Equivalent Ratios and Unit Rates | Grade 7 Math | LevelKite

A ratio compares two quantities by telling how much of one goes with a given amount of the other. In this lesson, both quantities are always positive, and each one keeps its unit.

Here is a drink recipe: mix $5$ cups of water with $2$ cups of juice. The ratio of water to juice is $5$ cups to $2$ cups. You can write it as $5 : 2$ and read it as 5 to 2.

Order matters. In the ratio of water to juice, the first number belongs to water and the second belongs to juice. The ratio $2 : 5$ would describe a different drink, with $2$ cups of water for $5$ cups of juice.

Suppose you make $3$ batches of the drink. Each batch needs $5$ cups of water and $2$ cups of juice, so $3$ batches need $3 \times 5 = 15$ cups of water and $3 \times 2 = 6$ cups of juice. The drink tastes the same, because every cup of juice still gets the same amount of water.

The ratios $5 : 2$ and $15 : 6$ are equivalent ratios. Two ratios are equivalent when they compare the same two quantities, such as water and juice, in the same order and the same units, and you can get one from the other by multiplying both positive amounts by the same positive number. That number is the scale factor. Here the scale factor is $3$ .

Batches of the same drink

Scale factor	Water (cups)	Juice (cups)	Water : juice
$\frac{1}{2}$	$\frac{1}{2} \times 5 = 2.5$	$\frac{1}{2} \times 2 = 1$	$2.5 : 1$
$1$	$1 \times 5 = 5$	$1 \times 2 = 2$	$5 : 2$
$2$	$2 \times 5 = 10$	$2 \times 2 = 4$	$10 : 4$
$3$	$3 \times 5 = 15$	$3 \times 2 = 6$	$15 : 6$
$10$	$10 \times 5 = 50$	$10 \times 2 = 20$	$50 : 20$

Every ratio in the table is equivalent to $5 : 2$ , because each row multiplies both $5$ cups of water and $2$ cups of juice by the same scale factor.

A scale factor must be positive, so that both amounts stay positive, but it does not have to be a whole number. The first row is half a batch. Its scale factor is $\frac{1}{2}$ , and multiplying by $\frac{1}{2}$ is the same as dividing by $2$ .

To decide whether two ratios are equivalent, first make sure they compare the same two quantities in the same order and the same units. Then look for one positive scale factor that works for both quantities. You find a scale factor by dividing a new amount by the matching original amount, which is positive and so is not zero. If the two quantities need different scale factors, the ratios are not equivalent.

Worked example

Write and check equivalent ratios

A trail mix uses $12$ cups of oats for every $8$ cups of raisins. Write the equivalent ratio of oats to raisins for a mix that uses only $2$ cups of raisins. Then decide whether $30$ cups of oats to $20$ cups of raisins is also equivalent to $12 : 8$ .

Name the quantities and fix their order. The ratio is oats to raisins, and both positive amounts are in cups. Oats stays first and raisins stays second in every ratio you write, so matching quantities stay in matching positions.
$12 \text{ cups of oats} : 8 \text{ cups of raisins}$
Find the scale factor that turns 8 cups of raisins into 2 cups. Divide the new amount by the original amount. You are dividing by 8, which is not zero, so the division is allowed.
$2 \div 8 = \frac{2}{8} = \frac{1}{4}$
Multiply the oats by the same scale factor, one fourth, which is the same as dividing by 4. Scaling both positive amounts by the same positive number keeps the same amount of oats for each cup of raisins. The new ratio is 3 cups of oats to 2 cups of raisins.
$\frac{1}{4} \times 12 = 12 \div 4 = 3$
Now test 30 cups of oats to 20 cups of raisins. Its order and units match the original ratio. Find the scale factor for the oats by dividing the new amount, 30, by the original amount, 12, which is not zero.
$30 \div 12 = \frac{30}{12} = \frac{5}{2} = 2.5$
Use the same scale factor on the raisins. It gives exactly 20 cups, so one positive scale factor works for both quantities and the ratios are equivalent.
$2.5 \times 8 = 20$

Answer

A mix with $2$ cups of raisins needs $3$ cups of oats, so $3 : 2$ is equivalent to $12 : 8$ . The ratio $30 : 20$ is also equivalent to $12 : 8$ , because the scale factor $2.5$ works for both quantities.

Some questions ask about just one unit: how much water goes with $1$ cup of juice? The half batch in the table answers this. It has $2.5$ cups of water for exactly $1$ cup of juice.

That amount is a unit rate: the amount of one quantity that goes with exactly $1$ unit of the other quantity. You say it with the word per, which means for each one: $2.5$ cups of water per cup of juice. The quantity named after per is the per-one quantity. Here it is the juice.

A ratio of two quantities with different units, such as kilometers and hours, is a rate. A rate has a unit rate too, such as kilometers per hour.

To find the drink's unit rate, divide the $5$ cups of water by the $2$ cups of juice: $5 \div 2 = 2.5$ . You divide by the amount of the per-one quantity.

Here is why this works. Dividing both quantities of $5 : 2$ by $2$ is the same as multiplying both by the positive scale factor $\frac{1}{2}$ . So you get an equivalent ratio, $2.5 : 1$ , in which the per-one quantity is exactly $1$ .

The amount of the per-one quantity is always the divisor, the number you divide by, and it must not be zero. Every quantity in this lesson is positive, so the divisor is never zero. If you divide the other way, $2 \div 5 = 0.4$ , you get a different unit rate: $0.4$ cups of juice per cup of water. So check which quantity comes after per.

Keep both units in your answer, because $2.5$ alone does not say what you are comparing. The same steps work for any ratio or rate of two positive quantities:

\text{unit rate} = \frac{\text{amount of the other quantity}}{\text{amount of the per-one quantity}} \qquad (\text{amounts positive, divisor} \neq 0)

Worked example

Find a unit rate

A hiker walks $18$ kilometers in $4$ hours at a steady pace. What is the hiker's unit rate in kilometers per hour?

Name the quantities with their units and find the per-one quantity. The question asks for kilometers per hour, so hours is the per-one quantity. Its amount, 4, is the divisor. It is positive, so it is not zero.
$18 \text{ kilometers} : 4 \text{ hours}$
Divide both quantities by 4. This multiplies both by the same positive scale factor, one fourth, so the new ratio is equivalent to the original one and has exactly 1 hour.
$18 \div 4 = \frac{18}{4} = \frac{9}{2} = 4.5 \qquad 4 \div 4 = 1$
Write the unit rate with both units, so it says what you are comparing.
$\frac{18 \text{ kilometers}}{4 \text{ hours}} = 4.5 \text{ kilometers per hour}$
Check by scaling back up. At 4.5 kilometers for each hour, 4 hours should give the original 18 kilometers, and it does.
$4 \times 4.5 = 18$

Answer

The hiker's unit rate is $4.5$ kilometers per hour.

Key concept

Equivalent ratios share one unit rate

Equivalent ratios are the same comparison at different sizes. Multiplying both positive amounts of a ratio by the same positive scale factor changes the amounts but not the unit rate.

In symbols, let $a$ and $b$ be the positive amounts in a ratio $a : b$ , each with its own unit, and let $k$ be a positive scale factor. Take the second quantity as the per-one quantity, so $b$ is the divisor and is not zero. The unit rate is $a \div b$ . The scaled ratio $(k \cdot a) : (k \cdot b)$ has the same unit rate, because the nonzero factor $k$ is in both parts of the division and divides out.

So ratios that compare the same two quantities in matching positions, with matching units, are equivalent exactly when their unit rates are equal.

\frac{k \cdot a}{k \cdot b} = \frac{a}{b} \quad \text{for } a > 0,\ b > 0,\ k > 0

Worked example

Use unit rates to find equivalent ratios

Each of three taps runs at a steady pace. Tap A delivers $45$ liters in $6$ minutes, Tap B delivers $60$ liters in $8$ minutes, and Tap C delivers $72$ liters in $10$ minutes. Which taps have equivalent ratios of liters to minutes?

Match the ratios before you compare them. Write each one as liters to minutes, so liters is first and minutes is second every time, with the same units. Minutes is the per-one quantity. The divisors 6, 8 and 10 are positive, so none of them is zero.
$45 : 6 \qquad 60 : 8 \qquad 72 : 10$
Find Tap A's unit rate. Divide its liters by its minutes, because minutes is the per-one quantity. Check: 6 times 7.5 is 45.
$\frac{45 \text{ liters}}{6 \text{ minutes}} = 7.5 \text{ liters per minute}$
Do the same for Tap B. Check: 8 times 7.5 is 60.
$\frac{60 \text{ liters}}{8 \text{ minutes}} = 7.5 \text{ liters per minute}$
Do the same for Tap C. Check: 10 times 7.2 is 72.
$\frac{72 \text{ liters}}{10 \text{ minutes}} = 7.2 \text{ liters per minute}$
Compare the unit rates. Taps A and B match, so their ratios are equivalent. Confirm it with a scale factor: Tap B's liters are four thirds of Tap A's, and four thirds of 6 minutes is 8 minutes. Tap C's unit rate is different, so its ratio is not equivalent to the other two.
$60 \div 45 = \frac{60}{45} = \frac{4}{3} \qquad \frac{4}{3} \times 6 = 8$

Answer

Taps A and B have equivalent ratios, $45 : 6$ and $60 : 8$ , because both have a unit rate of $7.5$ liters per minute. Tap C's ratio, $72 : 10$ , is not equivalent to them, because its unit rate is $7.2$ liters per minute.

Here is the idea to keep. Multiplying both positive quantities of a ratio by the same positive scale factor gives an equivalent ratio, and it keeps the unit rate the same. To find a unit rate, divide by the amount of the per-one quantity, which must not be zero, and keep the units in your answer. Ratios that compare the same two quantities in matching positions, with matching units, are equivalent exactly when they have the same unit rate for the same per-one quantity.

Next, you will compute unit rates with fractions and unit rates for quantities measured in different units.