## Easy Mathematics Step-by-Step (2012)

### Chapter 8. Units of Measurement

In use in the United States are two different measurement systems: the US customary system and the metric system. In this chapter, you learn how to work with both systems.

All other countries have adopted the metric system, although the imperial system—which is similar to the US customary system—is also used in the United Kingdom.

**Metric System Prefixes**

You are no doubt familiar with the US customary system. __Table 8.1__ contains the metric system prefixes you will find useful to know.

**Table 8.1 Metric System Prefixes**

**US Customary and Metric Units**

__Table 8.2__ contains a list of the US customary and metric units most commonly used in everyday activities.

**Table 8.2 Units of Measurement**

**Denominate Numbers**

You express measurements using denominate numbers. A *denominate number* is a number with units attached. Numbers without units attached are *abstract numbers*.

**Problem** Identify the denominate numbers in the following list: 1000, 1200 m, $400, , , 180d, 4840 yd^{2}, 8 gal, 5 kg, 3.785

**Solution**

*Step 1*. Recalling that denominate numbers have units attached, identify the denominate numbers in the list.

1200 m, $400, yd, 180 d, 4840 yd^{2}, 8 gal, and 5 kg

**Converting Units of Denominate Numbers**

You convert measurement units of denominate numbers to different measurement units by using “conversion fractions.” You make conversion fractions from conversion facts given in tables like __Table 8.2__. You have two conversion fractions for each conversion fact. For example, for , you have and . Each of these fractions is equivalent to the number 1 because the numerator and denominator are different names for the same length. Therefore, multiplying a quantity by either of these fractions, does not change the value of the quantity.

To change the units of a denominate number to different units, multiply by the conversion fraction whose *denominator is the same as the units of the quantity to be converted*. When you multiply, the units you started out with will divide (“cancel”) out, and you will be left with the new units.

When you’re converting from one measurement unit to another, if the original units don’t cancel out when you multiply, then you picked the wrong conversion fraction. Go back and do the multiplication over again with the other conversion fraction.

You always should assess your result to see if it makes sense. Here is a helpful guideline.

When you convert from a larger unit to a smaller unit, it will take more of the smaller units to equal the same amount. When you convert from a smaller unit to a larger unit, it will take less of the larger units to equal the same amount.

**Problem** Change the given amount to the units indicated.

**Solution**

*Step 1*. Using __Table 8.2__, determine the conversion fractions.

*Step 2*. Write 5 yd as a fraction with denominator 1, select the conversion fraction that has “yd” in the denominator, and then multiply.

(The “yd” units cancel out, leaving “ft” as the units for the answer.)

*Step 3*. State the main result.

*Step 4*. Assess the result.

A yard is longer than a foot, so the number in front of “ft” should be greater than the number in front of “yd.”

** b**.

*Step 1*. Using __Table 8.2__, determine the conversion fractions.

*Step 2*. Write 360 min as a fraction with denominator 1, select the conversion fraction that has “min” in the denominator, and then multiply.

(The “min” units cancel out, leaving “hr” as the units for the answer.)

*Step 3*. State the main result.

*Step 4*. Assess the result.

A minute is shorter than an hour, so the number in front of “hr” should be less than the number in front of “min.”

*Step 1*. Using __Table 8.2__, determine the conversion fractions.

. Do not make this common error. A square yard measures 1 yd by 1 yd. Thus, .

*Step 2*. Write 2 yd^{2} as a fraction with denominator 1, select the conversion fraction that has “yd^{2}” in the denominator, and then multiply.

(The “yd^{2}” units cancel out, leaving “ft^{2}” as the units for the answer.)

*Step 3*. State the main result.

*Step 4*. Assess the result.

A square yard is larger than a square foot, so the number in front of “ft^{2}” should be greater than the number in front of “yd^{2}.”

*Step 1*. Using __Table 8.2__, determine the conversion fractions.

*Step 2*. Write yd with “yd” as part of the numerator, select the conversion fraction that has “yd” in the denominator, and then multiply.

(The “yd” units cancel out, leaving “ft” as the units for the answer.)

or 2.25ft. When you have a choice of expressing an answer using fractions or decimals, you should use decimals because working with decimals is easier when you use a calculator.

*Step 3*. State the main result.

*Step 4*. Assess the result.

A yard is longer than a foot, so the number in front of “ft” should be greater than the number in front of “yd.”

*Step 1*. Using __Table 8.2__, determine the conversion fractions.

*Step 2*. Write 7 qt as a fraction with denominator 1, select the conversion fraction that has “qt” in the denominator, and then multiply.

(The “qt” units cancel out, leaving “gal” as the units for the answer.)

*Step 3*. State the main result.

*Step 4*. Assess the result.

A quart is less than a gallon, so the number in front of “gal” should be less than the number in front of “qt.”

*Step 1*. Using __Table 8.2__, determine the conversion fractions.

*Step 2*. Write 8 in as a fraction with denominator 1, select the conversion fraction that has “in” in the denominator, and then multiply.

(The “in” units cancel out, leaving “cm” as the units for the answer.)

*Step 3*. State the main result.

*Step 4*. Assess the result.

An inch is longer than a centimeter, so the number in front of “cm” should be greater than the number in front of “in.”

*Step 1*. Using __Table 8.2__, determine the conversion fractions.

*Step 2*. Write 1200 m as a fraction with denominator 1, select the conversion fraction that has “m” in the denominator, and then multiply.

(The “m” units cancel out, leaving “km” as the units for the answer.)

*Step 3*. State the main result.

*Step 4*. Assess the result.

A meter is shorter than a kilometer, so the number in front of “km” should be less than the number in front of “m.”

*Step 1*. Using __Table 8.2__, determine the conversion fractions.

*Step 2*. Write 5 kg as a fraction with denominator 1, select the conversion fraction that has “kg” in the denominator, and then multiply.

(The “kg” units cancel out, leaving “g” as the units for the answer.)

*Step 3*. State the main result.

*Step 4*. Assess the result.

A kilogram is heavier than a gram, so the number in front of “g” should be greater than the number in front of “kg.”

**Shortcut for Converting Within the Metric System**

*If the base unit in the problem is the meter, liter, or gram*, you have a shortcut to convert within the metric system. You use the mnemonic “**K**ing **H**enry **D**oesn’t **U**sually **D**rink **C**hocolate **M**ilk,” which helps you remember the following metric prefixes:

**k**ilo-, **h**ecto-, **d**eca-, (base) **u**nit (no prefix), **d**eci-, **c**enti-, **m**illi-

The metric system is a decimal-based system, so the prefixes are based on powers of 10. You convert from one unit to another by either multiplying or dividing by a power of 10. If you move from left to right on the above list, then you *multiply* by the power of 10 that corresponds to the number of times you moved. If you move from right to left, then you *divide* by the power of 10 that corresponds to the number of times you moved. *Note:* When you use this shortcut, don’t carry the units along when you do the calculations.

**Problem** Change the given amount to the units indicated.

**Solution**

*Step 1*. Using the list of metric prefixes, determine how many moves you make, and in what direction, to go from meters to kilometers.

*Step 2*. Divide 1200 by 10^{3}, the power of 10 that corresponds to the number of moves.

*Step 3*. State the main result.

*Step 4*. Assess the result.

A meter is shorter than a kilometer, so the number in front of “km” should be less than the number in front of “m.”

*Step 1*. Using the list of metric prefixes, determine how many moves you make, and in what direction, to go from kilograms to grams.

*Step 2*. Multiply 5 by 10^{3}, the power of 10 that corresponds to the number of moves.

*Step 3*. State the main result.

*Step 4*. Assess the result.

A kilogram is heavier than a gram, so the number in front of “g” should be greater than the number in front of “kg.”

*Step 1*. Using the list of metric prefixes, determine how many moves you make, and in what direction, to go from liters to milliliters.

*Step 2*. Multiply 3.5 by 10^{3}, the power of 10 that corresponds to the number of moves.

*Step 3*. State the main result.

*Step 4*. Assess the result.

A liter is larger than a milliliter, so the number in front of “mL” should be greater than the number in front of “L.”

**Using a “Chain” of Conversion Fractions**

For some conversions, you may need to use a “chain” of conversion fractions to obtain your desired units. Select the conversion facts that help you obtain your desired units and then multiply one after the other.

**Problem** Change the given amount to the units indicated.

**Solution**

*Step 1*. Using __Table 8.2__, determine the conversion fractions.__Table 8.2__ does not have a fact that shows the equivalency between gallons and pints. However, the table shows that and . These two facts yield four conversion fractions: and and and .

*Step 2*. Start with and keep multiplying by conversion fractions until you obtain your desired units.

*Step 3*. State the main result.

*Step 4*. Assess the result.

A gallon is larger than a pint, so the number in front of “pt” should be greater than the number in front of “gal.”

*Step 1*. Using __Table 8.2__, determine the conversion fractions.

__Table 8.2__ does not have a fact that shows the equivalency between weeks and minutes. However, the table shows that , , and . These three facts yield six conversion fractions: and , and , and and .

*Step 2*. Start with and keep multiplying by conversion fractions until you obtain your desired units.

*Step 3*. State the main result.

*Step 4*. Assess the result.

A week is longer than a minute, so the number in front of “min” should be greater than the number in front of “wk.”

**Converting Money to Different Denominations**

When converting with denominations of money, it is helpful to change the original amount to cents and then to the denomination to which you are converting.

It is customary to speak of “denominations” of money rather than “units” of money.

**Problem** Change the given amount to the denomination indicated.

**Solution**

*Step 1*. Convert 15 quarters to cents.

*Step 2*. Convert 375¢ to nickels.

*Step 3*. State the main result.

*Step 4*. Assess the result.

Quarters have more value than nickels, so the number in front of “nickels” should be greater than the number in front of “quarters.”

*Step 1*. Convert 75 dimes to cents.

*Step 2*. Convert 750¢ to quarters.

*Step 3*. State the main result.

*Step 4*. Assess the result.

Dimes have less value than quarters, so the number in front of “quarters” should be less than the number in front of “dimes.”

**Rough Equivalencies for the Metric System**

If you are not very familiar with the metric system, here are some “rough” equivalencies of the more common units for your general knowledge.

**Determining Unit Price**

Measurement skills include computing unit price. The *unit price* is the amount per unit. Unit price is used in many real-life situations.

**Problem** Which is a better buy for a certain cut of beef, 3 lb for $2.00 or 4 lb for $3.50?

**Solution**

*Step 1*. Compute the unit price for 3 lb for $2.00.

The unit price for 3 lb for (rounded to the nearest cent).

*Step 2*. Compute the unit price for 4 lb for $3.50.

The unit price for 4 lb for (rounded to the nearest cent).

*Step 3*. Compare the unit prices and select the better buy.

$0.67 per pound is less than $0.88 per pound, so, assuming the quality is the same, the better buy is 3 lb for $2.00.

**Exercise 8**

*For 1–14, change the given amount to the units indicated*.

__15.__ Which is a better buy for shelled peanuts, 8 oz for $4.50 or 9 oz for $4.99?

__16.__ A runner ran 250 m. How many kilometers did the runner run?

__17.__ First-class postage is charged by the ounce. A package weighs 3 lb 12 oz. How many ounces does the package weigh?

__18.__ Carpet is sold by the square yard. The surface of the floor of a 9 ft by 12 ft room is 108 ft^{2}. How many square yards of carpet are needed to cover the floor?

__19.__ A recipe calls for 3 tbsp of oil. How many fluid ounces is 3 tbsp?

__20.__ A large container holds 5 gal of water. How many cups of water does the container hold?