Testing and Solving Proportions | Grade 7 Math | LevelKite

You already know that in a proportional relationship, every pair of positive values has the same unit rate: the amount of one quantity for each $1$ of the other. In this lesson, you'll use that idea to compare just two ratios at a time.

A ratio compares two quantities, such as $6$ cups of flour for every $4$ cups of sugar. A proportion is a statement that two ratios are equal, written as an equation with an equals sign between the ratios.

Every ratio in this lesson meets three conditions, and each rule you learn applies to ratios like these:

The quantities are positive (greater than $0$ ).
You keep the units with the numbers, so you always know what each number measures.
Both ratios put the same kind of quantity, in the same unit, in the same position.

Under these conditions, two ratios are equal when they describe the same relationship: the same amount of the first quantity for each $1$ of the second. Here is a proportion that meets all three conditions.

\frac{6 \text{ cups of flour}}{4 \text{ cups of sugar}} = \frac{15 \text{ cups of flour}}{10 \text{ cups of sugar}}

Read the positions first. Flour is in both numerators (the top quantities) and sugar is in both denominators (the bottom quantities), each measured in cups. If one ratio listed sugar on top, you would rewrite it so the positions match before comparing. These ratios are ready to compare. Is the equation true?

Go back to what a ratio means. Dividing the first quantity by the second gives the unit rate. The second quantity is the per-one quantity, so it is the divisor (the number you divide by), and a divisor can never be $0$ . Here both divisors are positive, so you can divide:

$6 \div 4 = 1.5$ cups of flour per cup of sugar
$15 \div 10 = 1.5$ cups of flour per cup of sugar

Both ratios give $1.5$ cups of flour for each $1$ cup of sugar. They describe the same relationship, so the equation is true.

Unit-rate test. Suppose two ratios compare positive quantities, with the same units in the same positions. They form a proportion exactly when their unit rates are equal.

Unit rates are hard to compare when they are long decimals. A second test uses only multiplication, and it comes straight from the equation.

The units match position by position, so work with the numbers: $\frac{6}{4} = \frac{15}{10}$ . Multiply both sides by both denominators, $4 \cdot 10$ . The denominators are positive, so $4 \cdot 10$ is not $0$ . Multiplying both sides of an equation by the same nonzero number (any number except $0$ ) never changes whether the equation is true.

Left side: $\frac{6}{4} \cdot 4 \cdot 10 = 6 \cdot 10 = 60$
Right side: $\frac{15}{10} \cdot 4 \cdot 10 = 15 \cdot 4 = 60$

The fractions are gone, and each numerator now multiplies the other ratio's denominator. These two products have a name: cross products. Each one is a flour amount times a sugar amount, so comparing them is fair. Dividing both by $4 \cdot 10$ takes you back to the two ratios. So for ratios of positive quantities in matching positions, the ratios are equal exactly when the cross products are equal.

Key concept

A proportion means equal ratios

A proportion is an equation stating that two ratios are equal. When both ratios compare positive quantities, with the same units in the same positions, two tests decide whether they are equal, and the tests always agree:

Same unit rate. Divide each first quantity by its second quantity. The second quantity is the per-one quantity, a divisor that is never $0$ . Equal unit rates mean a proportion.
Equal cross products. Multiply each numerator by the other ratio's denominator. Equal products mean a proportion.

If a test fails, the ratios do not form a proportion. In the formula, $a$ and $b$ are the quantities of one ratio, and $c$ and $d$ are the matching quantities of the other.

\text{For positive } a, b, c, d\text{:} \quad \frac{a}{b} = \frac{c}{d} \quad \text{exactly when} \quad a \cdot d = b \cdot c

Worked example

Testing two ratios

One printer prints $8$ pages in $12$ seconds. Another prints $10$ pages in $15$ seconds. Do the ratios $\frac{8 \text{ pages}}{12 \text{ seconds}}$ and $\frac{10 \text{ pages}}{15 \text{ seconds}}$ form a proportion?

Check the conditions first. Pages are in both numerators, seconds are in both denominators, and all four quantities are positive. The ratios are ready to compare, so write the equation you want to test.
$\frac{8 \text{ pages}}{12 \text{ seconds}} \overset{?}{=} \frac{10 \text{ pages}}{15 \text{ seconds}}$
The unit rates are decimals that never end ( $8 \div 12 = 0.666\ldots$ ), so they are awkward to compare. Use cross products instead. The denominators $12$ and $15$ are positive, so the test works. Multiply the first numerator by the second denominator.
$8 \cdot 15 = 120$
Multiply the first denominator by the second numerator. Each cross product is a number of pages times a number of seconds, so you can compare them fairly.
$12 \cdot 10 = 120$
Compare the cross products. They are equal, so the ratios are equal. This fits the meaning of a proportion. Dividing both quantities of a ratio by the same positive number gives an equal ratio, so $8$ to $12$ becomes $2$ to $3$ (divide by $4$ ), and $10$ to $15$ becomes $2$ to $3$ (divide by $5$ ).
$8 \cdot 15 = 12 \cdot 10 \quad \text{so} \quad \frac{8}{12} = \frac{10}{15}$

Answer

Yes. Both cross products equal $120$ , so the two ratios form a proportion. Each printer prints $2$ pages every $3$ seconds.

Both tests agree in every row. In row 2 the rates are close but not equal, so there is no proportion.

Ratios (same units, same positions)	Unit rates	Cross products	Proportion?
$\frac{9 \text{ meters}}{6 \text{ seconds}}$ and $\frac{12 \text{ meters}}{8 \text{ seconds}}$	$9 \div 6 = 1.5$ and $12 \div 8 = 1.5$ meters per second	$9 \cdot 8 = 72$ and $6 \cdot 12 = 72$	Yes
$\frac{5 \text{ dollars}}{2 \text{ pounds}}$ and $\frac{12 \text{ dollars}}{5 \text{ pounds}}$	$5 \div 2 = 2.5$ and $12 \div 5 = 2.4$ dollars per pound	$5 \cdot 5 = 25$ and $2 \cdot 12 = 24$	No
$\frac{7 \text{ kilometers}}{4 \text{ hours}}$ and $\frac{21 \text{ kilometers}}{12 \text{ hours}}$	$7 \div 4 = 1.75$ and $21 \div 12 = 1.75$ kilometers per hour	$7 \cdot 12 = 84$ and $4 \cdot 21 = 84$	Yes

Adding is not multiplying. Compare $\frac{2 \text{ cups juice}}{3 \text{ cups water}}$ and $\frac{4 \text{ cups juice}}{5 \text{ cups water}}$ . Units and positions match, and all quantities are positive. It is tempting to call them equal because both quantities went up by $2$ . But the cross products are $2 \cdot 5 = 10$ and $3 \cdot 4 = 12$ . They differ, so this is not a proportion.

It fails because the unit rate changed: $2 \div 3 \approx 0.67$ , but $4 \div 5 = 0.8$ cups of juice per cup of water. Adding the same positive amount to two different positive quantities changes the unit rate. Equal ratios come from multiplying both quantities by the same positive number. Doubling gives $4$ cups of juice for $6$ cups of water, and $2 \cdot 6 = 12 = 3 \cdot 4$ .

Now suppose one of the four quantities in a proportion is unknown. Write a letter, called a variable, for the unknown number, and say exactly what it counts and in what unit. Solving a proportion means finding the value of the variable that makes the two ratios equal. That value is the solution.

The conditions stay the same: positive quantities, with the same units in the same positions. The reasoning behind cross products then does the work. Multiplying or dividing both sides of an equation by the same nonzero number never changes whether the equation is true. Every quantity here is positive, so every number you multiply or divide by is positive and can't be $0$ .

Use these moves to get the variable alone on one side. Then check your answer by putting it in place of the variable and testing the two ratios.

Worked example

Solving for an unknown numerator

Apples cost $5$ dollars for $4$ pounds. Let $x$ be the number of dollars that $12$ pounds cost at the same rate. Solve the proportion $\frac{x \text{ dollars}}{12 \text{ pounds}} = \frac{5 \text{ dollars}}{4 \text{ pounds}}$ .

Check the setup. Dollars are in both numerators, pounds are in both denominators, and the quantities are positive, so $x$ is a positive number of dollars. Because the units match position by position, you can solve with the numbers and bring the units back at the end.
$\frac{x}{12} = \frac{5}{4}$
On the left, $x$ is divided by $12$ . Multiply both sides by $12$ to undo that division. Because $12$ is not $0$ , the new equation is true for exactly the same value of $x$ .
$\frac{x}{12} \cdot 12 = \frac{5}{4} \cdot 12$
Simplify each side. On the left, dividing by $12$ and then multiplying by $12$ leaves $x$ . On the right, multiply $5 \cdot 12 = 60$ , then divide $60 \div 4 = 15$ .
$x = \frac{5 \cdot 12}{4} = \frac{60}{4} = 15$
Check by putting $15$ in place of $x$ and comparing unit rates. Both ratios give $1.25$ dollars per pound, so they are equal.
$15 \div 12 = 1.25 \quad \text{and} \quad 5 \div 4 = 1.25$

Answer

The solution is $x = 15$ . At $1.25$ dollars per pound, $12$ pounds of apples cost $15$ dollars.

Worked example

Solving for an unknown denominator

A tap pours $8$ liters of water in $5$ minutes. Let $t$ be the number of minutes it takes to pour $14$ liters at the same rate. Solve the proportion $\frac{8 \text{ liters}}{5 \text{ minutes}} = \frac{14 \text{ liters}}{t \text{ minutes}}$ .

Liters are in both numerators and minutes are in both denominators, so you can work with the numbers. The quantities are positive, so $t$ is positive. That matters because $t$ is a divisor, and a divisor cannot be $0$ .
$\frac{8}{5} = \frac{14}{t}$
Both denominators, $5$ and $t$ , are positive, so they are not $0$ . That means the ratios are equal exactly when the cross products are equal. Multiply each numerator by the other ratio's denominator.
$8 \cdot t = 5 \cdot 14$
Multiply the two known numbers, $5 \cdot 14 = 70$ , and write $8 \cdot t$ as $8t$ .
$8t = 70$
Now $t$ is multiplied by $8$ . Divide both sides by $8$ , which is not $0$ , to get $t$ alone. Then $70 \div 8 = 8.75$ , because $8 \cdot 8 = 64$ leaves $6$ , and $6 \div 8 = 0.75$ .
$\frac{8t}{8} = \frac{70}{8} \quad \text{so} \quad t = 8.75$
Check by putting $8.75$ in place of $t$ . The cross products match, so the ratios are equal.
$8 \cdot 8.75 = 70 \quad \text{and} \quad 5 \cdot 14 = 70$

Answer

The solution is $t = 8.75$ . The tap takes $8.75$ minutes to pour $14$ liters at the same rate.

Takeaway. A proportion says that two ratios are equal. When both ratios compare positive quantities, with the same units in the same positions, you can test them two ways: equal unit rates or equal cross products. To solve a proportion with an unknown, multiply or divide both sides by the same nonzero number until the variable is alone. Then put your answer back in to check that the two ratios really are equal.

Next, you'll set up proportions from word problems and use them to estimate the size of a population.