In This Chapter
Identifying classic pitfalls in human thought ^ Correcting your thinking
^ Getting to know the thinking errors you make most
\M Ou probably don’t spend a lot of time mulling over the pros and cons of the way you think. Most people don’t – but to be frank, most people ideally ought to!
One of the messages of CBT is that the thoughts, attitudes, and beliefs you hold have a big effect on the way you interpret the world around you and on how you feel. So, if you’re feeling excessively bad, chances are that you’re thinking badly – or, at least, in an unhelpful way. Of course, you probably don’t Intend To think in an unhelpful way, and no doubt you’re largely unaware that you do.
Thinking errors Are slips in thinking that everyone makes from time to time. Just as a virus stops your computer from dealing with information effectively, so thinking errors prevent you from making accurate assessments of your experiences. Thinking errors lead you to get the wrong end of the stick, jump to conclusions, and assume the worst. Thinking errors get in the way of, or cause you to distort, the facts. However, you do have the ability to step back and take another look at the way you’re thinking and set yourself straight.
Months or years after the event, you’ve probably recalled a painful or embarrassing experience and been struck by how differently you feel about it at this later stage. Perhaps you can even laugh about the situation now. Why didn’t you laugh back then? Because of the way you were thinking at the time.
To err is most definitely human. Or, as American psychotherapist Albert Ellis is quoted as saying, ‘If the Martians ever find out how human beings think, they’ll kill themselves laughing.’ By understanding the thinking errors we outline in this chapter, you can spot your unhelpful thoughts and put them straight more quickly. Get ready to identify and respond in healthier ways to some of the most common ‘faulty’ and unhelpful ways of thinking identified by researchers and clinicians.
Catastrophising: Turning Mountains Back Into Molehills
Catastrophising Is taking a relatively minor negative event and imagining all sorts of disasters resulting from that one small event, as we sum up in Figure 2-1.
Consider these examples of catastrophising:
You’re at a party and you accidentally stumble headlong into a flower arrangement. After you extract yourself from the foliage, you scurry home and conclude that everyone at the party witnessed your little trip and laughed at you.
You’re waiting for your teenage daughter to return home after an evening at the cinema with friends. The clock strikes 10:00 p. m., and you hear no reassuring rattle of her key in the door. By 10:05 p. m., you start imagining her accepting a lift home from a friend who drives recklessly. At 10:10 p. m., you’re convinced she’s been involved in a head-on collision and paramedics are at the scene. By 10:15 p. m., you’re weeping over her grave.
Your new partner declines an invitation to have dinner with your parents. Before giving him a chance to explain his reasons, you put down the phone and decide that this is his way of telling you the relationship’s over. Furthermore, you imagine that right now he’s ringing friends and telling them what a mistake it was dating you. You decide you’re never going to find another partner and will die old and lonely.
Catastrophising leads many an unfortunate soul to misinterpret a social faux pas as a social disaster, a late arrival as a car accident, or a minor disagreement as total rejection.
Nip catastrophic thinking in the bud by recognising it for what it is – just thoughts. When you find yourself thinking of the worst possible scenario, try the following strategies:
Put your thoughts in perspective. Even if everyone at the party did see your flower-arranging act, are you sure no one was sympathetic? Surely you aren’t the only person in the world to have tripped over in public. Chances are, people are far less interested in your embarrassing moment than you think. Falling over at a party isn’t great, but in the grand scheme of things it’s hardly society-page news.
Consider less terrifying explanations. What other reasons are there for your daughter being late? Isn’t being late for curfew a common feature of adolescence? Perhaps the movie ran over, or she got caught up chatting and forgot the time. Don’t get so absorbed in extreme emotions that you’re startled to find your daughter in the doorway apologising about missing the bus.
Weigh up the evidence. Do you have enough information to conclude that your partner wants to leave you? Has he given you any reason to think this before? Look for evidence that contradicts your catastrophic assumption. For example, have you had more enjoyable times together than not?
Focus on what you can do to cope with the situation, and the people or resources that can come to your aid. Engaging in a few more social encounters can help you put your party faux pas behind you. You can repair a damaged relationship – or find another. Even an injury following an accident can be fixed with medical care.
No matter how great a travesty you create in your mind, the world’s unlikely to end because of it even if the travesty comes to pass. You’re probably far more capable of surviving embarrassing and painful events than you give yourself credit for – human beings can be very resilient.
Aii-or-Nothing Thinking: finding Somewhere in Between
All-or-nothing Or Black-or-white thinking (see Figure 2-2) is extreme thinking that can lead to extreme emotions and behaviours. People either love you or hate you, right? Something’s either perfect or a disaster. You’re either responsibility-free or totally to blame? Sound sensible? We hope not!
Unfortunately, humans fall into the all-or-nothing trap all too easily:
Imagine you’re trying to eat healthily in order to lose weight and you cave in to the temptation of a doughnut. All-or-nothing thinking may lead you to conclude that your plan is in ruins and then to go on to eat the other 11 doughnuts in the pack.
You’re studying a degree course and you fail one module. All-or-nothing thinking makes you decide that the whole endeavour is pointless. Either you get the course totally right or it’s just a write-off.
Consider the humble thermometer as your guide to overcoming the tendency of all-or-nothing thinking. A thermometer reads degrees of temperature, not only ‘hot’ and ‘cold’. Think like a thermometer – in degrees, not extremes. You can use the following pointers to help you change your thinking:
Be realistic. You can’t possibly get through life without making mistakes. One doughnut doesn’t a diet ruin. Remind yourself of your goal, forgive yourself for the minor slip, and resume your diet.
Develop ‘both-and’ reasoning skills. An alternative to all-or-nothing thinking is Both-and reasoning. You need to mentally allow two seeming opposites to exist together. You can Both Succeed in your overall educational goals And Fail a test or two. Life is not a case of being either a success or a failure. You can Both Assume that you’re an OK person as you are And Strive to change in specific ways.
All-or-nothing thinking can sabotage goal-directed behaviour. You’re far more likely to throw in the towel at the first sign of something blocking your goal when you refuse to allow a margin for error. Beware of ‘either/or’ statements and global labels such as ‘good’ and ‘bad’ or ‘success’ and ‘failure’. Neither people nor life situations are often that cut and dry.
Fortune-telling: Stepping AuJay from the Crystal Ball
Often, clients tell us after they’ve done something they were anxious about that the actual event wasn’t half as bad as they’d predicted. Predictions are the problem here. You probably don’t possess extrasensory perceptions that allow you to see into the future. You probably can’t see into the future even with the aid of a crystal ball like the one in Figure 2-3. And yet, you may try to predict future events. Unfortunately, the predictions you make may be negative:
You’ve been feeling a bit depressed lately and you aren’t enjoying yourself like you used to. Someone from work invites you to a party, but you decide that if you go you won’t have a good time. The food will unpalatable, the music will be irksome, and the other guests are sure to find you boring. So, you opt to stay in and bemoan the state of your social life.
You fancy the bloke who sells you coffee every morning on the way to the office, and you’d like to go out with him on a date. You predict that if you ask him, you’ll be so anxious that you’ll say something stupid. Anyway, he’s bound to say no thanks – someone that attractive must surely be in a relationship.
You always thought that hang-gliding would be fun, but you’ve got an anxious disposition. If you try the sport, you’re sure to lose your nerve at the last minute and just end up wasting your time and money.
You’re better off letting the future unfold without trying to guess how it may turn out. Put the dustcover back on the crystal ball and leave the tarot cards alone, and try the following strategies instead:
Test out your predictions. You really never know how much fun you might have at a party until you get there – and the food could be amazing. Maybe the chap at the coffee shop has got a partner, but you won’t be sure until you ask. To find out more about testing out your predictions, have a read through Chapter 4.
Be prepared to take risks. Isn’t it worth possibly losing a bit of cash for the opportunity to try a sport you’ve always been interested in? And can’t you bear the possibility of appearing a trifle nervous for the chance to get to know someone you really like? There’s a saying ‘a ship is safe in a harbour, but that’s not what ships are built for’. Learning to live experimentally and taking calculated risks is a recipe for keeping life interesting.
Understand that your past experiences don’t determine your future experiences. Just because the last party you went to turned out to be a dreary homage to the seventies, the last person you asked out went a bit green, and that scuba-diving venture resulted in a severe case of the bends doesn’t mean that you’ll never have better luck again.
Typically, fortune-telling stops you from taking action. It can also become a bit of a self-fulfilling prophecy. If you keep telling yourself that you won’t enjoy that party, you’re liable to make that prediction come true. Same goes for meeting new people and trying new things. So, put on your party gear, ask him out for dinner, and book yourself in for some hang-gliding.
Mind-Reading: Taking \lour Guesses u/ith a Pinch of Salt
So, you think you know what other people are thinking, do you? With Mind-reading (see Figure 2-4), the tendency is often to assume that others are thinking negative things about you or have negative motives and intentions.
Here are some examples of mind-reading tendencies:
You’re chatting with someone and they look over your shoulder as you’re speaking, break eye contact, and (perish the thought) yawn. You conclude immediately that the other person thinks your conversation is mind-numbing and that he’d rather be talking to someone else.
Your boss advises that you book some time off to use up your annual leave. You decide that he’s saying this because he thinks your work is rubbish and wants the opportunity to interview for your replacement while you’re on leave.
You pass a neighbour on the street. He says a quick hello but doesn’t look very friendly or pleased to see you. You think that he must be annoyed with you about your dog howling at the last full moon and is making plans to report you to environmental health.
You can never know for certain what another person is thinking, so you’re wise to pour salt on your negative assumptions. Stand back and take a look at all the evidence to hand. Take control of your tendency to mind-read by trying the following:
Generate some alternative reasons for what you see. The person you’re chatting with may be tired, be preoccupied with his own thoughts, or just have spotted someone he knows.
Consider that your guesses may be wrong. Are your fears really about your boss’s motives, or do they concern your own insecurity about your abilities at work? Do you have enough information or hard evidence to conclude that your boss thinks your work is substandard? Does it follow logically that ‘consider booking time off means ‘you’re getting the sack’?
I Get more information (if appropriate). Ask your neighbour whether your dog kept him up all night, and think of some ways to muffle your pet next time the moon waxes.
You tend to mind-read what you fear most. Mind-reading is a bit like putting a slide in a slide projector. What you Project Or imagine is going on in other people’s minds is very much based on what’s already in yours.
Emotional Reasoning: Reminding \loursel( That Feelings Aren’t Facts
Surely we’re wrong about this one. Surely your feelings are real hard evidence of the way things are? Actually, no! Often, relying too heavily on your feelings as a guide leads you off the reality path. Here are some examples of emotional reasoning:
Your partner has been spending long nights at the office with a coworker for the past month. You feel jealous and suspicious of your partner. Based on these feelings, you conclude that your partner’s having an affair with his co-worker.
You feel guilty out of the blue. You conclude that you must have done something wrong otherwise you wouldn’t be feeling guilty.
When you feel emotional reasoning taking over your thoughts, take a step back and try the following:
1. Take notice of your thoughts. Note thoughts such as ‘I‘m feeling nervous, something must be wrong’ and ‘I‘m so angry, and that really shows how badly you’ve behaved’, and recognise that feelings are not always the best measure of reality, especially if you’re not in the best emotional shape at the moment.
2. Ask yourself how you’d view the situation if you were feeling calmer.
Look to see if there is any concrete evidence to support your interpretation of your feelings. For example, is there really any hard evidence that something bad is going to happen?
3. Give yourself time to allow your feelings to subside. When you’re feeling calmer, review your conclusions and remember that it is quite possible that your feelings are the consequence of your present emotional state (or even just fatigue) rather than indicators of the state of reality.
The problem with viewing your feelings as factual is that you stop looking for contradictory information – or for any additional information at all. Balance your emotional reasoning with a little more looking at the facts that support and contradict your views, as we show in Figure 2-5.
Overqeneraiisinq: Avoiding the Part/Whole Error
Overgeneralising Is the error of drawing global conclusions from one or more events. When you find yourself thinking ‘always’, ‘never’, ‘people are. . .’, or ‘the world’s.. .’, you may well be overgeneralising. Take a look at Figure 2-6. Here, our stick man sees one black sheep in a flock and instantly assumes the whole flock of sheep is black. However, his overgeneralisation is inaccurate because the rest of the flock are white sheep.
You might recognise overgeneralising in the following examples:
You feel down. When you get into your car to go to work, it doesn’t start. You think to yourself, ‘Things like this are always happening to me. Nothing ever goes right’, which makes you feel even more gloomy.
You become angry easily. Travelling to see a friend, you’re delayed by a fellow passenger who cannot find the money to pay her train fare. You think, ‘This is typical! Other people are just so stupid’, and you become tense and angry.
You tend to feel guilty easily. You yell at your child for not understanding his homework and then decide that you’re a thoroughly rotten parent.
Situations are rarely so stark or extreme that they merit terms like ‘always’ and ‘never’. Rather than overgeneralising, consider the following:
Get a little perspective. How true is the thought that nothing Ever Goes right for you? How many other people in the world may be having car trouble at this precise moment?
Suspend judgement. When you judge all people as stupid, including the poor creature waiting in line for the train, you make yourself more outraged and are less able to deal effectively with a relatively minor hiccup.
Be specific. Would you be a Totally Rotten parent for losing patience with your child? Can you legitimately conclude that one incident of poor parenting cancels out all the good things you do for your little one? Perhaps your impatience is simply an area you need to target for improvement.
Shouting at your child in a moment of stress no more makes you a rotten parent than singing him a great lullaby makes you a perfect parent. Condemning yourself on the basis of making a mistake does nothing to solve the problem, so be specific and steer clear of global conclusions.
Labelling: GiVinq Up the Ratinq Game
Labels, and the process of labelling people and events, are everywhere. For example, people who have low self-esteem may label themselves as ‘worthless’, ‘inferior’, or ‘inadequate’ (see Figure 2-7).
If you label other people as ‘no good’ or ‘useless’, you’re likely to become angry with them. Or perhaps you label the world as ‘unsafe’ or ‘totally unfair’? The error here is that you’re globally rating things that are too complex for a definitive label. The following are examples of labelling:
You read a distressing article in the newspaper about a rise in crime in your city. The article activates your belief that you live in a thoroughly dangerous place, which contributes to you feeling anxious about going out.
You receive a poor mark for an essay. You start to feel low and label yourself as a failure.
You become angry when someone cuts in front of you in a traffic queue. You label the other driver as a total loser for his bad driving.
Strive to avoid labelling yourself, other people, and the world around you. Accept that they’re complex and ever-changing (see Chapter 12 for more on this). Recognise evidence that doesn’t fit your labels, in order to help you weaken your conviction in your global rating. For example:
Allow for varying degrees. Think about it: The world isn’t a dangerous place but rather a place that has many different aspects with varying degrees of safety.
Celebrate complexities. All human beings – yourself included – are unique, multifaceted, and ever-changing. To label yourself as a failure on the strength of one failing is an extreme form of overgeneralising. Likewise, other people are just as complex and unique as you. One bad action doesn’t equal a bad person.
When you label a person or aspect of the world in a global way, you exclude potential for change and improvement. Accepting yourself as you are is a powerful first step towards self-improvement.
Making demands: Thinking Flexibly
Albert Ellis, founder of rational emotive behaviour therapy, one of the first cognitive-behavioural therapies, places demands at the very heart of emotional problems. Thoughts and beliefs that contain words like ‘must’, ‘should’, ‘need’, ‘ought’, ‘got to’, and ‘have to’ are often problematic because they’re extreme and rigid (see Figure 2-8).
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In This Chapter ^ Embracing the International System of units ^ Relating base units and derived units ^ Converting between units Ave you ever been asked for your height in centimeters, your weight in kilograms, or the speed limit in kilometers per hour? These measurements may seem a bit odd to those folks who are used to feet, pounds, and miles per hour, but the truth is that scientists sneer at feet, pounds, and miles. Because scientists around the globe constantly communicate numbers to each other, they prefer a highly systematic, standardized system. The International System Of units, abbreviated SI From the French term Systeme International, Is the unit system of choice in the scientific community. You find in this chapter that the SI system is a very logical and well organized set of units. Despite what many of their hairstyles may imply, scientists love logic and order, so that’s why SI is their system of choice. As you work with SI units, try to develop a good sense for how big or small the various units are. Why? That way, as you’re doing problems, you have a sense for whether your answer is reasonable. Familiarizing Yourself With Base Units and Metric System Prefixes The first step in mastering the SI system is to figure out the base units. Much like the atom, the SI base units are building blocks for more complicated units. In later sections of this chapter, you find out how more complicated units are built from the SI base units. The five SI base units that you need to do chemistry problems (as well as their familiar, non-SI counterparts) are given in Table 2-1.
Feel free to refer to Table 2-2 as you do your problems. You may want to earmark this page because, after this chapter, we simply assume that you know how many meters are in one kilometer, how many grams are in one microgram, and so on. You measure a length to be 0.005m. How A. 5 mm. 0.005 is 5 x 10 3m, or 5 mm. might this be better expressed using a metric system prefix?
Building Derived Units from Base Units Chemists aren’t satisfied with measuring length, mass, temperature, and time alone. On the contrary, chemistry often deals in quantities. These kinds of quantities are expressed with Derived units, Which are built from combinations of base units. Area (for example, catalytic surface): Area = Length x Width and has units of length squared (meter2, for example). Volume (of a reaction vessel, for example): You calculate volume by using the familiar formula: Volume = Length x Width x Height. Because length, width, and height are all length units, you end up with length x length x length, or a length cubed (for example, meter3). Density (of an unidentified substance): Density, arguably the most important derived unit to a chemist, is built by using the basic formula, Density = Mass / Volume.
In the SI system, mass is measured in kilograms. The standard SI units for mass and length were chosen by the Scientific Powers That Be because many objects that you encounter in everyday life weigh between 1 and 100 kg and have dimensions on the order of 1 meter. Chemists, however, are most often concerned with very small masses and dimensions; in such cases, grams and centimeters are much more convenient. Therefore, the standard unit of density in chemistry is grams per cubic centimeter (g/cm3), rather than kilograms per cubic meter. The cubic centimeter is exactly equal to 1 milliliter, so densities are also often expressed in grams per milliliter (g/mL). Pressure (an example is of gaseous reactants): Pressure units are derived using the formula, Pressure = Force / Area. The SI units for force and area are Newtons (N) and square meters (m2), so the SI unit of pressure, the Pascal (Pa), can be expressed as N m-2.
Q. A physicist measures the density of a substance to be 20 kg/m3. His chemist colleague, appalled with the excessively large units, decides to change the units of the measurement to the more familiar g/cm3. What is the new expression of the density? A. 0.002 g/cm3. A kilogram contains 1,000 grams, so 20 kilograms equals 20,000 grams. Well, 100 cm = 1m, therefore (100 cm)3 = (1m)3. In other words, there are 1003 (or 106) cubic centimeters in 1 cubic meter. Doing the division gives you 0.002 g/cm3.
Converting between Units: The Conversion Factor So what happens when chemist Reginald Q. Geekmajor neglects his SI units and measures the boiling point of his sample to be 101 degrees Fahrenheit, or the volume of his beaker to be 2 cups? Although Dr. Geekmajor should surely have known better, he can still save himself the embarrassment of reporting such dirty, unscientific numbers to his colleagues: He can use Conversion factors.
A conversion factor simply uses your knowledge of the relationships between units to convert from one unit to another. For example, if you know that there are 2.54 centimeters in every inch (or 2.2 pounds in every kilogram, or 101.3 kilopascals in every atmosphere), then converting between those units becomes simple algebra. Peruse Table 2-3 for some useful conversion factors. And remember: If you know the relationship between any two units, you can build your own conversion factor to move between those units.
760mm Hg 1atm » , ,, * —;–or—-— Atmosphere 760 mm Hg* 1atm 760mm Hg One of the more peculiar units you’ll encounter in your study of chemistry is mm Hg, or millimeters of mercury, a unit of pressure. Unlike SI units, mm Hg don’t fit neatly into the base-10 metric system, but reflect the way in which certain devices like blood pressure cuffs and barometers use mercury to measure pressure.
A: As With many things in life, chemistry isn’t always as easy as it seems. Chemistry teachers are sneaky: They often give you quantities in non-SI units and expect you to use one or more conversion factors to change them to SI units — all this before you even attempt the "hard part" of the problem! We are at least marginally less sneaky than your typical chemistry teacher, but we hope to prepare you for such deception. So, expect to use this section throughout the rest of this book! The following example shows how to use a basic conversion factor to "fix" non-SI units. Dr. Geekmajor absent-mindedly measures the mass of a sample to be 0.75 pounds and records his measurement in his lab notebook. His astute lab assistant wants to save the doctor some embarrassment, and knows that there are 2.2 pounds in every kilogram; the assistant quickly converts the doctor’s measurement to SI units. What does she get? 0.35 kg. 0.75 lbs x 1kg = o 35 kg 1 0.75 lbs Notice that something very convenient just happened. Because of the way this calculation was set up, you end up with pounds on both the top and bottom of the fraction. In algebra, whenever you find the same quantity in a numerator and in a denominator, you can cancel them out. Canceling out the pounds is a lovely bit of algebra, because you didn’t want them around anyway. The whole point of the conversion factor is to get rid of an undesirable unit, transforming it into a desirable one — without breaking any rules. You had two choices of conversion factors to convert between pounds and kilograms; one with pounds on the top, and another with pounds on the bottom. The one to choose was the one with pounds on the bottom, so the undesirable pounds units cancel. Had you chosen the other conversion factor you would’ve ended up with X 2^ = L65lb2/kg 1 1kg This calculation doesn’t simplify your life at all, so it’s clearly the wrong choice. If you end up with more complicated units after employing a conversion factor, then try the calculation again, this time flipping the conversion factor. If you’re a chemistry student, you’re probably pretty familiar with the basic rules of algebra (nod your head in emphatic agreement. . . good). So, you know that you can’t simply multiply one number by another and pretend that nothing happened — you altered the original quantity when you multiplied, didn’t you? What in blazes is going on here? With all these conversion factors being multiplied willy-nilly, why aren’t the International Algebra Police kicking down doors to chemistry labs all around the world? Though a few would like you to believe otherwise, chemists can’t perform magic. Recall another algebra rule: You can multiply any quantity by 1 and you’ll always get back the original quantity. Now, look closely at the conversion factors in the example: 2.2 pounds and 1 kilogram are exactly the same thing! Multiplying by 2.2 lbs / 1 kg or 1 kg / 2.2 lbs is really no different than multiplying by 1.
Q. A chemistry student, daydreaming During lab, suddenly looks down to find that he’s measured the volume of his sample to be 1.5 cubic Inches. What does he get when he converts this quantity to cubic centimeters? A. 24.6 cm3. 1.5 in3 ., ( 2.54 cm 1 I 1in 24.6 cm3 You’ve already converted between inches and centimeters, but not between cubic inches and cubic centimeters. Rookie chemists often mistakenly assume that if there are 2.54 centimeters in every inch, then there are 2.54 cubic Centimeters in every cubic inch. No! Although this assumption seems logical at first glance, it leads to catastrophically wrong answers. Remember that cubic units are units of volume, and that the formula for volume is length x width x height. Imagine 1 cubic inch as a cube with 1-inch sides. The cube’s volume is 1 in x 1 in x 1 in = 1 in3. Now consider the dimensions of the cube in centimeters: 2.54 cm x 2.54 cm x 2.54 cm. Calculate the volume using these measurements and you get 2.54 cm x 2.54 cm x 2.54 cm = 16.39 cm3. This volume is much greater than 2.54 cm3! Square or cube everything in your conversion factor, not just the units, and everything works out just fine. 5. A sprinter running the 100-meter dash runs how many feet? Solve It 6. At the top of Mount Everest, the air pressure is approximately 0.33 atmospheres, or one third of the air pressure at sea level. A barometer placed at the peak would read how many millimeters of mercury? SolVe It 7. A "league" is an obsolete unit of distance used by archaic (or nostalgic) sailors. A league is equivalent to 5.6 kilometers. If the characters in Jules Verne’s novel 20,000 Leagues Under the Sea Actually travel to a depth of 20,000 leagues, how many kilometers under water are they? If the radius of the earth is 6,378 km, is this a reasonable depth? Why or why not? Solve It 8. The slab of butter slathered by Paul Bunyan onto his morning pancakes is 2 feet wide, 2 feet long, and 1 foot thick. How many cubic meters of butter does Paul consume each morning? Solve It Letting the Units Guide You In the earlier sections in this chapter, the problems have used just one conversion factor at a time. You may have noticed that Table 2-3 doesn’t list all possible conversions (between meters and inches, for example). Rather than bother with memorizing or looking up conversion factors between every type of unit, you can memorize just a handful and use them one after another, letting the units guide you each step of the way. Say you wanted to know the number of seconds in one year (clearly a very large number, so don’t forget about your scientific notation). Very few people have this conversion memorized — or will admit to it — but everyone knows that there are 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, and 365 days in a year. So, use what you know to get what you want! 1 yr x 365 days x 24 hr x 60 min x 60 sec = 315 x sec 1 1 yr 1 day 1 hr 1 min
You can use as many conversion factors as you need as long as you keep track of your units in each step. The easiest way to do this is to cancel as you go, and use the remaining unit as a guide for the next conversion factor. For example, examine the first two factors of the years-to-seconds conversion. The years on the top and bottom cancel, leaving you with days. Because days remain on top, the next conversion factor needs to have days on the bottom and hours on the top. Canceling days then leaves you with hours, so your next conversion
Q. Factor must have hours on the bottom and minutes on the top. Just repeat this process until you arrive at the units you want. Then do all the multiplying and dividing of the numbers and rest assured that the resulting calculation is the right one for the final units. A silly chemistry student measures the mass of a sample of mystery metal to be 0.5 pounds and measures the sample’s dimensions to be 1 cubic inch. When she realizes her error, she attempts to convert her measurements into proper SI units. What should her units be, and what’s the density of her sample (in those units)? A. The units should be g/mL or g/cm3, and the density is 1.4 X 102 g/mL. The proper SI units for density are g/mL (g/cm3), so she should use the following method: 0.5 lbs x / 1in 1in3 1cm3 1kg 1000g nn,. , , i x, " x „ „ „ x, ,°= 0.014g/mL 2.54 cm 1mL 2.2 lbs 1kg Notice that the conversion factors have been cleverly chosen so that all the non-SI units cancel, leaving only SI units behind. This answer is a little awkward and ought to be converted to scientific notation (see Chapter 1), such as 1.4 x 10-2. Or, you can be extra sneaky and use a metric conversion factor, which allows you to do away with the scientific notation. 0.014g 100 cg -r-2-x-2. = 1.4 cg/mL ML G How many meters are in 15 feet? Solve It 10. If Steve weighs 175 pounds, what’s his weight in grams? Solve It 11. How many liters are in 1 gallon of water? Solve It 12. If the dimensions of a cube of sample are 3 inches x 6 inches x 1 foot, what’s the volume of that cube in cubic centimeters? Give your answer in scientific notation or use a metric system prefix. Solve It 13. If there are 5.65 kilograms per every half liter of a particular substance, which of the following is that substance: liquid mercury (density 13.5 g/cm3), lead (11.3 g/cm3), or tin (7.3 g/cm3)? Solve It Answers to Questions on Using and Converting Units The following are the answers to the practice problems presented in this chapter. D 1 X 107 nm. Both 102 centimeters and 109 nanometers equal 1 meter. Set the two equal to one another (102 cm = 109 nm) and solve for centimeters by dividing. This conversion tells you that 1 cm = 109/102 nm, or 1 x 107 nm. Efl 2.5 kg. Because there are 1,000 grams in 1 kilogram, simply divide 2,500 by 1,000 to get 2.5. 1Pa= 1kg First, you write out the equivalents of Pascals and Newtons as explained in the problem: 1N 1 kgm M and 1N= Now, substitute Newtons (expressed in fundamental units) into the equation for the Pascal to 1 kgm . iPa = S2 Get 1Pa =-2— 1P = 1 kgm Simplify this equation to = m2s2 and cancel out the meter, which appears in both the top P = 1kg And the bottom, leaving = ms2 WM 12 G/mL. Because a milliliter is equivalent to 1 cubic centimeter, the first thing to do is to convert all the length measurements to centimeters: 1 cm, 1.5 cm, and 0.5 cm. Then, multiply the converted lengths to get the volume: 1 cm x 1.5 cm x 0.5 cm = 0.75 cm3, or 0.75 mL. The mass should be expressed in grams rather than Dg; there are 10 grams in 1 decogram, so 0.9 Dg = 9g. Using the formula D = m / V, you calculate a density of 9 grams per 0.75 milliliter, or 12 g/mL. KM 330m. Set up the conversion factor as shown. 100m x 3.3 ft = 330m 1 1m If 251 mm Hg. 0.33 atm x 760 mm Hg = 251mm Hg 1 1atm S Gg 1.12 X 105 km. 20,000 leagues x 5^ km = x 1 1 league The radius of the Earth is only 6,378 km. 20,000 leagues is 17.5 times that radius! So, the ship would’ve burrowed through the Earth and been halfway to the orbit of Mars if it had truly sunk to such a depth. Jules Verne intended the title to imply a distance traveled across the sea, not a depth! Efl 0.11 m3. The volume of the butter in feet is 2 ft x 2 ft x 1 ft, or 4 ft3. 4 ft3 1 1m 3.3 ft 0.11m3 H 4.572m. 15 ft x 12 in x 2.54 cm 1m 1 1ft 1in jj 7.95 X 104g. 100 cm 4.572m 175 lbs 1 1kg 1000g 4 ——2- X ——2 = 7.95 x 104g .2 lbs 1kg |f| 15.7L. Gal x 1 1 gal 1 cup 3 gal x 16c x 327 mL x 1L 15.7L 1000 mL U 3.54 X 103 cm3. First convert all of the inch and foot measurements to centimeters: 3in x 2.54 cm = 7 62 cm by 6in x 2.54 cm = 15 24 cm by 1ft x 12in x 2.54 cm = 30 48, 1 1in 1 1in 1 1ft 1in The volume is therefore 7.62 cm x 15.24 cm x 30.48 cm, or 3.54 x 103 cm3. MI The substance is lead. 5.65 kg 0.5 L " 1000i 1L, x 1mL3 x 1000 g = 11.3 g/cm3 , which is exactly the density of lead. 0 mL 1cm3 1kg X
^ Deciding between addition and multiplication ^ Determining whether it’s a difference or ratio ^ Mixing it up with the different operations /f you’re like most people, you’re leery of word problems because the tasks involved in solving them aren’t always immediately apparent. The task or operation doesn’t cooperate by standing up and shouting, "Here I am!" Sure, everyone is more comfortable with a simple addition problem or maybe a combination multiply-then-subtract problem. Give me a + or x any day, and I’m in hog heaven. (Well, that may be stretching it a bit.) The challenge of doing a word problem comes when you have to decide which operations to use and in what order. In this chapter, you get some down-to-earth tips — so you can change a wordy description or situation into a standard arithmetic problem. Here I show you how to spot the clues that lie along the path toward a word problem’s solution. Does It All Add Up? The easiest arithmetic operation you encounter is addition. The first things that kids master in school are addition rules and addition tables. So it’s always comforting to find a word problem that involves the operation of addition. A big clue that you’re probably dealing with an addition problem is the word And. (See Chapter 1 for more on the mathematical meanings of everyday words.) Determining when the sum is needed The word And Indicates that you want a sum, but you can’t assume that everything on one side of an And And everything on the other side of an And Gets added up. For example, if you’re talking about Jim and Jon going to a
The Problem: Michael and Owen went shopping. Michael bought new jeans that cost $49.95 And A T-shirt that cost $12.50. What was the total cost of his purchase? The last sentence tells you that you need a total; the word And Between the prices of the clothing suggests that you add. So, finding the sum of $49.95 + $12.50, you get $62.45. (Note: This sum doesn’t include any sales tax; I cover finding and adding in sales tax in Chapter 6.) Adding up two or more
Vjf. VLA* It’s probably pretty obvious that you want to add all the numbers together. Multiplying 12 numbers at a time or in a row doesn’t sound like a good idea — or like a lot of fun. The And Tells you to add, and the last digits in the weights are just begging for you to take advantage of grouping numbers together before adding. Specifically, group the numbers whose decimal values add up to 10. (Actually, they add up to 1: 0.3 and 0.7 = 1.0 or 1.) Rewriting the 12 numbers in groups of nicely-combining numbers, you get: (9.3 + 8.7) and (10.4 + 8.6) and (11.2 + 9.8) and (9.5 + 10.5) and (8.9 + 10.1) and (11.0 + 9.1). Actually, the An old counting trick Old Mother Hubbard had more problems than a bare cupboard. She also had to figure out where to put all her children when they went to bed. She had 11 children to put in 10 beds, and she had promised them that each would have his or her own bed. She accomplished this great feat by putting two children in the first bed — with the promise that the second would have a bed to himself later, after all the others were settled in. Then she put the third child in the second bed, the fourth child in the third bed, the fifth in the fourth bed, the sixth in the fifth bed, the seventh in the sixth bed, the eighth in the seventh bed, the ninth in the eighth bed, and the tenth in the ninth bed. She then put the eleventh child, who was temporarily located in the first bed, in the bed that was left. What an amazing woman! How did she do it? ‘slues em Buisq se p|iqo 4JU8A8|8 84J Pue p|iqo puooas eqj peiunoo peq 345 Buipueis ])8| |||is p|iL|o 4JU8A8|8 341 411M ‘speq emu jsji) 84) Ui U8JPH40 U8) Peq A||EnioE 345 ‘peq )sj|) 3q) ui p|iqo EJ1X8 3q) SEM p|iqo q)U8A3|8 3q) )Eq) pewnssE 3qs M8A8|3 Ej)x8 )ou ‘pejij ej)x8 Usaq 3AEq lsniu pjsqqriH J3q)ow P|o Jood JaMsuy Last two don’t have decimals that add up to 1, they’re just left over, and it seemed nice to let them form a group, too. Adding up the numbers in the parentheses and then summing the sums, you get 18.0 + 19.0 + 21.0 + 20.0 + 19.0 + 20.1 = 37.0 + 41.0 + 39.1 = 117.1 pounds. Did you notice how I grouped the groups to make the addition easier? I wish Mrs. Dopke had shown me that trick earlier, to avoid all the pain and agony of those huge columns of numbers. What’s the Difference — When You Subtract? Subtraction is an operation that’s just a little harder than addition. Subtraction is still a basic operation, though, and shouldn’t pose too much of a problem for you. The main challenge in subtraction is getting the order of the numbers right. Do you subtract 90 minus 47 or 47 minus 90? It’s pretty obvious, when doing word problems about practical matters, such as weights and money, which number you subtract from which — it’s more in the algebra word problems, where you can end up with negative numbers, that you have to be especially careful about the order of subtraction. VLA* Deciphering the subtraction lingo
The Problem: Avery lives 47 miles from Grandma’s house, and Reid lives 63 miles from Grandma’s house. How many fewer miles from Grandma does Avery live than Reid? The problem suggests subtraction, because there’s a comparison of numbers. You subtract 47 miles from 63 miles to get: 63 – 47 = 16 miles, and the answer is that Avery lives 16 miles Fewer From Grandma than Reid. Actually, the question could have been: "How many More Miles away does Reid live from Grandma than Avery?" Notice that the word Fewer Changed to More And that the names got reversed, too. It’s sort of like negating things twice. The answer is the same, it’s just stated differently. When combining signed (positive and negative) numbers in algebra, two negatives or reverses can make a positive. For instance, the opposite of negative 4, written -(-4), is equal to positive 4. Subtracting for the answer Sometimes a clue or key word isn’t really apparent in a word problem; it may be there and you don’t recognize it, or it just may not be there at all. You have to determine which operation or operations are needed without much help. In the case of problems that finally end up using subtraction, you’ll find that the situation has to do with finding the difference between two values — how much bigger one is than the other. Using four 4s Can you create the numbers 1, 2, 3, 4, and 5 using four 4s and the operations of addition, subtraction, multiplication, and division, only? ■st, Jno) Buisn 0l qBnojqj 9 Buijijm Ajj ‘os|v
■>|sEJ 8L|1 0P oj SAEM J8L|)0 PUIJ ueo hoa Jl 88S ‘St, jho) IJUISH S L|6nOJL|J t SJ8q -lunu eqj ojijm oj sAem A|uo eqj j. uejE eseqi ■■, + Y, + (t> x f)] qjm punoj si g jeqiunu eqj puv ■■, = (t> – t>) t> + t> :>|0!JJ eqj sejeH lusi jl jnq ‘jseisee eqj eq pinoqs t> jeqiunu eqi ■■, + (> + Y + t>) = £ ueqj puv jz = (fr + fr) + (fr x t>) jnoqe Moq ’3 Jeqiunu eqj ojijm 01 ■ 1 = V, + t>t> asn ’1 Jeqiunu eqj ojijm oijwwsup The Problem: Ruby is taking a mathematics class where 4 tests, 10 quizzes, and 20 homework assignments determine her grade. She has one test left to take and needs 920 points to get an A in the course. She currently has 847 points, and the test is worth 100 points. How many points does she have to get on the test to get an A? Is it possible for her to get an A? Vjf. VLA* This problem is an example of a situation where you have to wade through all the information to figure out what’s needed to answer the question. The fact that there are 4 tests, 10 quizzes, and 20 homework assignments has nothing to do with the question. Also, there aren’t any clue words such as Difference Or Less. To answer the question, Ruby needs to subtract 847 from 920 to see how many points she needs. Then she can compare that answer to the number of points possible. Doing the subtraction, 920 – 847 = 73. It looks like, if Ruby gets 73 percent of the points on the test, she’ll get her A. How Many Times Do I Have to Tell You? Multiplication is really just repeated addition. If you’re adding the same number to itself over and over again, then you’re really doing multiplication, and the multiplication facts are very helpful. Some of the key or clue words for multiplication are: Times, multiplied, twice, And Thrice. (Okay, so nobody says "thrice" anymore, but if they did, it would be a clue that multiplication was involved.) Doing multiplication instead of repeated addition Addition is a comfortable operation, but it can get tedious after a while, if you have to repeat the same task over and over again. Word problems involving 5,VLA*
Dealing with clear-cut multiplications You already know that you aren’t interested in repeated additions when multiplication is so much easier to handle. The multiplication problems can be fairly simple or bordering on the challenging. No matter what the situation, you can handle it. The Problem: Sydney has a new job and wants to draw up a timetable for every day of the week so she can get everything done at home as well as at her job. She wants to know, "How many hours are in the month of January?" With 24 hours in a day, and 31 days in the month of January, Sidney can add 24 plus 24 plus 24 a total of 31 times. Of course, it makes much more sense to multiply 31 x 24 to get 744 hours. The Problem: Have you ever been amazed that your heart keeps beating and beating without your thinking about it or doing anything to help? Just how hard does your heart work? An average adult’s heart beats 72 times each minute; a child’s heart beats faster than that — more like 90 times each minute. Use the heart rate of an average adult — 72 times each minute — to answer the question: How many times does the heart of an average adult beat in one day? Vjj. VUWV You want to multiply to get this answer, and there are actually two different multiplication problems to deal with. First, you need to know how many minutes there are in one day; you determine that by multiplying 60 minutes times 24 hours. Then you can multiply the number of minutes times the number of heart beats. For the number of minutes in one day, 60 x 24 = 1,440 minutes. Multiplying that by 72, you get 1,440 x 72 = 103,680 beats of the heart. That’s just in one day!
60 min. # 24 hours # 72 beats Hour day min.
60 minf.24 hours 72 beats —-■ X—x ■- Hour day min. = 60 Xd2ayX 72 = 103,680 beats/day Notice how the minutes and hours cancel out, leaving beats and day. Doubling or tripling your pleasure
The special math words that deal with multiplying something a number of times are somewhat recognizable. Here are the more commonly used multipliers: Double: Two times Twice: Two times Triple: Three times Thrice: Three times, in Shakespeare’s day Quadruple: Four times Quintuple: Five times The Problem: Juan was told that he had to quadruple the amount of his sales in the next six months in order to win that trip to Tahiti. His sales are currently at $230,000, so what level of sales does he have to reach in the next six months to get that vacation? The word Quadruple Means to multiply by 4, so $230,000 X 4 = $920,000. Aw, Juan can do it! Taking charge of the number of times
Writing numbers using scientific notation Scientific notation was developed to allow scientists, mathematicians, and people like you and me to write really, really large numbers or teeny, tiny small numbers without using up a full page to express the numbers for the reader. ToER The format for a number written in scientific notation is a number Between 1 and 10 times a power of 10: N X 10p. For example, the number 230,000,000,000,000,000,000,000,000 is written as 2.3 X 1026 in scientific notation. The number 2.3 is between 0 and 10, and it multiplies a power of 10. See how much shorter the number in scientific notation is than the number written out the long way? Also, comparing two large numbers is easier when they’re written in scientific notation. You look at the power of 10, first, and then compare the multiplier. Rolling out the barrel A man wants to fill a barrel exactly half full of oil, The man is very clever, though (he’s read this but there are no markings on the barrel to tell book), and he is able to fill the barrel exactly where the halfway mark is, and there are no halfway. How did he do it? measuring instruments to do any measurements. ■|8iieq eqj jo 8iun|0A eqj pq ApoExe ueqj sem |io eqi jeneq eqj jo wojjoq eqj pejeAoo jsnl ‘euiij ewes eqj je ‘pus |enEq eqj jo di| eqj peqosej jsnl |enEq eqj ui |io eqj |ijun peqojEM eH ‘(AABeq Ajjejd jo6 jl – d|eq ewos qjiM] epis euo oj peddij |eJJEq eqj jde>| ueui eqj ‘|enEq eqj ojui peMO|j |io eqj sV :i9MsuV To change a number into scientific notation, you move the decimal point from the right end of the number until there’s just one digit to the left of the decimal point. With just one digit in front of the decimal point, you automatically have a number between 1 and 10. The power that you put on the 10 is the number of places that you had to move the decimal point. Moving the decimal point to the left gives you a positive exponent, and moving the decimal point to the right requires a negative exponent.
JfttNfi/ Many scientific calculators write scientific notation using the letter E Instead of a power of 10. On your calculator screen, you’ll see 2.3E26 instead of 2.3 X 1026 or 1.23E-39 instead of 1.23 X 10-39. This isn’t a big problem — you just want to be aware of this cryptic notation so that you know what it means and can write it correctly. Changing units to make numbers smaller Keep in mind how units are related when working with a large number — that happens to be of one type of unit. For example, because there are 5,280 feet in a mile, you can say 45 miles instead of 23,760 feet. With 16 ounces in a pound and 2,000 pounds in a ton, you can say 9 tons instead of 288,000 ounces. The Problem: Don has 150 drafting rulers, each of which is 18 inches long. If he lays the rulers end to end, how long a line of rulers can he form? Multiply 150 X 18 to get 2,700 inches. That’s a perfectly good answer. But Don can also report that the line is 225 feet or 75 yards. How do you get the other units? There are 12 inches in a foot and 36 inches in a yard. So you can divide 2,700 either by 12 or by 36 to get the feet or yards, respectively. Another way to look at the problem is to change the 18 inches to feet or yards first, before you multiply by 150. Eighteen inches is 1.5 feet. Multiplying 1.5 X 150, you get 225 feet. Eighteen inches is also half a yard. Multiplying K X 150 you get 75 yards. The answers are the same, of course. You just have to choose how you want to deal with the numbers — and when to change them to smaller units. Dividing and Conquering Division is usually the last of the four basic operations that kids study in school. Why? Because many of the results are not whole numbers. When you add, subtract, or multiply whole numbers together, you always get a whole number as a result (or an integer — in the case of subtracting a larger number from a smaller number). Not so with division. Not every division problem comes out evenly, and dealing with a remainder can be a bit unsettling or even daunting. Using division instead of subtraction Just as multiplication is used instead of repeated addition, you can say that division is used instead of repeated subtraction. For example, if Keisha wants to hand out 4 pieces of candy to each of her friends and she has 38 pieces of candy, she can give 4 pieces to the first friend and 4 pieces to the second friend and so on until 9 friends have 4 pieces of candy each — and Keisha has 2 left over. Of course, you would do the division problem 38 4 = 9 with a remainder of 2. The remainder can also be written as a fraction (K which can be simplified to 34).
The world’s smallest computer This morning, I bought a word processor small divide. It has a delete capability that will correct enough to fit in my pocket. It can write in any all errors. No battery or power outlet is needed. language that I want and use any alphabet I And, believe it or not, it only cost me a quarter. need. It can also add, subtract, multiply, and How is this possible? ■|IOU8d B SI J0SS800jd pjOM AW
When a division problem doesn’t come out even — when the number you’re dividing into isn’t a multiple of the number you’re dividing by — you have several options: You can report the remainder as a number, which is the actual value of the remainder. I You can report the remainder as a fraction. I You can round the number up to the next higher number. You can just lop off the remainder and ignore it. The following problems give examples of when and how to use the different options. Reporting the amount remaining
The Problem: Pete works at a doughnut shop and is packing the morning’s supply of doughnuts into boxes containing one dozen doughnuts each. The owner of the doughnut shop tells Pete that he can keep any that are left after filling as many boxes as possible. Pete started with 4,000 doughnuts. How many doughnuts does he get to keep? ^VLA* Pete has to put one dozen doughnuts in each box. It takes 12 donuts to make a dozen. Divide 4,000 by 12 and determine the remainder — if there is one: 4,000 ■ 12 = 333 with 4 left over. The owner of the shop has 333 boxes of donuts to sell, and Pete gets the last 4. This is a case where it didn’t make sense to change the remainder to a fraction or decimal. The last four doughnuts weren’t going to be sold — but the number of doughnuts remaining was pretty important to Pete! Creating a fraction or decimal of the remainder When you’re dividing one number by another, you often have the situation where the division doesn’t come out even. This happens when the number Being divided isn’t a multiple of the number doing the dividing. The remainder may or may not have importance. Examples of cases where the remainder Does Have importance are when vendors sell fractions of pounds of candy or meat or some other product. In these cases, the vendors include the remainder in the computations — they don’t want to lose even a small fraction of the amount. The Problem: Chuck, the candy-store owner, has a large box of jelly beans that weighs 480 pounds. He wants to divide this large box into 100 smaller boxes and sell each of the smaller packages for $2.50 per pound. How much does each of the smaller boxes weigh? ^VLA* The 480 pounds of candy divided into 100 smaller units doesn’t come out to a whole number of pounds of candy in smaller boxes. Chuck changes the remainder into a fraction and uses it in determining the cost of each small box. To answer the question, you don’t need to know the cost per pound — that’s just extra information needed by Chuck. Just do the division: 480 100 = 4 with a remainder of 80. Change the 80 to a fraction or decimal, and the result of the Division is 4 + = 4 4 or 4.8 pounds per box. Rounding up — numbers, not cattle Rounding The answer of a division problem to a whole number results in a new answer that’s an approximation of the real, exact answer. Rounding answers is practical and makes sense in many situations.
The standard rule for rounding numbers is to round Down When the fractional or decimal part is less than half (when the remainder is less than half of the divisor), round Up When the fractional part is more than half, and round to the closer even number when the fractional part is exactly half. Some businesses can’t (or don’t choose to) use the standard rule of rounding. They round all fractions up to the next higher whole number. For example, with some phone companies, you pay for a whole number of minutes, and any fraction of a minute is considered to be the whole thing.
If Stefanie has 64 ounces of jelly and divides that among ten friends, then you divide 64 ■ 10 = 6.4 ounces per friend. You can’t force 6.4 ounces of jelly into a 6 ounce jar, so you round up to 7 and leave a little room at the top of each of those jars. The 8-ounce jars are overkill. Rounding up does have its practical applications. Pocketing the difference The story goes (and it may just be one of those urban legends) that an enterprising bank employee figured out how to program the bank’s computers so that any fractions of cents that occurred when the interest was computed on a savings account got deposited in his own personal account. So, if a person was supposed to earn $37.43255 in interest, the $0.00255 was Lopped off the interest amount and placed in the employee’s account. This fraction of a dollar doesn’t seem like much, but, if you add up the fractions of dollars from every account, the amount grows and grows. Again, according to the story, he got away with it for a few years. But, as happens to those who go astray, he finally got caught. Rounding down or truncating the remainder Most scientific calculators round up or down automatically when an answer has more decimal places than are available in the display. Back in the good old days of early, handheld calculators, you were more apt to see long decimal answers being Truncated — meaning that the extra decimal values were just dropped off with no consideration given for rounding one way or the other. The Problem: Ebenezer announced to his employees that he would be doing some profit-sharing with them. The nine employees would each get one-ninth of the $75 profit each month — and be paid in silver dollars. So the employees each got $8 per month, for a total of $96 for the year. How could Ebenezer have been just a bit more fair to his employees? .iPUUV First, dividing 75 by 9, you get 8 with a remainder of 3. In decimals, this is 8.3. F #S\ (lhat line over the 3 indicates that the number repeats forever.) It looks like AW ] Ebenezer just lopped off the remainder and kept it himself — because he paid
In silver dollars and no other coins, he couldn t give his employees the fraction of the total. But if Ebenezer really wanted to be fair, he would multiply the 75 times the 12 months of the year to get 900. Dividing that by 9, each employee would get 100 dollars for the whole year instead of 96 dollars (8 X 12). He could give the extra as a holiday bonus. Mixing Up the Operations Most of the interesting problems — the ones that you find in real life — are made up of words describing a situation or question that is solved using one or more operations. The Best Problems are those that involve more than one operation. Okay, that’s from my perspective, but I’m going to try to win you over to my side with some scintillating examples. When more than one operation is involved, you need to be cautious about which operations to use and in what order to use them. Making decisions about the operations and the order requires a mixture of good old common sense and a willingness to tackle a problem, slightly seasoned with some good old math rules. Doing the operations in the correct order One challenge of dealing with two or more operations is determining the order in which you do these operations. Sometimes the path or order is clear cut. Sometimes the problem is loaded with pitfalls or opportunities to flub. You need to keep in mind the Order of operations From algebra when doing multiple operations in story problems. The Order of operations Says that, when you have more than one operation to do, first perform any powers or roots, then do any multiplication or division, and end by doing any addition or subtraction. The order of operations can be interrupted, though, by parentheses or brackets or other grouping symbols. Here are some examples of using the order of operations: 4 + 12 3 – 1 = 7 (62 – 10) 13 + 5 = 7 First you do the division, 12 ■ 3 = 4. Then you have the problem 4 + 4 – 1. Do these operations in order, moving from left to right: 8 – 1 = 7. First you have to simplify the terms in the parentheses. Square the 6 to get 36. Subtract 10 from 36 to get 26. The problem now reads: 26 ■ 13 + 5. Do the division and then add. 2 + 5 = 7. This is just a quick review of the order of operations. If you need more of an explanation, refer to my book Algebra For Dummies (Wiley), where I give a more thorough explanation. Now I’ll show you how the order of operations works in a practical situation. Consider a problem where you have to figure out the cost of lunch, and see how it works. The Problem: You have a buy-one-get-one-free coupon from a local restaurant. The coupon is good for the cheaper of the two lunches. The amount of that meal is deducted after the tax is added. You and a friend go to lunch. You have the $6.95 chef salad, and she has the $6.50 fruit plate (you’re both eating so healthy). The tax comes to $1.08, and you want to leave a tip of $2.80. You’re going to split the total cost of the lunch equally, so what will lunch cost you?
Whew! It’s going to take several operations to solve this problem. You have to Add Up all the costs, SubtracT the amount for the coupon, and then Divide The resulting amount by 2. You’ll use parentheses and brackets so that the correct amount is divided by 2. Here’s what the problem looks like using the opera - Tions and grouping symbols:
Even though parentheses are around all the addition, you can drop the parentheses and include the subtraction of 6.50 with that group. You want to do this, because you get a 0 by cleverly adding and subtracting the same number (+ 6.50 – 6.50). The new version of the problem is: [6.95 + 1.08 + 2.80] 2 = [10.83] 2 = 5.415 You can’t split a penny in half, so your friend very generously offers to pay the extra penny because you brought the coupon. You pay $5.41 and she pays $5.42. (It doesn’t make sense to round up or down with that K of a cent, because you’d be paying too little or too much if you both paid the same amount after rounding.) Determining which of the many operations to use Using the operations of addition, subtraction, multiplication, and division isn’t all that taxing. It’s deciding When To use Which Operation — and how often. You pick up on the word clues and try out something that makes sense. Check to see if your answer is what you expected or is something very surprising. Surprising isn’t bad, but it should make you go back to see if you’ve made ^tASfi? some sort of calculation error or if it’s just your estimate that was wrong. Operating on the digits
■1« si Jeqiunu Aiu jeqj mou>| noA mon L S| 31 Aq pepiAip t>8 Jaqwnu eqj – sjiBip sji jo urns eqj Aq e|qisiAip A, ueAe si sjeqiunu jiBip-OAAj eqj jo euo A|u0 ■AieAijoedsej – n puE – u ’0l ’8 ’9 :sjeqiunu oai, eqj ui sjiBip eqj, o siuns eqj jeB oj uoijippe esn n»oN fsjequmu esoqj jno ejnBij oj uoijoejjqns esn noA) – S6 P"= ‘W ‘zl ’39 ‘IS :eJE sjeqiunu esoqi -t Aq jsjij eqj ueqj jepius si jiBip puooes eqj ejeqn» sjeqiunu ji6ip-on»j eqj p jo >|uiqj ‘siqj op Oi:jaMsuV The following examples incorporate several operations in each, for your perusal. The first problem uses multiplication and division. The Problem: Ted can use a disposable razor six times before it gets too dull to do a good job. He shaves twice on Saturday — to look his very best on Saturday night. He buys the razors in packs of 10. Each pack of razors costs $4.95. How much does Ted spend on razors every year? A? LA*V First, you want to wade through all the information to get to the pertinent /ISSN facts. And, for a problem like this, make an estimate in your head. If he buys a ^_VL ) Pack of razors every month (which is more than he needs) and spends $5 per ^t^^^ Pack (rounding up to make the math easier), then that’s 12 X $5 = $60. You know that the estimate is high, but it gives you a ballpark figure to aim at. Now you need to determine how many Shaves Ted has per year, divide that number by 6 to get the number of razors, divide that number by 10 to get the number of packs, and multiply the result by $4.95. Yes, you could divide the number of shaves by 60 instead of doing two separate divisions, but I want to show every step and not make big leaps. Start with the number of shaves. You take the number of weeks in a year, 52, and multiply that by 8. (Why 8? Because Ted shaves twice on Saturday, and you add 1 to the number of days in a week.) So 52 X 8 = 416 shaves per year. Divide the number of shaves by 6 (the number of shaves per razor), and you get 416 ■ 6 = 69 and a remainder of 2, which is the number of razors needed. How do you handle the remainder in this case? (If you need help, see "Making use of pesky remainders," earlier in this chapter.) Because this is a number of razors needed, and you can’t break apart a razor, you choose to round up to 70 razors. Now divide 70 by 10, the number of razors in a pack. Ted needs 70 ■ 10 = 7 packs. At $4.95 per pack, that’s $4.95 X 7 = $34.65 per year for razors. That’s much cheaper than going to the barber every day — and it’s much lower than the estimate of $60. This doesn’t necessarily fit in the Surprising Category. The estimate in these types of problems are especially useful if you make an error in the decimal point. For instance, if you came out with an answer of $346.50, you’d know that something was very wrong with your answer.
The next problem uses addition — with a tad of multiplication — and subtraction. You need to be careful about the grouping — keeping the additions separate before subtracting. The Problem: Twins Jake and Jill are always competing with each another. They’re in the same math class and want to compare their quiz scores to see who has done the best on quizzes. Jake’s scores are 19, 18, 17, 16, 17, 18, 19, 20, 20, and 20; Jill’s scores are 20, 20, 20, 20, 14, 13, 14, 15, 16, 20. Who has the higher total, and by how much? VLAiV You first want to total each person’s quizzes and take advantage of repeated scores to cut down on the number of computations. By grouping and multiplying, Jake’s scores look like the following: ([2 x 19] + [2 x 18] + [2 x 17] + 16 + [3 x 20]) = (38 + 36 + 34 + 16 + 60) = 184 Now, grouping and multiplying Jill’s scores you get:
([5 x 20] + [2 x 14] + 13 + 15 + 16) = (100 + 28 + 13 + 15 + 16) = 172 It looks like Jake has the better total. You subtract 184 – 172 = 12, which means that Jake’s score is better than Jill’s by 12 points. This last problem uses addition, multiplication and division. Again, you have to group the correct values and then do the division at the end.
The Problem: Five friends — Adam, Ben, Charlie, Duncan, and Eduardo — have decided to work together on a project and then split up the proceeds evenly. A neighbor has offered to pay them $6.50 per hour to paint the fence around his pasture. (If you’ve never painted a fence — with all the surfaces and nooks and crannies — you’ve never lived. Oh, yeah.) The boys were to come when they could and report the number of hours that they spent painting. They’d all be paid at the end. Adam worked for 8K hours, Ben worked for 6 hours, Charlie worked for 9>4 hours, Duncan worked for 7 hours, and Eduardo worked for 9 hours. How much did each boy make? First, you need to total all the hours by adding them up. Then multiply the number of hours by $6.50. Last, divide the total amount paid by 5 to get each person’s share. Adding, 8K+ 6 + 7 + 9 = 39K hours. Then multiply 39K x $6.50 and you get $258.375. Now, dividing the total by 5, you get $258.375 ■ 5 = $51.675, which rounds up to $51.68. That seemed like good money to the boys, but I think that the neighbor made out pretty well, too. Chapter 6
^ Taking the insurance factor into account ^ Finding a possible compromise — coverage at the lowest cost Ere you are in the pink of health and lucky enough not to be taking any prescription drugs — or perhaps only an antibiotic now and again. Maybe you’ve never needed anything beyond the occasional aspirin. And — even though you have no other drug coverage — you’re asking yourself why on earth you should pay out good money every month to a Medicare drug plan when you’d be getting zip in return. No doubt about it, this is a major dilemma for healthy people. Of course, you have every right not to join Part D. Maybe even the threat of a late penalty — the higher rate you’d pay for drug coverage if you delay enrolling in Part D, as explained in Chapter 8 — doesn’t cut much ice with you. But I’m guessing that you also know, in your heart of hearts, that this isn’t really the point. The real question is whether you can afford to be without drug coverage when you also lack a crystal ball — or any way of peering into the future to see whether you’ll fall prey to some unforeseen disease or injury that requires expensive meds to treat. That’s the insurance factor, and although Part D provides less-than-comprehensive coverage (unless you qualify for Extra Help, described in Chapter 5), it does offer protection against cata-strophically high drug expenses that you may face if you’re diagnosed with a serious illness. It covers you If You need it and When You need it. In this chapter, I assume you’re healthy, have no other drug coverage, and aren’t inclined to enroll in Part D, or at least not for now. I highlight factors to think about when weighing the current state of your health against future risks. And I also suggest a compromise — how to find a Part D plan that will cost you the least money while still giving you an umbrella for insurance. Balancing Today’s Good Health against Tomorrow’s Risks On the whole, human beings are optimistic by nature, and it’s often said that Americans are the most optimistic on earth. We don’t expect bad things to happen to our bodies, and we’re confident that, if ill health does hit us sometime in the far-off future, medical science will be able to fix it. Well, increasingly, medical research is finding answers — and very often they come in the form of prescription drugs. In this section, I play devil’s advocate. If you’re on the fence about joining Part D, then you’re figuring on being without any insurance against possible drug costs for months and perhaps years to come. So here I raise several "what ifs" about your chances of becoming ill and the potential costs of being without coverage. The odds of getting sick Around the time when the Part D drug benefit went into effect in 2006, I talked with a man in his 70s who passionately — and sincerely — argued against signing up. He described how healthily he’d lived, never smoking or drinking alcohol, always eating natural foods, and getting plenty of exercise and sound sleep. He believed that this long-held regimen would see him through, and he’d never need prescription drugs. I don’t know what happened to this man, though I hope his excellent health continues. But a few months later, a good friend of mine who’d pursued an equally healthy lifestyle — to the extent of never letting red meat pass his lips — suddenly developed Parkinson’s disease in his mid-60s. It seemed tragically unfair. But that’s the point of this parable. Life can be unfair. Lightning strikes out of the blue. Stuff happens. Living healthily is always the best way of preventing or postponing the common maladies that come with the aging process. And yes, it’s also the best method of averting the need for many kinds of prescription drugs! Yet even the healthiest lifestyle can’t completely protect you against all medical setbacks. These certainly include Genetic diseases that medications can help alleviate Physical injuries that are treated with painkillers, muscle relaxants, and other medications to help restore body parts to working order Surgeries that require medications for postoperative care and complications But let’s face it; most of us don’t lead rigorously healthy lives. We smoke; we drink; we sit on our fannies all day; we choose a salad and pour bacon bits and fatty glop all over it. Along the way, our bodies are silently clocking the damage and doing their best to repair it — until one day, in our later years, The bill comes due. Warning signs appear. And then the doctor confirms we have some condition we need to do something about. Sometimes that just means changing our bad habits. Often it means taking prescription drugs to minimize the problem’s effects or hold at bay a more serious medical event, such as a heart attack or stroke, that may otherwise occur. I don’t want to belabor this point unduly, but when you’re figuring the odds against getting sick over the age of 65, it’s useful to know what the odds are. Here are just a few facts, culled from statistics collected by the Alliance for Aging Research on its Web site, Www. silverbook. org: At least 80 percent of Americans age 65 and over have at least one chronic condition (such as high blood pressure, heart disease, diabetes, arthritis, or vision disorders), and more than half have at least two. More than 1 in 5 Americans age 60 or older has diabetes. Among Americans age 65 to 74, 60 percent of men and 73 percent of women have high blood pressure.
About 77 percent of all cancers are diagnosed in people age 55 and over, most often around age 70. More than two-thirds of prostate cancer cases occur in men age 65 and older. The greatest risk for ovarian cancer is to women in their late 70s.
The cost of going without drug coverage One of the costs of waiting to enroll in Part D until the time you think you need it is the late penalty, which increases the amount you’d pay for Part D coverage for every month that you delay. You may think that delaying offsets the penalty if you save money by not having to pay premiums in the meantime. I delve into this very point, along with everything else you need to know about the late penalty, in Chapter 8. But you have to remember that — except for certain circumstances, also explained in Chapter 8 — after you’ve missed your deadline for enrolling in a Part D plan, you can sign up only once a year during the open enrollment period in November and December. Missing that window doesn’t just mean another 12 months of added penalties. More importantly, it means another 12 months Without coverage. If you fall sick and need prescription drugs during that time, you’ll pay the full cost out of your own pocket. That’s obvious — duh! But do you have any idea how much you’d pay compared to the cost of being in a Part D plan? Of course not — it would depend on the drugs you had to take. (Where’s that crystal ball when you need one?) But I can give you examples that illustrate the cost differences, and they may surprise you. Retail drug costs versus out-of-pocket Part D costs In Table 7-1, I show four brand-name drugs. U I chose Fosamax because it’s commonly used to prevent or treat osteoporosis, a condition that makes bones more fragile and likely to break. It affects 44 million Americans (1 in 2 women, 1 in 8 men), most often after the age of 60. U I deliberately chose the other three drugs as examples of those used to treat "lightning-strike" medical problems — the serious kind that can creep up on you without warning, until you get the diagnosis. Gleevec, for example, is used mainly to treat chronic myelogenous leukemia (CML), a cancer that attacks bone marrow. Most of the 4,500 Americans who are diagnosed with CML each year are middle-aged or older, according to the National Cancer Institute. Gleevec is the most expensive drug on this list, but many anticancer meds cost a lot more, especially the new biologic ones that target cancers more specifically than older drugs. Table 7-1 Out-of-Pocket Costs of Four Drugs Bought Retail or through a Part D Plan in 2008
Source: Part D costs from Www. medicare. gov. Chain pharmacy prices from Www. cvs. com, Www. costco. com, And Www. drugstore. com, June 2008. I looked up the mail-order cost of each of these brand-name drugs at three online chain pharmacies, where you may well be buying your meds if you have no drug insurance. The results, in column three of Table 7-1, reflect the lowest cost for a full year’s supply. Then I ran the numbers to see what you’d pay for the same drugs (whether purchased from a retail pharmacy or by mail order) under the Part D plan that showed up as the least expensive in my own state of Maryland. (Costs vary among states because the same plan may charge a different premium in different states. Also, the plan that proved the least expensive turned out to be different for each drug.) And, believe me, I played fair! The results, shown in column four, reflect not only what you’d pay for just one of these drugs over a whole year (including co-pays, deductible, and full price in the coverage gap, which I cover in Chapter 15), but also the plan premiums. (And remember, the premium stays the same, no matter how many drugs you buy.) In other words, Table 7-1 shows the difference between the price of each drug when purchased retail and what you’d pay for the whole insurance caboodle under Part D. The results speak for themselves. In each case, the Part D plan works out much cheaper — and for the most expensive medicine, you’d save more than half the cost. I also took the comparison one step further, in a way not shown in Table 7-1. I wondered whether the price of these drugs would be lower at Canadian online pharmacies than the American ones shown in the table. So I checked out a reputable Canadian pharmacy that advertises itself as having the lowest mail-order prices (and often does). Here are its prices for four batches of 90-day supplies over a year (including $40 a year for shipping charges): U Fosamax: $476 U Prograf: $1,480 U Avastin: Not available U Gleevec: $13,027 As you can see, only Fosamax is less expensive at the Canadian mail-order price ($476) than under the Part D plan ($556). Yet this plan’s overall cost would drop to $459 if the drug were bought in similar 90-day supplies by mail order instead of at a retail pharmacy. For the two other available drugs, Part D worked out far cheaper than Canada, even without mail order. And the price of Gleevec was nearly three times as much as the total out-of-pocket costs under the Part D plan. (If you’re curious about buying prescription drugs abroad, flip to Appendix C for details on how to do it safely.) The high cost of stopping maintenance drugs without coverage
Ignoring Part D brings another potential cost. Without coverage, people are less likely to take the drugs they may need to actually keep themselves healthy. These so-called Maintenance drugs Are taken regularly to keep certain conditions in check — for instance, high blood pressure, cholesterol, diabetes, osteoporosis, arthritis, and so on. Without those treatments, the risk of serious illness or injury rises hugely. To take just one example: Untreated osteoporosis, which weakens bone tissue and greatly increases the risk of falling, is the most common cause of hip fractures. Such an injury can cause permanent disability or require hip replacement surgery that is extremely expensive. That too is a possible future cost — both physically and financially — of being without drug insurance. And ultimately a much higher one. Compromising on Coverage at the Lowest Cost If you read the earlier sections in this chapter, you know that turning your back on Part D is at best a gamble on your future good health. But, naturally enough, you still may be disgruntled when you think about forking over money each month — possibly out of a Social Security check that buys less and less each passing year — and not getting anything back. Here’s a possible compromise you can consider: Purchase coverage at the least possible cost by choosing a Part D plan that has the lowest premium in your area. How low is low? That depends on where you live and the kind of Part D coverage you choose:
U Part D stand-alone plans (PDPs) — the kind that provide coverage Only For drugs and are usually purchased as an add-on to traditional Medicare medical coverage — vary a lot in the monthly premiums they charge. The same plan can charge different premiums in different states, but each plan must offer the same premium to all enrollees within a state. In 2008, every state has at least one plan with a premium of under $20 a month, and 28 states have at least one under $15 a month. Only Arizona has a plan costing less than $10. (Puerto Rico has a PDP with a premium of $2.60, and the U. S. Virgin Islands has one costing $4.30.) U Medicare Advantage plans that include drug coverage (MAPDs) — the Kind that combine medical and drug coverage in a single package — also vary a lot in their premiums. Many MAPDs charge zero premiums for both medical and drug benefits. In 2008, all states, the District of Columbia, and Puerto Rico have at least one (and often many) MAPDs with zero premiums, though these plans aren’t offered in all counties or zip codes. (None of the other U. S. territories have any Medicare Advantage plans.) You can easily find out the lowest premiums of plans in your area for 2009 and subsequent years. This information is contained in the Medicare & You Handbook that is sent to everybody on Medicare in October each year. You’ll find the list of PDP and MAPD plans that will be available to you in the following year toward the end of the handbook. You can also find these details by going to the Medicare Web site at Www. medicare. gov and clicking "Learn More About Plans in Your Area." See Chapters 9 and 10 for more details about deciding on an MAPD or a PDP. To give you a general idea, Table 7-2 shows the lowest premiums for PDPs in several states and how many MAPDs offered zero premiums in 2008. (Note: The asterisks in the MAPD columns indicate that the plans weren’t available in all counties.) Table 7-2 Lowest Monthly Part D Premiums in Selected States in 2008
(continued) Table 7-2 (continued)
Source: Part D plan details by state, 2008, at Www. medicare. gov. Among PDPs in the states, the lowest premiums range from $9.40 a month in Arizona to $18.00 in Alaska and Tennessee in 2008. These minimums are likely to rise in future years. Gone are the days of the $1.87 premium, which one plan offered in seven states in 2006, the first year of the Part D program, as a come-on to people to enroll. MAPDs are likely to continue offering zero premiums as long as they keep getting the extra federal subsidies that allow them to do so. Paying no premiums (except for the standard Part B premium) is attractive to people who can’t (or don’t want to) pay premiums for drug coverage, including those who don’t use prescription drugs right now. If you consider joining an MAPD plan for this reason, remember that the way you receive medical benefits is different from traditional Medicare, usually with restrictions on the doctors and hospitals you can go to and sometimes higher costs for certain services, as explained in Chapter 9. One more thing: If paying the kind of premiums listed among PDPs in Table 7-2 would be a hardship to you, you should look into whether you qualify for the Part D Extra Help program. If you qualify for full Extra Help benefits, you wouldn’t pay a premium at all, and qualifying for even partial benefits would reduce the premium. I explain the Extra Help program in detail in Chapter 5. You can also check out whether you’re eligible for help under a State Pharmacy Assistance Program (SPAP) if your state has one. These too provide prescription drug coverage at a low cost. Chapter 8 |
In This Chapter
Hockey game, you don’t add Jim and Jon together. You can add their ages or their weights or the number of hamburgers that they eat, but you don’t add people or colors. You just add numbers.
You probably remember your second-grade teacher (mine was Mrs. Dopke — haven’t thought about her in years) and those splendid columns of numbers that you added for days and days and days. For example, you got to find the sum of 299 And 401 And 650 And 850 And 76 And 24. Notice the repeated use of the word And? That’s your signal to add and add and add.
As you do the addition problem, you want to take advantage of any grouping of numbers that makes the sum more convenient. Notice in the listing in the preceding paragraph that the 99 in 299 and the 1 in 401 add up to an even 100. The 50s in the 650 and 850 add up to an even 100. Nice combinations like these don’t happen all the time, but you can take advantage of the nice sums if you catch them in time.
The Problem: Liam and Cassidy spent the day picking strawberries — a hot and backbreaking job, but someone has to do it. Every time they filled a flat with fresh fruit, they took it back to the weighing shed and picked up a new, empty box to fill. In all, they filled 12 flats of strawberries weighing: 9.3 pounds, 10.4 pounds, 8.7 pounds, 11.2 pounds, 9.5 pounds, 10.5 pounds, 11.0 pounds, 8.9 pounds, 9.8 pounds, 8.6 pounds, 10.1 pounds, and 9.1 pounds. What is the total weight of all the strawberries that they picked that day?
Here are some words that tell you to use the operation of subtraction: Less, lower, fewer, difference, minus, And, of course, Subtract. Someone can weigh 10 pounds Less Than another person (fie on them). Your colleague can make a Lower Salary than you (but you deserve it). To make comparisons of things such as weights or money or temperatures, you subtract.
Multiplication may be spelled out pretty clearly, or they may be masked with some math jargon such as Thrice Or Quadruple.
The problem involving minutes and hours and heart beats involved several ratios of units. There are minutes per hour, hours per day, and beats per minute. And your answer comes out to be beats per day. How does the answer work out to be in the correct number of units? Look at the preceding problem, written as fractions multiplied together.
Multiplying numbers can result in quite large results. This isn’t a bad thing — especially if you’re talking about your bank account. It’s just that large numbers can get unwieldy — to write and to deal with in a calculator. Here are two options for handling large numbers: Write the numbers in scientific notation, or change the units.
For really small numbers, the power on the 10 is a negative number. So, the number 0.00000000000000000000000000000000000000123 is written as 1.23 X 10-39 in scientific notation.
The Problem: Clara ordered a barrel of M&M’s online and was billed for V«tASQ>} 11,000 ounces of candy. How many pounds of M&M’s did Clara order?
One pound is equivalent to 16 ounces, so divide 11,000 by 16. This doesn’t come out even: 11,000 16 = 687 with a remainder of 8. Clara ordered 687!816 (or 687>2) pounds of M&M’s.
Making use of pesky remainders
You report the amount of a remainder if the amount left over is to be treated in a different way — that is, if it doesn’t figure into the answer to the problem. An example involves a doughnut shop and a worker putting a certain number of doughnuts into boxes.
The Problem: Stefanie made some homemade jelly and wants to divide it among ten of her best friends. She has a total of 64 ounces of jelly. She has some jars that hold 6 ounces of jelly, other jars that hold 7 ounces of jelly, and still other jars that hold 8 ounces of jelly. What size jars will she use if she wants to distribute all the jelly evenly, and not have any left over?
%>/Y^uZ^ / lin i i i ill i i ir
[(6.95 + 6.50 + 1.08 + 2.80) - 6.50] 2
I’m thinking of a number in which the second divide the number by the sum of the digits, the digit is smaller than the first digit by 4. When you quotient is 7. What is the number I’m thinking of?
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