6. Health Option
6.3 Proportions; Health Applications
Learning Objectives
By the end of this section it is expected that 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.
EXAMPLE 1
Write the sentence as a proportion:
72 heartbeats in 1 minute is the same as 216 heartbeats in 3 minutes.
| 72 is to 1 as 216 is to 3. | |
| Write as a proportion. |
TRY IT 1
Write the sentence as a proportion:
is to as is to .
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 the cross products. |
|
Since the cross products are equal, , the equation is a proportion.
TRY IT 2
Determine whether each equation is a proportion:
Show answer
- no
- yes
Solve Proportions
To solve a proportion containing a variable, we remember that the proportion is an equation. All of the techniques we have used so far to solve equations still apply. In the next example, we will solve a proportion by multiplying by the Least Common Denominator (LCD) using the Multiplication Property of Equality.
EXAMPLE 3
Solve: .
| To isolate , multiply both sides by the LCD, 63. | |
| Simplify. | |
| Divide the common factors. | |
| Check: To check our answer, we substitute into the original proportion. | |
| Show common factors. | |
| Simplify. |
TRY IT 3
Solve the proportion: .
Show answer
77
When the v