6.5 Solve proportions and their applications  (Page 2/12)

Determine whether each equation is a proportion:

1. ⓐ $\frac{4}{9}=\frac{12}{28}$
2. ⓑ $\frac{17.5}{37.5}=\frac{7}{15}$

Solution

To determine if the equation is a proportion, we find the cross products. If they are equal, the equation is a proportion.

ⓐ
Find the cross products.	$28\cdot 4=1129\cdot 12=108$

Since the cross products are not equal, $28·4\ne 9·12,$ the equation is not a proportion.

ⓑ
Find the cross products.	$15\cdot 17.5=262.537.5\cdot 7=262.5$

Since the cross products are equal, $15·17.5=37.5·7,$ the equation is a proportion.

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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 .

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

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 the variable is in a denominator, we’ll use the fact that the cross products    of a proportion are equal to solve the proportions.

We can find the cross products of the proportion and then set them equal. Then we solve the resulting equation using our familiar techniques.

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

Solution

Notice that the variable is in the denominator, so we will solve by finding the cross products and setting them equal.

Find the cross products and set them equal.
Simplify.
Divide both sides by 9.
Simplify.

Check your answer.

Show common factors..
Simplify.

Another method to solve this would be to multiply both sides by the LCD, $4a.$ Try it and verify that you get the same solution.

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Solve: $\frac{52}{91}=\frac{\mathrm{-4}}{y}.$

Solution

Find the cross products and set them equal.
Simplify.
Divide both sides by 52.
Simplify.

Check:
Show common factors.
Simplify.

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Solve applications using proportions

The strategy for solving applications that we have used earlier in this chapter, also works for proportions, since proportions are equations. When we set up the proportion    , we must make sure the units are correct—the units in the numerators match and the units in the denominators match.

When pediatricians prescribe acetaminophen to children, they prescribe $5$ milliliters (ml) of acetaminophen for every $25$ pounds of the child’s weight. If Zoe weighs $80$ pounds, how many milliliters of acetaminophen will her doctor prescribe?

Solution

Identify what you are asked to find.	How many ml of acetaminophen the doctor will prescribe
Choose a variable to represent it.	Let $a=$ ml of acetaminophen.
Write a sentence that gives the information to find it.	If 5 ml is prescribed for every 25 pounds, how much will be prescribed for 80 pounds?
Translate into a proportion.
Substitute given values—be careful of the units.
Multiply both sides by 80.
Multiply and show common factors.
Simplify.
Check if the answer is reasonable.
Yes. Since 80 is about 3 times 25, the medicine should be about 3 times 5.
Write a complete sentence.	The pediatrician would prescribe 16 ml of acetaminophen to Zoe.

You could also solve this proportion by setting the cross products equal.

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