CHAPTER 4 Ratio, Proportion, and Percent
4.3 Solve Proportions and their Applications
Learning Objectives
By the end of this section, you will be able to:
- Use the definition of proportion
- Solve proportions
- Solve applications using proportions
- Write percent equations as proportions
- Translate and solve percent proportions
Use the Definition of Proportion
When two ratios or rates are equal, the equation relating them is called a proportion.
Proportion
A proportion is an equation of the form , where .
The proportion states two ratios or rates are equal. The proportion is read is to , as is to
The equation is a proportion because the two fractions are equal. The proportion is read is to as is to
If we compare quantities with units, we have to be sure we are comparing them in the right order. For example, in the proportion we compare the number of students to the number of teachers. We put students in the numerators and teachers in the denominators.
EXAMPLE 1
Write each sentence as a proportion:
- is to as is to .
- hits in at bats is the same as hits in at-bats.
- for ounces is equivalent to for ounces.
| a) | |
| 3 is to 7 as 15 is to 35. | |
| Write as a proportion. |
| b) | |
| 5 hits in 8 at-bats is the same as 30 hits in 48 at-bats. | |
| Write each fraction to compare hits to at-bats. | |
| Write as a proportion. |
| c) | |
| $1.50 for 6 ounces is equivalent to $2.25 for 9 ounces. | |
| Write each fraction to compare dollars to ounces. | |
| Write as a proportion. |
TRY IT 1.1
Write each sentence as a proportion:
- is to as is to .
- hits in at-bats is the same as hits in at-bats.
- for ounces is equivalent to for ounces.
Show answer
TRY IT 1.2
Write each sentence as a proportion:
- is to as is to .
- adults for children is the same as adults for children.
- for ounces is equivalent to for ounces.
Show answer
Look at the proportions and . From our work with equivalent fractions we know these equations are true. But how do we know if an equation is a proportion with equivalent fractions if it contains fractions with larger numbers?
To determine if a proportion is true, we find the cross products of each proportion. To find the cross products, we multiply each denominator with the opposite numerator (diagonally across the equal sign). The results are called a cross products because of the cross formed. The cross products of a proportion are equal.
Cross Products of a Proportion
For any proportion of the form , where , its cross products are equal.
Cross products can be used to test whether a proportion is true. To test whether an equation makes a proportion, we find the cross products. If they are the equal, we have a proportion.
EXAMPLE 2
Determine whether each equation is a proportion:
To determine if the equation is a proportion, we find the cross products. If they are equal, the equation is a proportion.
| a) | |
| Find the cross products. |
Since the cross products are not equal, , the equation is not a proportion.
| b) | |
| Find t