In This Chapter
^ Adjusting units for ease in computations
^ Changing from English to metric units and back again
^ Squaring off with square and cubic units
Any mathematical problems involve units of length, weight, volume, or money. You incorporate the units into your computations and
Then report them in the answers so the solution makes sense and is useful. Sometimes you’re confronted with problems that have two or more different units — such as feet and inches or pounds and ounces — and you have to make a decision as to which unit to use.
In this chapter, I offer suggestions on how to choose the unit or units and then how to work with the unit or units you’ve chosen. This chapter also covers the tricky conversions of square feet to square inches or cubic yards to cubic feet. And, of course, no discussion of units is complete without introducing meters and kilograms, so you get conversions involving metric and English measures.
Choosing the Best Measure
When a problem involves two different measures, you choose one or the other measure to work with and convert the unchosen measure to the unit you want so that they’re all the same. You may even decide to change measures when they’re already all the same — just because you think that another measure may work better.
Using miles instead of inches
A mile is much longer than an inch. In fact, there are 12 X 5,280 = 63,360 inches in 1 mile. If you’re measuring how far it is from one side of the desk to another, you’ll use inches. If you’re measuring how far it is from your home to your workplace, you’ll usually measure in miles.
The Problem: Train tracks are made of metal, and metal expands when it gets hot and shrinks when the weather is cold. When the tracks are put in place, a gap should be left between the adjacent tracks to allow for expansion. A metal track that’s 1 mile long expands with the heat and increases in length by 12 inches. There was no gap between the tracks, however, so it buckled in the middle and formed a V shape. (See Figure 3-1 for a picture of what the track looks like.) How high up did the track rise?
Figure 3-1
The train track expanded with the heat.
1 mile
^VLA/J! Two different measures are given: 1 mile and 12 inches. You can work with inches, and change the 1 mile to 63,360 inches, or you can work with miles
And write the 12 inches (1 foot) as 5 mile. The choice here is between
Using really large numbers and using fractions or decimals.
I choose to go with the fractions — to work with pieces of a mile. To find out how high the rise in the track is, I use the Pythagorean theorem (Chapter 18 is completely devoted to that theorem of Pythagoras) and one right triangle going halfway down the track. The bottom segment of the triangle is >2 Mile
Long, and the Hypotenuse (longest side) is >2 Mile plus 6 inches or 10 650 mile
Long. To solve for the rise, which I’ll represent with X, I solve the equation for X.
/1\2 /-« 1 \
‘ T ) + x2 = D 2 + Tn^ 2 2 10,560
This looks pretty nasty, but a scientific calculator makes short work of the problem, and you get
0.25 + X2. 0.2500947 X2 = 0.0000947 X = 0.009732
The height or rise of 0.009732 doesn’t seem like much, but, remember, this is in miles. Multiply by 5,280 feet and you get over 51 feet. Whoa! That’s quite a rise!
Working with square feet instead of square yards
When you buy carpeting, you usually buy it in square yards — 3 feet by 3 feet. But you probably bought your last tile floor in terms of a number of square feet.
The Problem: Using your yardstick, you measure the length of a room to be 6 yards and its width to be 5 yards. You plan on putting in 1-foot-square tiles. How many tiles will you need?
Before determining the area of the room, first change the yards to feet using 1 yard = 3 feet. So 6 yards is 6 x 3 = 18 feet. Five yards is 5 x 3 = 15 feet. A room that’s 18 feet by 15 feet is 18 x 15 = 270 square feet.
But what if you preferred finding the area in square yards, first, and then changing the area to square feet? The area of the room is 6 x 5 yards or 30 square yards. A square yard is equal to 9 square feet (3 feet x 3 feet). So multiply 30 x 9 to get 270 square feet.
Converting from One Measure to Another
When a problem contains more than one measure, you change everything to the same measure before doing the computing on the problem or solving the equation. You can’t add 6 inches to 4 feet and get 10 — you have to change the inches to feet or feet to inches. Knowing when to multiply and when to divide sometimes gets confusing, so your best bet is to write down the equivalence or change of units and then work from the equation.
Changing linear measures
First, here’s a list of some common equivalences used when working with lengths. I cover the English and metric equivalences later, in "Mixing It Up with Measures."
1 foot = 12 inches 1 yard = 3 feet 1 mile = 5,280 feet
The measure equivalences are used to convert from one measure to another. You may need to do more than one computation if there isn’t a direct equivalence between units — such as changing inches to yards or yards to miles.
The Problem: Cheryl has 48 rolls of satin ribbon, each containing 15 yards of ribbon. She plans to wrap packages to send overseas as gifts, and each package requires 30 inches of ribbon. How many packages can she wrap?
First, determine how many yards of ribbon are in those 48 rolls. Then change the yards to feet using 1 yard = 3 feet and the feet to inches using 1 foot = 12 inches. After you have the total number of inches, you can divide by 30 to get the number of packages that can be wrapped.
Multiplying 48 rolls x 15 yards you get 720 yards. Start with the equivalence involving yards and feet. To change 720 yards to feet, you multiply each side of the equation 1 yard = 3 feet by 720.
1 yard x 720
3 feet
X720
720 yards = 2,160 feet
You have 2,160 feet of ribbon. Change this to inches by using the equivalence involving feet and inches, 1 foot = 12 inches.
1 foot = 12 inches X 2,160 x 2,160 2,160 feet = 25,920 inches
That’s 25,920 inches of ribbon. Divide 25,920 by 30 to get 864 packages that Cheryl can wrap.
Adjusting area and volume
Area is a two-dimensional measure. You’re counting up how many squares — all the same size — fit into some flat region. You use area measures for floors in buildings and spaces in parking lots, as well as when you want to find out how much room there is in a backyard.
Volume is a three-dimensional measure and tells you how many cubes of a particular size fit into an object. Volume measures tell you about the inside of a refrigerator or the size of a cardboard carton.
IBE# 1 square foot = 144 square inches (12 inches X 12 inches)
1 square yard = 9 square feet (3 feet X 3 feet) 1 square mile = 640 acres
1 cubic foot = 1,728 cubic inches (12 inches X 12 inches X 12 inches) 1 cubic yard = 27 cubic feet (3 feet X 3 feet X 3 feet)
The Problem: Jimmy is going to play a prank on his dad and fill the refrigerator with ice cubes that are 1 inch on a side. The refrigerator can hold 6 cubic feet. How many ice cubes will Jimmy need?
Determine the number of cubic inches in 6 cubic feet by using the equivalence 1 cubic foot = 1,728 cubic inches and multiplying each side of the equation by 6.
1 cubic foot = 1,728 cubic inches
X 6 X 6
6 cubic feet = 10,368 cubic inches
That’s over 10,000 ice cubes. Jimmy had better rethink his plan. He’ll never get the ice cubes all stacked inside the refrigerator before they start melting — and he gets frostbite.
The Problem: Timothy bought 3,200 acres of land and intends to plant seedling trees on it. If each seedling requires an area of 9 square yards to grow properly, how many seedlings can he plan on his new acreage?
YUUV First, change the acres to square miles using 1 square mile = 640 acres and then the square miles to square feet. Determine how many square feet are in 9 square yards using 1 square yard = 9 square feet and dividing the result into the number of square feet in the acreage.
Changing the acres to square miles:
1 square mile = 640 acres X Square miles = 3,200 acres
Make a proportion of the equivalences, lining up the numbers and the X Exactly as they appear — opposite one another. Solve for X.
1 640
- = -
X 3,200 3,200 = 640X 3,200 640
Ran = 77777 X
640 640 5 = X
So 3,200 acres is equivalent to 5 square miles. Change the square miles to square feet by multiplying each side of the equation 1 square mile = 5,280 X 5,280 square feet by 5. You get that 5 square miles = 5,280 X 5,280 X 5 = 139,392,000 square feet.
Now find the number of square feet that each tree needs. If each seedling needs 9 square yards, use the equivalence that 1 square yard = 9 square feet and multiply each side of the equation by 9 to get 9 square yards = 81 square feet.
Now divide 139,392,000 square feet by 81 square feet to get the number of trees that will fit on the acreage. 139,392,000 divided by 81 = 1,720,888.89 trees. That’s a lot of seedlings.
Keeping It All in English Units
Many countries, including the United States, use primarily the English units of measurement. Pressure to change to metric hasn’t been strong enough, even though advocates have proposed changing to metric for over 40 years. The awkwardness of the English units is that they have all sorts of different numbers in their equivalences — as compared to the metric system where all the numbers are multiples of 10.
Comparing measures with unlikely equivalences
As disjointed as the English measurement system seems to be, it has a long tradition and some interesting and charming equivalences. Here are some more uncommon but historic measures, plus, to finish it off, a rate.
1 rod = 16>2 Feet 1 fathom = 6 feet 1 furlong = 220 yards 1 hand = 4 inches 1 league = 3 miles 1 pica = 12 points
1 mile per hour = 88 feet per minute = 1 -JF Feet per second
The Problem: The Preakness, one of the horse races in the Triple Crown, has a distance of 9.5 furlongs. How many miles is that?
Change the furlongs to yards using 1 furlong = 220 yards and the yards to feet using 1 yard = 3 feet. Then change the feet to miles using 1 mile = 5,280 feet. The race is 9.5 furlongs, so multiply 9.5 x 220 to get 2,090 yards. Multiply the number of yards by 3 to get the number of feet: 2,090 x 3 = 6,270. Now
A – a ,u Co7n( ,u Coon. , ,u u t I 6,270 .,
Divide the 6,270 feet by 5,280 to get the number of miles: r OOA = 1.1875 miles.
3 3 5,280
The decimal 0.1875 is equal to Tf, so the race is 1Tf.
16 16
To change a terminating decimal to its fractional equivalent, create a fraction that has all the digits to the right of the decimal point in the numerator and, in the denominator, a power of 10 that has as many zeros as there are digits in the numerator. Then reduce the fraction. For the decimal 0.1875, you write 1875 in the numerator and a 1 followed by four zeroes in the denominator.
0.1875 =
1875 10,000
Now reduce the fraction. You can first divide both numerator and denominator by 25 and then divide the resulting numerator and denominator by 25.
1875 75 10,000 400
_3_ 16
The Problem: A popular method for determining how far away a bolt of lightning has struck is to count the number of seconds between the lightning flash and the sound of the thunder. If sound travels at about 1,100 feet per second, and if it’s 6 seconds between the flash of lightning and the roar of the thunder, then how far away was the lightning strike in miles? And what is the speed of the sound in miles per hour?
First, determine how many feet the sound traveled by multiplying 1,100 feet by 6 to get 6,600 feet. Determine the number of miles using 1 mile = 5,280 feet. You divide 6,600 feet by 5,280 and you get 1.25 miles. According to what I was told, the number of seconds you count is the number of miles away. I don’t think I was told right.
Now, to the speed of sound in miles per hour, use the equivalence that 1 mile
Per hour is equal to 1 feet per second. The speed of sound is about 1,100
Feet per second. Write a proportion using these figures, letting the speed of sound in miles per hour be represented by X. Then solve for X.
1 mile per hour
1feet per second
X Miles per hour 1,100 feet per second
1 – _J5_ _ 22 _
X
1,100 15 (1H)0100) 1,500 750 11
- — ———
X 750
The speed of sound comes out to be about 750 miles per hour. This is a bit over the speed you usually see quoted. The textbooks say that the speed of sound is actually 1,088 feet per second at 32° F. I rounded the number up to 1,100 feet for ease in computation and assumed that the temperature during a thunderstorm would be a bit warmer than 32°F.
Working for a bar of gold
Television’s favorite billionaire is interviewing yet another group of potential employees. To keep from being told, "You’re fired!" the finalists have the following problem posed to them, and the first person to come up with the solution will not hear the dreaded words. The problem: You’ve hired someone to work for you for the
Next seven days. You must pay him VJ Of a bar of gold per day, but he requires a daily payment of VJ Of that bar of gold — no credit. It’s expensive to cut through a bar of gold, so what are the fewest number of cuts necessary to meet his requirements?
■808ld 44 Jse| eqj Uliq 8AI6 ‘f AeQ U0 ^eid L/z Eqj LUIM 8Al6 pue 808ld 44 Eqj
,0Eq 8,e, ’9 Ae0 u0 ■eoeid 44 Eq, puE eoeid 44 Eq, eAEq ||,eq os ‘eoeid 44 Eq, >|3Eq je>|j0M eqj saiB ‘g Aeq u0 ‘seoeid 44 Pus 44 Eqj >|oeq e>|Ej pus JEq e (o 44 BuiuiEinej eqj je>|joM Eqj baiB > Ae0 u0 ‘SEq ApBej|b eq 4/, Eqj oj peppE eq oj 44 |eui6ijo eqj je>|jOM Eqj eAi6 ‘£ Aeq u0 ^osq I/l Jsjij eqj e>|Ej pus je>|jOM Eqj oj jsqj saiB ‘JEq eqj jo 44. Jjo jno ’1 Aeq u0 ‘je^jOM Eqj oj ji saiB pus >|jeiu 4/j eqj je JEq eqj jno ’1 Aeq uq ‘sino onnj A|uo e>|Eiu Peeu NoAMmsuy
Loving you a bushel and a peck
Volumes and weights take on some historically interesting values when working with equivalences. Do you buy your apples by the bushel, or is a peck all you need?
1 quart = 2 pints = 4 cups 1 gallon = 4 quarts = 32 gills 1 bushel = 4 pecks 1 pound = 16 ounces
1 ton = 2,000 pounds = 20 hundredweights
The Problem: According to a recent census, the typical American ate 13.8 pounds of turkey in a recent year. If this represents the total amount of turkey eaten at 12 different meals, then how many ounces of turkey were consumed at each meal?
VLAiV Change the number of pounds to ounces using 1 pound = 16 ounces by multiplying 16 X 13.8. Then divide the total number of ounces by 12. The product of 16 X 13.8 = 220.8 ounces. Divide 220.8 12 and you get 18.4 ounces per meal. That’s over a pound of turkey at a sitting!
The next problem involves a phenomenon of agriculture and requires that fruit growers adopt a good balance in their orchards. Consider an orchard where a certain number of apples are produced by each tree in an average year. If the number of trees in the orchard is increased, there’ll be more trees producing apples, but the crowding causes a reduction in the number of apples per tree. The balancing act for growers amounts to adding enough trees to increase production but not too many to decrease the production per tree by too much.
The Problem: An orchard contains 240 apple trees, which produce, on average, 2 bushels of apples per year. If the orchard manager increases the number of trees in the orchard by 60, she calculates that the amount of apples will be reduced by 1 peck per tree. If she goes ahead and plants those 60 additional trees, will the total crop be greater or smaller than the crop obtained from the original 240 trees?
^VLA* First, determine how many bushels of apples are produced by 240 apple trees by multiplying 240 x 2 = 480 bushels. Increasing the number of trees by 60 results in 300 apple trees. The production of 300 trees will be 2 bushels less 1 peck. Change pecks to bushels with 1 bushel = 4 pecks, giving you that 1 bushel is >4 Peck. Subtract >4 Bushel from 2 bushels, and the yield per tree
Will be 13 bushels. Multiply the number of trees by the new yield, and you 43
Get a total of 300 x 1-4 = 300 x 1.75 = 525 bushels. There’s an increase of 525 – 480 = 45 bushels of apples.
Mixing It Up with Measures
Most of the problems in this book use the English measures of length, volume, and weight. But metric measures are very important to know, because of the great incidence of foreign travel and trade with other countries that use metrics.
Matching metric with metric
The metric measurement system is extremely easy to use, because all the units and equivalents are powers of 10. A kilogram is 1,000 times as big as a gram, and a centimeter is 0.01 as big as a meter. The multiplication and division problems using metric measures are really a piece of cake. When you learn what the different prefixes stand for, you can navigate your way through the metric measurement system.
In the metric system, Kilo Means 1,000 times as much, Hecto Means 100 times as much, Deca Means 10 times as much, Deci Means K0 As much, Centi Means M™ As much, and Milli Means K000 As much.
The Problem: Stephen is the track manager at a race-car competition that’s 16 kilometers long. If he wants to put a spotter every 25 meters for the length of the race, then how many spotters will he need?
Change kilometers to meters using 1 kilometer = 1,000 meters and then divide the total number of meters by 25. The 16-kilometer race is 16 x 1,000 = 16,000 meters long. Divide 16,000 ■ 25 to get 640 spotters. That’s a lot of people!
The Problem: Stephanie works for a candy company and got permission to produce a piece of licorice that’s 12 meters long. She’s going to take the licorice to a party (just for the effect) and then divide it into individual pieces that are each 3 centimeters long. How many pieces of licorice will she have?
A centimeter is M™ Of a meter, so there are 100 centimeters in a meter. Multiply the 12 meters by 100, and you get 1,200 centimeters of licorice. Divide that total by 3, and Stephanie will have 400 pieces of candy.
Changing from metric to English
You’ve decided to go to Europe and you want to be sure that you order the right size beverage, know how far you’ll be traveling by car, and dress appropriately for the weather on any particular day. All these functions relate to changing from English units of measure to metric measure. Here are some of the more useful conversion equivalences you’ll need for your travels. For help with the temperatures, refer to Chapter 10 for conversions from Celsius to Fahrenheit and back again.
1 meter = 39.37 inches 1 kilometer = 0.621 mile 1 liter = 1.057 quarts 1 kilogram = 2.205 pounds
The Problem: You’re in Europe and about to take a day trip with a rented car and trusty map. You’re going to drive from your hotel to a famous cathedral. According to the map, the distance is 500 kilometers. How far is that in miles?
Use the equivalence 1 kilometer = 0.621 mile and multiply each side of the equation by 500. You get that 500 kilometers = 310.5 miles. That’s a pretty long trip, depending on what kinds of roads you’re going to find. You may want to check on an overnight stop.
The Problem: You’re driving along and notice that you’ll be needing fuel very soon. You spot a service station and pull over to buy fuel. The price on the sign is $2.25. You gulp after you realize that the price is for 1 liter of fuel. What is the price per gallon?
First, use the equivalence 1 liter = 1.057 quarts in a proportion with the price of 1 liter = $2.25 to determine how much the fuel costs per quart. Then you multiply that price by 4 because 1 gallon = 4 quarts.
1 liter 1.057 quarts
$2.25 1
2.25
X Dollars
1.057
X
2.25 (1.057) = 2.37825
The fuel is about $2.38 per quart. Multiply by four, and 2.37825 x 4 = 9.513 or about $9.51 per gallon. And you thought gas prices were bad in the United States!
Zz
X
Changing from English to metric
You’re on your European tour and you’ve brought some fabric samples to make curtains for your hostess and a recipe so you can cook up a thank-you dinner. Now you have the challenge of converting some of your measures into the measures of the country you’re visiting.
1 yard = 0.9144 meter 1 pound = 0.454 kilogram 1 cup = 0.2365 liter
Part of the challenge of cooking in another country is trying to find the ingredients that you’re used to working with at home. The other challenge comes when you need to measure those ingredients.
The Problem: Your recipe for lasagna calls for a 16-ounce jar (2 cups) of tomato sauce. You find a can of tomato puree (which you’ll have to spice up a bit), and the can contains % liter of puree. How many cans of the puree will you need to buy?
Create a proportion using 1 cup = 0.2365 liter and 2 cups = X Liters. Solve for X, Which will be the amount of tomato sauce you need in terms of liters. Then compare that amount to % or 0.75 liter.
1 cup 0.2365 liter
2 cups X Liters 1 0.2365
—— — -—-
— :
2
2 (0.2365)=0.473
The can contains 0.473 liter of tomato sauce. You need 0.75 liter. You’ll have to buy 2 cans, giving you 0.946 liter, and just save the extra for the next project. Good luck with the measuring part!
You’ve had way too good of a time on your trip to Europe. You’ve been avoiding getting on the scale to see if you’ve gained any weight, but you finally decide, the day before leaving for home, to get on the scale to see what the damage is. Omigosh! You’ve lost weight! You’ve lost a Lot Of weight! Then you realize that the scale is in kilograms.
The Problem: You weigh yourself on a metric scale and it says 68. How many pounds do you really weigh?
Use the equivalence 1 kilogram = 2.205 pounds and multiply each side of the equation by 68. You get that 68 kilograms = 149.9 or 150 pounds.
X
Chapter 4