8.7 Solve proportion and similar figure applications

Solve proportions

When two rational expressions are equal, the equation relating them is called a proportion .

The equation $\frac{1}{2}=\frac{4}{8}$ is a proportion because the two fractions are equal. The proportion $\frac{1}{2}=\frac{4}{8}$ is read “1 is to 2 as 4 is to 8.”

Proportions are used in many applications to ‘scale up’ quantities. We’ll start with a very simple example so you can see how proportions work. Even if you can figure out the answer to the example right away, make sure you also learn to solve it using proportions.

Suppose a school principal wants to have 1 teacher for 20 students. She could use proportions to find the number of teachers for 60 students. We let x be the number of teachers for 60 students and then set up the proportion:

We are careful to match the units of the numerators and the units of the denominators—teachers in the numerators, students in the denominators.

Since a proportion is an equation with rational expressions, we will solve proportions the same way we solved equations in Solve Rational Equations . We’ll multiply both sides of the equation by the LCD to clear the fractions and then solve the resulting equation.

So let’s finish solving the principal’s problem now. We will omit writing the units until the last step.

Multiply both sides by the LCD, 60.
Simplify.
	The principal needs 3 teachers for 60 students.

Now we’ll do a few examples of solving numerical proportions without any units. Then we will solve applications using proportions.

Solve the proportion: $\frac{x}{63}=\frac{4}{7}.$

Solution

To isolate $x$ , 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.

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When we work with proportion    s, we exclude values that would make either denominator zero, just like we do for all rational expressions. What value(s) should be excluded for the proportion in the next example?

Solve the proportion: $\frac{144}{a}=\frac{9}{4}.$

Solution

Multiply both sides by the LCD.
Remove common factors on each side.
Simplify.
Divide both sides by 9.
Simplify.
Check.
Show common factors.
Simplify.

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Solve the proportion: $\frac{n}{n+14}=\frac{5}{7}.$

Solution

Multiply both sides by the LCD.
Remove common factors on each side.
Simplify.
Solve for $n$ .
Check.
Simplify.
Show common factors.
Simplify.

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