100 percent or All. The X And the 65 each represent a Part Of the whole thing. To solve the proportion and determine Henry’s share, first reduce across the bottom of the proportion, and then cross-multiply and solve for X.

65 X

-= -——-

100 2.500:000 ,

1* —25000

65 X

1 — -

25,000 65 X 25,000 = X x 1 1,625,000 = X

Henry gets $1,625,000 of the $2.5 million in their father’s estate.

Figuring each person’s share

Another possible scenario when dividing things up is that three or more people are involved, and they each get a different share or fraction of the total amount. A situation like this occurs when people do different amounts of the total work or when they are different ages or different weights or whatever makes them different from one another.

The proportion or proportions used to solve problems where three or more people get differing shares all have a common theme. You’re always concerned with the total amount — and all the parts must add up to the total amount.

If a pie is to be divided among four people, and the shares are one-twelfth, one-sixth, one-fourth, and one-half of the pie, you have to be sure that these fractions all add up to 1 — which is the whole pie.

1 + J_ + J_ + J_ = J_ + _2_ + _3_ + _6_ 12 6 4 2 12 12 12 12

= 1+2 + 3 + 6 = 12 = 1 12 12

Consider a situation where contestants share in the total prize depending on the number of points that they’ve scored.

The Problem: In a fishing tournament, a sports-equipment company has offered a prize of $100,000 to be divided among the top five winners in proportion to their scores. The points earned by the top five winners are: 60, 40, 30, 20, and 10. How much money does each get?

Determine the total number of points, and then figure out the proportion or part of the prize that each gets. The total number of points is: 60 + 40 + 30 + 20 + 10 = 160 points.

Writing a proportion with the 60 points for the top scorer, = Tjtt^aa . N i ti i ^ it – i iii 160 100,000

Reduce across the bottom, cross-multiply and solve for X.

60 X

160 100000 ,

\s -— 625

60 X625 = 1 XX 37,500 = X

The top scorer gets $37,500. The rest of the winners get amounts as shown here:

The second-place winner gets $25,000, determined from

40 = X 40 = X 4x = 100 000 160 100,000 160 100,000 ’ ‘

The third-place winner gets $18,750, from

30 = X 30 = X = 300 000

160 100,000 160 100,000 ’ ‘

16

The fourth-place w1inner gets $12,500, using

20 = X 20 = X 8x = 100 000 160 100,000 160 100,000 ’ ‘

‘ 8

The fifth-place winner gets $6,250, which you determine by just subtracting all the other prizes from $100,000 and seeing what’s left.

Comparing the proportions for differing amounts of money

Some very interesting problems occur when an estate is divided up between all the heirs. And then there’s the executor’s share to be considered in the mix. Proportions are very handy when you’re determining who gets how much money.

Diophantus’s life

Diophantus was a Greek mathematician who had an impact on algebra back in the day. A puzzle that’s attributed to his life goes: "Diophantus’s boyhood lasted for one-sixth of his life. He married after one-seventh more. His

Beard grew after one-twelfth more of his life, and his son was born 5 years later. The son lived to half his father’s age, and Diophantus died four years after the son. How long did Diophantus live?

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The Problem: A woman’s estate totals $5 million. She leaves KO of her estate to her butler, %> to her chauffer, XO to her daughter, !4 To her son, and KO is to be put in a trust to take care of her dog, Puddles. But all these bequests aren’t to be made until After The executor of her will gets 10 percent of the original amount. The others get their share of the net amount. How much does each person get?

■AVLAW You first deduct the amount that the executor gets. Because she gets 10 percent of the estate, you multiply $5,000,000 X 0.10 = $500,000. Subtract that from the estate, and it leaves $4,500,000 to be divided among the others. The other shares are determined using proportions, letting the fractional share be one side of the proportion and an X Divided by $4,500,000 be the other side of the proportion. The value of X In each case is that person’s share.

Butler: -XR = . ,Ax AAA, – Xr = . „ fL^-, X = 225,000

20 4,500,000 20 4,500:000

‘ 1 ^jj-—-’ 225,000

Chauffer: -k^ = . ,Ax AAA, – Mr = , rr, A -, X = 3 • 225,000 = 675,000

20 4,500,000 20 4J500000

1 225, 000

Daughter: Mr = A r^ nnn, "^r ^ ^-, X = 450,000

0 10 4,500,000 10 4.51M0000

Son: = 4 500 000, "4~ = 4 5OO-00t)-, X = 1,125,000

‘ ’ * 1 ^.JJ-1*"7, 1,125,000

Puddles: – J77 = Tft^h, -Mr = . rAA -, X = 9 $ 225,000 = 2,025,000

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20 4,500,000 – 20 4,500000

12

Comparing Apples and Oranges

One of the nicest things about proportions is that they can be used to solve problems involving items that don’t seem to have anything in common except for their ratios to one another. For instance, if you’re told that three apples can be traded for four oranges, then you can figure out how many apples you can get for 28 oranges by using a proportion.

The Problem: How many apples can you trade for your 28 oranges if the current trading rate is that three apples are worth four oranges?

Write the apples over the oranges in one fraction, and then put the 28 oranges on the bottom of the other fraction — opposite the oranges in the first fraction. Let the unknown number of apples be represented by X.

3 apples _ X Apples 3 _ X

4 oranges 28 oranges 4 28

This proportion is solved by reducing across the bottom and cross-multiplying.

3 = X 21 = X 41 287′

It takes 21 apples to get 28 oranges. As long as you have apples across from apples and oranges across from oranges, the proportion will work. Another format for this is to have apples over apples and oranges over oranges, with the equivalence of 3 and 4 across from one another. (Refer to "Working with the Math of Proportions," earlier in this chapter, if you need help with the manipulations of proportions.)

Determining the amounts in recipes

If you’re a cook — or even if you don’t have much interest in the culinary arts — you’re apt to come across a situation where you need to double a recipe for a bigger crowd, halve a recipe that makes too much food, or compute some such multiple or part of a recipe. Even if the recipe is for cement, getting the amounts correct is important. Otherwise, the chili will taste too salty or the cake won’t rise or the cement will never harden. Proportions are a huge help with these recipe challenges.

The Problem: Your favorite chili recipe calls for 2 pounds of hamburger and 3 onions. You’re going to make enough chili for your whole fraternity and plan to use 28 pounds of hamburger. How many onions do you need?

^VLA* Set up a proportion with pounds of hamburger divided by number of onions in one fraction and pounds of hamburger divided by onions in the other fraction. Place an X For the unknown number of onions. Be sure to put the 28 pounds of hamburger in the same fraction as the X Number of onions.

2 pounds _ 28 pounds 2 _28

3 onions X Onions ‘ 3 X

Solving this proportion is easier if you reduce across the top by dividing each numerator by 2. Then flip the proportion before cross-multiplying. (Refer to "Working with the Math of Proportions," earlier in this chapter, if you need a refresher on these techniques.)

H = .2814 3 = X_ = =

3 X 1 14

You will need 42 onions. I feel a crying session coming on.

I inherited my grandmother’s recipe box, and there are some wonderful, old recipes in it. One of my favorites (just because I’m trying to imagine my very proper grandmother, Marion Jones Roby Ingersoll, making this much food) is for sausage. The recipe calls for 100 pounds of pork, >4 pound of sage, >4 pound of pepper, 2 pounds of salt, 1 tablespoon of mustard and a "little" summer savory. The recipe says to put the stuff in a crock by layers and "weigh it down."

The Problem: I want to try my grandmother’s recipe, but not in that huge quantity. If I start with 5 pounds of pork instead of 100, how much sage and mustard will I need?

^VLA* Even though the quantities are in pounds and tablespoons, I can still use proportions to solve for the amounts needed. Write a proportion with the original number of pounds of pork and sage in the numerator and denominator of the first fraction, and then write the 5 pounds of pork in the other fraction across from the 100 pounds. Solve for the reduced pounds of sage. Do the same thing with the original measures to solve for the amount of mustard.

100 pounds = 5 pounds 100 = 5 20 = 1 = 1 1 pound p ^

When I divided each side by 20, I got >8o pound of sage. It makes more sense to change the fraction of a pound to ounces. If there are 16 ounces in a pound, then multiply 16 X >8o pound to get >5 ounce. And now, for the mustard, write a new proportion and solve it.

100 pounds = 1 T 100 = 1 100 = 1 = 1 4 pound 4

The measure of >4oo of a tablespoon is probably better known as a "dash."

Truck on a bridge

A large truck is crossing a bridge that measures of the truck and all its cargo. The truck makes it 1 mile in length. The bridge can only hold 14,000 halfway across the bridge and stops. A bird pounds, which happens to be the exact weight lands on the truck. Does the bridge collapse?

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Figuring out weighted averages

Weighted averages are used to give more importance or emphasis to one thing than another. A prime example of weighted averages is when they’re used to determine your grade in a college course. The weighting can go something like this: Tests count three times as much as papers, the final exam counts twice as much as a test, and attendance counts one-fourth as much as a paper. In general, to find a weighted average, you set up a proportion and multiply the weights times their respective amounts. Look at these next two problems, and you’ll see what I mean.

The Problem: An Astrodollars Coffee shop sells several different types of whole coffee beans. Last Monday they sold 100 pounds of Honduran coffee beans, 70 pounds of Guatemalan coffee beans, 40 pounds of Nicaraguan coffee beans, and 40 pounds of Chilean coffee beans. The Honduran beans cost $8 per pound, the Guatemalan beans cost $9 per pound, the Nicaraguan beans cost $10.50 per pound, and the Chilean beans cost $13.50 per pound. What was the average cost per pound of the coffee beans sold on Monday?

£?LAiV Set up a proportion where each poundage multiplies its respective price. Put ‘ <ES\ the sum of all the products you get in the numerator of a fraction, and divide Hp \f ) by the total number of pounds. That’s one side of the proportion. Set that

Fraction equal to another fraction with a 1 in the denominator (opposite the

Total number of pounds) and an X In the numerator.

(100 X + (70 X 9.00) + (40 X 10.50) + (40 X 13.50) = X 100 pounds + 70 pounds + 40 pounds + 40 pounds 1 pound

Now simplify the proportion by doing the multiplications and additions; then reduce the fraction on the left. Cross-multiply and solve for X, And you get $9.56 for the average cost of the coffee beans sold on Monday.

800 +630 +420 +540

100 +70 +40 +40 1 pound

2,390 X

=

250 1

X

2

2,390 250 1

25

239 = 25X 239 X = "25"

9.56

A college grade point average (GPA) is usually a weighted average. Different courses are a different number of hours or quarters or other units, which serve to determine their relative worth. Consider a student who attends a college that measures courses in semester hours and uses grades of A, B, C, D, and F. The corresponding point values for the grades are: 4, 3, 2, 1, and 0 points.

The Problem: Nick took five courses last semester and needs to determine his GPA. He got an A in his 4-semester-hour calculus course, an A in his 1-semester-hour computer course, a B in his 5-semester-hour biology course, a C in his 3-semester-hour English course, and a B in his 3-semester-hour Spanish course. What is his GPA for the semester?

Multiply each number of semester hours by its worth in terms of points for that grade, and add up all the points; divide by the total number of semester hours. Set that fraction equal to X Divided by 1.

(A x 4) + (A x 1) + (B x 5) + (C x 3) + (B x 3) = X 4 + 1 + 5 + 3 + 3 1 (4 X 4) + (4 X 1) + (3 X 5) + (2 X 3) + (3 X 3) = X 16 1 16 +4 +15 +6 +9 X

-t~7*- = ~t~

16 1

50 X

— = ——

16 1

25

50 X

16 1

25 8X

8

3.125 = X

X

After simplifying the proportion, reducing the fraction on the left, cross-multiplying, and solving for X, You get that Nick’s GPA is 3.125.

Computing Medicinal Doses Using Proportions

Modern medicine offers many wonderful options to reduce pain and suffering. The amount of the medication for a particular person has to be correct, though. The amount of medicine prescribed may depend on a person’s weight, age, or current health status — or, often, a mixture of all these things. Proportions are used to determine dosages of many medications and the number of tablets needed per dose.

Figuring the tablets for doses

A Scored tablet Is a medicine tablet that is designed so that it can be broken into halves or quarters, making it possible to administer a dose that is less than the amount in the tablet. For purposes of these problems, assume that the tablets are scored into quarters. (You can break the tablet into four equal pieces.)

The Problem: A doctor prescribes 0.375 mg of Digoxin, and the scored tablets that are available contain 0.25 mg each. How many tablets should be administered?

Set up a proportion with the amount per tablet in one fraction and the needed dosage and number of tablets in the other fraction.

0.25 mg 0.375 mg

—

- = -

1 tablet X Tablets

Reduce the fractions across the top by dividing each numerator by 0.125. Then solve for X.

0252 = 03753 1 = X

2X= 3

X= 1.5

The patient needs 1>2 tablets.

The Problem: A patient is to take 0.5 g of Ampicillin, and the capsules available are 250 mg. How many capsules are needed?

From the metric system: 1 gram = 1,000 milligrams or 1 g = 1,000 mg.

Change the 0.5 grams to milligrams using a proportion with 1 gram over 1,000 milligrams in one fraction and 0.5 grams over X Milligrams in the other fraction. Solve for X.

1g 0.5 g

- _ -

1,000 mg X Mg

1 _ 0.5 1,000 X

X _ 0.5 X 1000 _ 500

So 0.5 g = 500 mg. If the patient is to take 500 mg of Ampicillin, and the capsules are 250 mg, then the patient will need two capsules.

Making the Weight count

A person’s weight can affect the dosage of the medication they’re given. You use a proportion to determine the amount of medication based on that weight.

The Problem: Robert has been taking 80 mg of a medication every day. When it was prescribed, he weighed 170 pounds. He’s lost 30 pounds, so how much should the new dosage be?

Create a proportion with 80 mg and 170 pounds in one fraction and 140 pounds in the other fraction. The 140 is found by subtracting 170 – 30 to determine Robert’s new weight. Be sure to put the 140 pounds opposite the 170 pounds in the proportion.

80 mg X Mg

- zz -

170 lb 140 lb

Reduce either vertically in the left fraction or horizontally across the bottom. Then cross-multiply and solve for X.

You can "drop" the zeros in only one direction. Reducing by dividing by 10 in the proportion can be done either vertically or horizontally, but not both.

80 X

- zz -

170 140

80 X

=

170 140

S 17 * 14

1,120 17

The dosage is about 65.88 mg of the medication. The doctor will have to round up or down to find a suitable tablet to use.

Chapter 8