Introduction to Proportion Word Problems
There is a difference between not knowing and not knowing yet.
—SHEILA TOBIAS (1935– )
One important ingredient in the recipe for success in solving math word problems is persistence; another is practice. This lesson will concentrate on the concepts of proportion and scale. Practice solving word problems on these topics is provided. Use this lesson to become more skilled in this important area of math problem solving by applying persistence and practice.
Proportions
While a ratio is a comparison of two numbers, and a rate is a comparison of two units, a proportion is a comparison of two ratios.
Tip:Like ratios, proportions can be written in different ways. The most common ways are using a colon a:b = c:d or fraction form |
.Although there are a number of ways to test if a proportion is a true proportion, one way is to cross multiply. To cross multiply, find the product of the numerator of one ratio and the denominator of the other ratio. Then, set this product equal to the result of multiplying the other numerator by the other denominator.
For example, use cross multiplication to see if the following proportion is a true proportion.
In this proportion, multiply 2 × 6 and 3 × 4. Each product is 12, so the proportion is a true proportion.
- To test the proportion that follows, cross multiply 5 × 11 and 8 × 7.
The cross products are not equal, so this is not a true proportion.
Tip:Another name for cross multiplying is the means-extremes property. In the proportion a:b = c:d, the terms a and d are known as the extremes and b and c are known as the means. |
Solving Proportions
In other cases, one or more terms of the proportion are unknown. These proportions can be solved by cross multiplying, or by the means-extremes property, in order to find the value or values to make it a true proportion. To do this, find the product of the means and set it equal to the product of the extremes. Then, solve the equation.
For example, to solve the proportion 
, multiply the means, and set it equal to the product of the extremes. This is equal to 15x = 60. Divide each side of the equation by 15:
The value x = 4 makes this a true proportion.
Scale Drawings
Applying the scale on a map or scale drawing is another way that proportions can be very useful. When the scale is given, and the distance on the map or drawing can be measured, the actual lengths can be calculated. Use a proportion to set the scale measurements equal to the actual measurements.
For example, if the scale on a map is 1 inch = 5 miles, and a school and a house on the map are 12 inches apart, set up a proportion comparing the measures.
- The proportion could be set up as:
Notice how the labels match up to help place the values correctly.
- Cross multiply the means and the extremes, and solve the equation.
- 1x = 5 × 12
- x = 60 miles
The actual distance between the two locations is 60 miles.
Tip:When you are solving proportion and scale word problems, use the labels to help you line up the values in the correct places. For example, in the previous problem, the labels inch and inches are lined up across from each other, and the labels miles are also lined up across from each other. |
Divide each side of the equation by 2.
Divide each side of the equation by 3.
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