Proportions allow us to convert measurements with different units. We can use a conversion factor, which is written as a ratio in fraction form with equivalent measurements in the numerator and denominator. The first conversion values we may see are the metric system relationships:

1 \text{ km} = 1000 \text{ m}

1 \text{ m} = 100 \text{ cm}

1 \text{ cm} = 10 \text{ mm}

The conversion factors for the metric system are summarized in the conversion chart below:

Conversion factors between the units of length from millimetres to kilometres. Ask your teacher for more information.

In the United States, we will more regularly see relationships from the U.S. customary units:

1 \text{ mi} = 1760 \text{ yards}

1 \text{ yard} = 3 \text{ ft}

1 \text{ ft} = 12 \text{ in}

Conversion factors between the units of length from inches to feet. Ask your teacher for more information.

If we want to convert 8 feet into inches, the conversion value is 1 \text{ ft} = 12 \text{ in}.

\displaystyle 8 \text{ } \cancel{\text{ft}} \cdot \dfrac{12 \text{ in} }{1 \text{ } \cancel{\text{ft}}}

\displaystyle =

\displaystyle 96 \text{ in}

Multiply the given by the conversion factor

Notice that the unit of the given number is \text {ft} and the conversion factor has the unit \text {ft} in the denominator. This causes the \text {ft} to divide out leaving just the unit, \text {in}.

However, if we want to convert 96 inches into feet, the conversion value is the same, but we put the feet in the numerator.

\displaystyle 96\text{ in}\cdot \dfrac{1 \text{ ft}}{12\text{ in}}	\displaystyle =	\displaystyle \dfrac {96 \text{ } \cancel{\text{in}}} {12\text{ } \cancel{\text{in}}} \cdot 1\text{ ft}	Multiply the given by the conversion factor
	\displaystyle =	\displaystyle 8\text{ ft}	Simplify the fraction by dividing and cancelling the units

We can also solve by setting up a proportion:

\displaystyle \dfrac{96 \text{ in}}{x \text{ ft}}	\displaystyle =	\displaystyle \dfrac{12 \text{ in}}{1 \text{ ft}}	Setting up our proportion
\displaystyle 96 \cdot 1	\displaystyle =	\displaystyle x \cdot 12	Means Extremes Property
\displaystyle 96	\displaystyle =	\displaystyle 12x	Simplify the multiplication
\displaystyle 8	\displaystyle =	\displaystyle x	Divide by 12

Fortunately, we don't have to memorize these conversions. As long as we are given the conversion factor, we can use what we know about proportions to convert between units of measurement.

Examples

Example 1

To convert meters to inches, we can use the following table:

Meters	Inches
1	39.4
	78.8
3
4
	197

Complete the table.

Worked Solution

Create a strategy

Use the relationship 1\text{ m} = 39.4\text{ in} that was given in the table.

Apply the idea

We can use the conversion factor of \dfrac{1 \text{ m}}{39.4 \text{ in}} to find othe missing values in meters.

78.8 \text{ }\cancel{\text{in}} \cdot \dfrac{1 \text{ m}}{39.4 \text{ }\cancel{\text{ in}}} = 2 \text{ m}

197 \text{ }\cancel{\text{in}} \cdot \dfrac{1 \text{ m}}{39.4 \text{ }\cancel{\text{ in}}} = 5 \text{ m}

We can use the reciprocal of the conversion factor \dfrac{39.4 \text{ in}}{1 \text{ m}} to find the missing values in inches.

3 \text{ }\cancel{\text{m}} \cdot \dfrac{39.4 \text{ in}}{1 \text{ }\cancel{\text{ m}}} = 118.2 \text{ in}

4 \text{ }\cancel{\text{m}} \cdot \dfrac{39.4 \text{ in}}{1 \text{ }\cancel{\text{ m}}} = 157.6 \text{ in}

Our completed table should look like this:

Meters	Inches
1	39.4
2	78.8
3	118.2
4	157.6
5	197

Using the table, convert 13 meters to inches.

Worked Solution

Create a strategy

Use the relationship 1\text{ m} = 39.4\text{ in} to set up a proportion.

Apply the idea

Ensure the units match in the numerators and the units match in the denominators.

\displaystyle \dfrac{13 \text{ m}}{x \text{ in}}	\displaystyle =	\displaystyle \dfrac{1 \text{ m}}{39.4 \text{ in}}	Write a proportion
\displaystyle 13 \cdot 39.4	\displaystyle =	\displaystyle x \cdot 1	Means Extremes Property
\displaystyle 512.2	\displaystyle =	\displaystyle x	Evaluate the multiplication

So 13 meters is equivalent to 512.2 inches.

Reflect and check

We can also convert the units using a conversion factor. Since we are starting with units in meters, we want to make sure meters is in the denominator of the conversation factor so it will divide out and leave inches. We will use \dfrac{39.4 \text{ in}}{1 \text{ m}} to find the missing values in inches.

13 \text{ }\cancel{\text{m}} \cdot \dfrac{39.4 \text{ in}}{1 \text{ }\cancel{\text{ m}}} = 512.2 \text{ in}

Using the table, convert 709.2 inches to meters.

Worked Solution

Create a strategy

Using the conversion factor from inches to meters, we calculate the equivalent amount in meters.

Apply the idea

We can use the conversion factor of \dfrac{1 \text{ m}}{39.4 \text{ in}} to find othe missing values in meters.

709.2 \text{ }\cancel{\text{in}} \cdot \dfrac{1 \text{ m}}{39.4 \text{ }\cancel{\text{ in}}} = 18 \text{ m}

So, 709.2\text{ in} is equal to 18\text{ m}.

Example 2

To ride the scariest rollercoasters in an amusement park, Wynsleth needs to be over 160\text{ cm} tall.

If Wynsleth is 4\text{ ft } 11\text{ in} and 1 \text{ in} = 2.54 \text{ cm}, is she tall enough?

Worked Solution

Create a strategy

First, convert the 4 feet and 11 inches to be measured in only inches. Then convert to centimeters and compare it to 160\text{ cm}.

Apply the idea

First, we will convert 4\text{ ft} to inches to convert the whole height to inches. We will use the converstion factor \dfrac{12 \text{ in}}{1 \text{ ft}}:

4 \text{ } \cancel{\text{ft}} \cdot \dfrac{12 \text{ in}}{1 \text{ }\cancel{\text{ft}}} = 48 \text{ in}

So, 4\text{ ft } 11\text{ in} is equivalent to 48 \text{ in} + 11 \text{ in} = 59 \text{ in} To convert from inches to centimeters, we will use the conversion factor \dfrac{2.54 \text{ cm}}{1 \text{ in}}:

59 \text{ } \cancel{\text{in}} \cdot \dfrac{2.54 \text{ cm}}{1 \text{ }\cancel{\text{in}}} = 149.9 \text{ cm}

Since Wynsleth's height is 149.9\text{ cm} and the height of the rollercoaster is 160\text{ cm} tall, Wynsleth is not tall enough to ride the rollercoaster.

Reflect and check

Notice we did not need the precision of measurements to the tenths place with this problem. Oftentimes, we can use the context to determine the precision of our answer. In this problem, rounding to the nearest whole number would have been sufficient.

Idea summary

Conversion of units can be done by applying ratios and proportional relationships. If we know a relationship between the units, we can create a conversion factor equal to 1 where the units of the denominator match the given units in the problem.

\cancel{\text{Given units}} \cdot \dfrac{\text{Desired units}}{ \cancel{\text{Given units}}} = \text{Desired units}

We can also set up a proportion to convert between units.

Outcomes

7.CE.2

The student will solve problems, including those in context, involving proportional relationships.

7.CE.2c

Apply proportional reasoning to solve problems in context, including converting units of measurement, when given the conversion factor.

2.03 Unit conversions

Ideas

Unit conversions

Examples

Example 1

Example 2

Outcomes

7.CE.2

7.CE.2c

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