6.1.2: Weight

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The NROC Project

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Learning Objectives

Define units of weight and convert from one to another.
Perform arithmetic calculations on units of weight.
Solve application problems involving units of weight.

Introduction

When you mention how heavy or light an object is, you are referring to its weight. In the U.S. customary system of measurement, weight is measured in ounces, pounds, and tons. Like other units of measurement, you can convert between these units and you sometimes need to do this to solve problems.

In 2010, the post office charged $0.44 to mail something that weighed an ounce or less. The post office charged $0.17 for each additional ounce, or fraction of an ounce, of weight. How much would it have cost to mail a package weighing two pounds three ounces? To answer this question, you need to understand the relationship between ounces and pounds.

Units of Weight

You often use the word weight to describe how heavy or light an object or person is. Weight is measured in the U.S. customary system using three units: ounces, pounds, and tons. An ounce is the smallest unit for measuring weight, a pound is a larger unit, and a ton is the largest unit.

Whales are some of the largest animals in the world. Some species can reach weights of up to 200 tons: that’s equal to 400,000 pounds.	$Screen Shot 2021-05-03 at 12.35.08 PM.png$
Meat is a product that is typically sold by the pound. One pound of ground beef makes about four hamburger patties.	$Screen Shot 2021-05-03 at 12.36.28 PM.png$
Ounces are used to measure lighter objects. A stack of 11 pennies is equal to about one ounce.	$Screen Shot 2021-05-03 at 12.38.04 PM.png$

You can use any of the customary measurement units to describe the weight of something, but it makes more sense to use certain units for certain purposes. For example, it makes more sense to describe the weight of a human being in pounds rather than tons. It makes more sense to describe the weight of a car in tons rather than ounces.

1 pound = 16 ounces

1 ton = 2,000 pounds

Converting Between Units of Weight

Four ounces is a typical serving size of meat. Since meat is sold by the pound, you might want to convert the weight of a package of meat from pounds to ounces in order to determine how many servings are contained in a package of meat.

The weight capacity of a truck is often provided in tons. You might need to convert pounds into tons if you are trying to determine whether a truck can safely transport a big shipment of heavy materials.

The table below shows the unit conversions and conversion factors that are used to make conversions between customary units of weight.

Unit Equivalents	Conversion Factors (heavier to lighter units of measurement)	Conversion Factors (lighter to heavier units of measurement)
$\ 1 \text { pound }=16 \text { ounces }$	$\ \frac{16 \text { ounces }}{1 \text { pound }}$	$\ \begin{array}{cc} 1 & \text { pound } \\ \hline 16 & \text { ounces } \end{array}$
$\ 1 \text { ton }=2000 \text { pounds }$	$\ \frac{2000 \text { pounds }}{1 \text { ton }}$	$\ \begin{array}{c} 1 \text { ton } \\ \hline 2000 \text { pounds } \end{array}$

You can use the factor label method to convert one customary unit of weight to another customary unit of weight. This method uses conversion factors, which allow you to “cancel” units to end up with your desired unit of measurement.

Each of these conversion factors is a ratio of equal values, so each conversion factor equals 1. Multiplying a measurement by a conversion factor does not change the size of the measurement at all, since it is the same as multiplying by 1. It just changes the units that you are using to measure it in.

Two examples illustrating the factor label method are shown below.

Example

How many ounces are in $\ 2 \frac{1}{4}$ pounds?

Solution

$\ 2 \frac{1}{4} \text { pounds }=? \text { ounces }$	Begin by reasoning about your answer. Since a pound is heavier than an ounce, expect your answer to be a number greater than $\ 2 \frac{1}{4}$.
$\ 2 \frac{1}{4} \text { pounds } \cdot \frac{16 \text { ounces }}{1 \text { pound }}=? \text { ounces }$	Multiply by the conversion factor that relates ounces and pounds: $\ \frac{16 \text { ounces }}{1\text { pound }}$
$\ \frac{9 \text { pounds }}{4} \cdot \frac{16 \text { ounces }}{1 \text { pound }}=? \text { ounces }$	Write the mixed number as an improper fraction.
$\ \frac{9 \cancel{\text { pounds }}}{4} \cdot \frac{16 \text { ounces }}{1 \cancel{\text { pound }}}=? \text { ounces }$	The common unit “pound” can be cancelled because it appears in both the numerator and denominator.
$\ \frac{9}{4} \cdot \frac{16 \text { ounces }}{1}=? \text { ounces }$ $\ \frac{9 \cdot 16 \text { ounces }}{4 \cdot 1}=? \text { ounces }$ $\ \frac{144 \text { ounces }}{4}=? \text { ounces }$ $\ \frac{144 \text { ounces }}{4}=36 \text { ounces }$	Multiply and simplify.

Multiply and simplify 36 ounces in $\ 2 \frac{1}{4}$ pounds.

Example

How many tons is 6,500 pounds?

Solution

$\ 6,500 \text { pounds }=? \text { tons }$	Begin by reasoning about your answer. Since a ton is heavier than a pound, expect your answer to be a number less than 6,500.
$\ 6,500 \text { pounds } \cdot \frac{1 \text { ton }}{2,000 \text { pounds }}=? \text { tons }$	Multiply by the conversion factor that relates tons to pounds: $\ \begin{array}{c} 1 \text { ton } \\ \hline 2,000 \text { pounds } \end{array}$
$\ \frac{6,500 \text { pounds }}{1} \cdot \frac{1 \text { ton }}{2,000 \text { pounds }}=? \text { tons }$ $\ \frac{6,500 \cancel{\text { pounds }}}{1} \cdot \frac{1 \text { ton }}{2,000 \cancel{\text { pounds }}}=? \text { tons }$	Apply the factor label method.
$\ \frac{6,500}{1} \cdot \frac{1 \text { ton }}{2,000}=? \text { tons }$ $\ \frac{6,500 \text { tons }}{2,000}=? \text { tons }$ $\ \frac{6,500 \text { tons }}{2,000}=3 \frac{1}{4} \text { tons }$	Multiply and simplify.

6,500 pounds is equal to $\ 3 \frac{1}{4}$ tons.

Exercise

How many pounds is 72 ounces?

$\ 4 \frac{1}{2} \text { pounds }$
$\ 6 \text { pounds }$
$\ 24 \text { pounds }$
$\ \text { 1, } 152 \text { pounds }$

Answer

Correct. There are ounces in one pound, so $\ 72 \text { ounces } \cdot \frac{1 \text { pound }}{16 \text { ounces }}=4 \frac{1}{2} \text { pounds }$.
Incorrect. There are 16 ounces in a pound, not 12. The correct answer is $\ 4 \frac{1}{2} \text { pounds }$.
Incorrect. There are 16 ounces in a pound, not 3. The correct answer is $\ 4 \frac{1}{2} \text { pounds }$.
Incorrect. Pounds are heavier than ounces, so the answer must be less than 72. Multiply by $\ \frac{1 \text { pound }}{16 \text { ounces }}$, not $\ \frac{16 \text { ounces }}{1\text { pound }}$. The correct answer is $\ 4 \frac{1}{2} \text { pounds }$.

Applying Unit Conversions

There are times when you need to perform calculations on measurements that are given in different units. To solve these problems, you need to convert one of the measurements to the same unit of measurement as the other measurement.

Think about whether the unit you are converting to is smaller or larger than the unit you are converting from. This will help you be sure that you are making the right computation. You can use the factor label method to make the conversion from one unit to another.

Here is an example of a problem that requires converting between units.

Example

A municipal trash facility allows a person to throw away a maximum of 30 pounds of trash per week. Last week, 140 people threw away the maximum allowable trash. How many tons of trash did this equal?

Solution

$\ 140 \cdot 30 \text { pounds }=4,200 \text { pounds }$	Determine the total trash for the week expressed in pounds. If 140 people each throw away 30 pounds, you can find the total by multiplying.
$\ 4,200 \text { pounds }=? \text { tons }$	Then convert 4,200 pounds to tons. Reason about your answer. Since a ton is heavier than a pound, expect your answer to be a number less than 4,200.
$\ \frac{4,200 \text { pounds }}{1} \cdot \frac{1 \text { ton }}{2,000 \text { pounds }}=? \text { tons }$ $\ \frac{4,200 \cancel{\text { pounds }}}{1} \cdot \frac{1 \text { ton }}{2,000 \cancel{\text { pounds }}}=? \text { tons }$	Find the conversion factor appropriate for the situation: $\ \begin{array}{c} 1 \text { ton } \\ \hline 2,000 \text { pounds } \end{array}$
$\ \frac{4,200}{1} \cdot \frac{1 \text { ton }}{2,000}=? \text { tons }$ $\ \frac{4,200 \cdot 1 \text { ton }}{1 \cdot 2,000}=? \text { tons }$ $\ \frac{4,200 \text { tons }}{2,000}=? \text { tons }$ $\ \frac{4,200 \text { tons }}{2,000}=2 \frac{1}{10} \text { tons }$	Multiply and simplify.

The total amount of trash generated is $\ 2 \frac{1}{10}$ tons.

Let’s revisit the post office problem that was posed earlier. We can use unit conversion to solve this problem.

Example

The post office charges $0.44 to mail something that weighs an ounce or less. The charge for each additional ounce, or fraction of an ounce, of weight is $0.17. At this rate, how much will it cost to mail a package that weighs 2 pounds 3 ounces?

Solution

$\ 2 \text { pounds } 3 \text { ounces }=? \text { ounces }$	Since the pricing is for ounces, convert the weight of the package from pounds and ounces into just ounces.
$\ \frac{2 \text { pounds }}{1} \cdot \frac{16 \text { ounces }}{\text { pound }}=? \text { ounces }$ $\ \frac{2 \cancel{\text { pounds }}}{1} \cdot \frac{16 \text { ounces }}{\cancel{\text { pound }}}=? \text { ounces }$	First use the factor label method to convert 2 pounds to ounces.
$\ \frac{2}{1} \cdot \frac{16 \text { ounces }}{1}=32 \text { ounces }$	$\ 2 \text { pounds }=32 \text { ounces }$
$\ 32 \text { ounces }+3 \text { ounces }=35 \text { ounces }$	Add the additional 3 ounces to find the weight of the package. The package weighs 35 ounces. There are 34 additional ounces, since 35-1=34.
$\ \$ 0.44+\$ 0.17(34)$ $\ \$ 0.44+\$ 5.78$ $\ \$ 0.44+\$ 5.78=\$ 6.22$	Apply the pricing formula. $0.44 for the first ounce and $0.17 for each additional ounce.

It will cost $6.22 to mail a package that weighs 2 pounds 3 ounces.

Exercise

The average weight of a northern bluefin tuna is 1,800 pounds. The average weight of a great white shark is $\ 2 \frac{1}{2}$ tons. On average, how much more does a great white shark weigh, in pounds, than a northern bluefin tuna?

$\ 5,000 \text { pounds }$
$\ 3,200 \text { pounds }$
$\ 182 \frac{1}{2} \text { pounds }$
You cannot answer this because the units of weight are different.

Answer

Incorrect. A great white shark has an average weight of 5,000 pounds, and you have to subtract 1,800 to find the difference in the weights of the shark and the tuna. The correct answer is 3,200 pounds.
Correct. $\ 2 \frac{1}{2} \text { tons }=5,000 \text { pounds }$.
Incorrect. You cannot subtract these weights because they are given in different units. To find the difference, first convert the weight of the shark to pounds and then subtract the weight of the tuna in pounds. The correct answer is 3,200 pounds.
Incorrect. The units are different, but you can express the weights in the same unit and then compute. First convert the weight of the shark to pounds and then subtract the weight of the tuna in pounds. The correct answer is 3,200 pounds.

Summary

In the U.S. customary system of measurement, weight is measured in three units: ounces, pounds, and tons. A pound is equivalent to 16 ounces, and a ton is equivalent to 2,000 pounds. While an object’s weight can be described using any of these units, it is typical to describe very heavy objects using tons and very light objects using an ounce. Pounds are used to describe the weight of many objects and people. Often, in order to compare the weights of two objects or people or to solve problems involving weight, you must convert from one unit of measurement to another unit of measurement. Using conversion factors with the factor label method is an effective strategy for converting units and solving problems.

Unit Equivalents	Conversion Factors (heavier to lighter units of measurement)	Conversion Factors (lighter to heavier units of measurement)
\(\ 1 \text { pound }=16 \text { ounces }\)	\(\ \frac{16 \text { ounces }}{1 \text { pound }}\)	\(\ \begin{array}{cc} 1 & \text { pound } \\ \hline 16 & \text { ounces } \end{array}\)
\(\ 1 \text { ton }=2000 \text { pounds }\)	\(\ \frac{2000 \text { pounds }}{1 \text { ton }}\)	\(\ \begin{array}{c} 1 \text { ton } \\ \hline 2000 \text { pounds } \end{array}\)

\(\ 2 \frac{1}{4} \text { pounds }=? \text { ounces }\)	Begin by reasoning about your answer. Since a pound is heavier than an ounce, expect your answer to be a number greater than \(\ 2 \frac{1}{4}\).
\(\ 2 \frac{1}{4} \text { pounds } \cdot \frac{16 \text { ounces }}{1 \text { pound }}=? \text { ounces }\)	Multiply by the conversion factor that relates ounces and pounds: \(\ \frac{16 \text { ounces }}{1\text { pound }}\)
\(\ \frac{9 \text { pounds }}{4} \cdot \frac{16 \text { ounces }}{1 \text { pound }}=? \text { ounces }\)	Write the mixed number as an improper fraction.
\(\ \frac{9 \cancel{\text { pounds }}}{4} \cdot \frac{16 \text { ounces }}{1 \cancel{\text { pound }}}=? \text { ounces }\)	The common unit “pound” can be cancelled because it appears in both the numerator and denominator.
\(\ \frac{9}{4} \cdot \frac{16 \text { ounces }}{1}=? \text { ounces }\) \(\ \frac{9 \cdot 16 \text { ounces }}{4 \cdot 1}=? \text { ounces }\) \(\ \frac{144 \text { ounces }}{4}=? \text { ounces }\) \(\ \frac{144 \text { ounces }}{4}=36 \text { ounces }\)	Multiply and simplify.

\(\ 6,500 \text { pounds }=? \text { tons }\)	Begin by reasoning about your answer. Since a ton is heavier than a pound, expect your answer to be a number less than 6,500.
\(\ 6,500 \text { pounds } \cdot \frac{1 \text { ton }}{2,000 \text { pounds }}=? \text { tons }\)	Multiply by the conversion factor that relates tons to pounds: \(\ \begin{array}{c} 1 \text { ton } \\ \hline 2,000 \text { pounds } \end{array}\)
\(\ \frac{6,500 \text { pounds }}{1} \cdot \frac{1 \text { ton }}{2,000 \text { pounds }}=? \text { tons }\) \(\ \frac{6,500 \cancel{\text { pounds }}}{1} \cdot \frac{1 \text { ton }}{2,000 \cancel{\text { pounds }}}=? \text { tons }\)	Apply the factor label method.
\(\ \frac{6,500}{1} \cdot \frac{1 \text { ton }}{2,000}=? \text { tons }\) \(\ \frac{6,500 \text { tons }}{2,000}=? \text { tons }\) \(\ \frac{6,500 \text { tons }}{2,000}=3 \frac{1}{4} \text { tons }\)	Multiply and simplify.

\(\ 140 \cdot 30 \text { pounds }=4,200 \text { pounds }\)	Determine the total trash for the week expressed in pounds. If 140 people each throw away 30 pounds, you can find the total by multiplying.
\(\ 4,200 \text { pounds }=? \text { tons }\)	Then convert 4,200 pounds to tons. Reason about your answer. Since a ton is heavier than a pound, expect your answer to be a number less than 4,200.
\(\ \frac{4,200 \text { pounds }}{1} \cdot \frac{1 \text { ton }}{2,000 \text { pounds }}=? \text { tons }\) \(\ \frac{4,200 \cancel{\text { pounds }}}{1} \cdot \frac{1 \text { ton }}{2,000 \cancel{\text { pounds }}}=? \text { tons }\)	Find the conversion factor appropriate for the situation: \(\ \begin{array}{c} 1 \text { ton } \\ \hline 2,000 \text { pounds } \end{array}\)
\(\ \frac{4,200}{1} \cdot \frac{1 \text { ton }}{2,000}=? \text { tons }\) \(\ \frac{4,200 \cdot 1 \text { ton }}{1 \cdot 2,000}=? \text { tons }\) \(\ \frac{4,200 \text { tons }}{2,000}=? \text { tons }\) \(\ \frac{4,200 \text { tons }}{2,000}=2 \frac{1}{10} \text { tons }\)	Multiply and simplify.

\(\ 2 \text { pounds } 3 \text { ounces }=? \text { ounces }\)	Since the pricing is for ounces, convert the weight of the package from pounds and ounces into just ounces.
\(\ \frac{2 \text { pounds }}{1} \cdot \frac{16 \text { ounces }}{\text { pound }}=? \text { ounces }\) \(\ \frac{2 \cancel{\text { pounds }}}{1} \cdot \frac{16 \text { ounces }}{\cancel{\text { pound }}}=? \text { ounces }\)	First use the factor label method to convert 2 pounds to ounces.
\(\ \frac{2}{1} \cdot \frac{16 \text { ounces }}{1}=32 \text { ounces }\)	\(\ 2 \text { pounds }=32 \text { ounces }\)
\(\ 32 \text { ounces }+3 \text { ounces }=35 \text { ounces }\)	Add the additional 3 ounces to find the weight of the package. The package weighs 35 ounces. There are 34 additional ounces, since 35-1=34.
\(\ \$ 0.44+\$ 0.17(34)\) \(\ \$ 0.44+\$ 5.78\) \(\ \$ 0.44+\$ 5.78=\$ 6.22\)	Apply the pricing formula. $0.44 for the first ounce and $0.17 for each additional ounce.