Solve, Interpret, and Check Proportions

This proportion is true because both ratios have the same value: $3 \div 4 = 0.75$ and $15 \div 20 = 0.75$. The divisors $4$ and $20$ are nonzero, so both quotients exist. You can also see it by scaling: multiplying $3$ and $4$ by the same nonzero number, $5$, gives $15$ and $20$ without changing the value of the ratio.

To solve a proportion means to find the value of the unknown that makes the two ratios equal. Take $\frac{x}{20} = \frac{3}{4}$, where $x$ stands for an unknown number and both denominators are nonzero. The denominator $4$ is multiplied by $5$ to make $20$, so the numerator $3$ must be multiplied by $5$ as well: $x = 15$.

Scaling is quick when the multiplier is easy to spot. When it is not, you need a method that does not depend on spotting it.

Let $a$, $b$, $c$ and $d$ stand for numbers, with $b \neq 0$ and $d \neq 0$, and suppose $\frac{a}{b} = \frac{c}{d}$.

Multiply both sides by $bd$. Because $b$ and $d$ are both nonzero, $bd$ is nonzero too. Multiplying both sides of an equation by the same nonzero number, positive or negative, gives an equation with exactly the same solutions.

On the left, $b$ divides out and leaves $ad$. On the right, $d$ divides out and leaves $bc$. The products $ad$ and $bc$ are called cross products, because each numerator is multiplied by the denominator of the other ratio.

A proportion fits a word problem only when the two quantities are in a proportional relationship: every pair of matching values gives the same ratio, so the unit rate is constant. The quantity you divide by must be nonzero. Look for words such as at a constant rate or at the same rate. If the rate changes, a proportion does not apply.

When it does apply:

• Define the variable. Say what it counts, its unit, and which values make sense.
• Write both ratios with the same units in the same positions.
• Solve, then check the answer against the situation.

The table organizes a problem about a pump that runs at a constant rate. It moves $36$ liters in $8$ minutes, and you want the number of minutes, $t$, that it needs to move $99$ liters. A pumping time is positive, so $t > 0$.

Checking also helps you judge a finished solution, including your own. Errors can hide in the setup, as in the warning above, or in the solving steps. To find and correct an error:

• Test the answer. Put it into the original proportion. Use the exact value, because a rounded value is only an approximation and can make equal ratios look slightly different. If the two ratios are not equal, there is an error.
• Trace the lines. For each line, ask what was done to both sides and whether that step keeps the equation true.
• Fix the first wrong line, then finish and check again.

Dev solved $\frac{8}{x} = \frac{12}{21}$, where $x$ stands for an unknown nonzero number. His work is shown below.

Takeaway. A proportion says that two ratios are equal, and every denominator in it must be nonzero. Under that condition the cross products are equal, so you can rewrite the proportion as a simpler equation and divide both sides by the number multiplying the unknown, as long as that number is not zero. In a word problem, use a proportion only when the rate is constant, define your variable with its unit, and keep matching units in matching positions. Always test an answer in the original proportion with its exact value. That test is also how you find errors in a solution.

Next, you will use proportions to solve word problems about scale drawings.