Ratios and Unit Rates | Grade 8 Math | LevelKite

Suppose your playlist has $18$ pop songs and $12$ rock songs. You could say it has more pop than rock, but a ratio makes the comparison exact. A ratio compares two quantities by telling you how much of one there is for a given amount of the other. Here, the ratio of pop songs to rock songs is $18$ to $12$ .

You can write a ratio three ways:

with the word to: $18$ to $12$
with a colon: $18:12$
as a fraction: $\frac{18}{12}$

In the fraction form the second number is the denominator, so it must not be zero, because division by zero is undefined.

The two numbers are the terms of the ratio, and their order follows the wording. In the ratio of pop songs to rock songs, pop songs come first, so $18$ is the first term. The ratio of rock songs to pop songs is $12:18$ , a different comparison.

Next, see what happens to the ratio when you split the playlist into equal groups.

The same playlist split into equal groups

Number of equal groups	Pop songs in each group	Rock songs in each group	Ratio of pop to rock	Pop divided by rock
$1$	$18 \div 1 = 18$	$12 \div 1 = 12$	$18:12$	$18 \div 12 = 1.5$
$2$	$18 \div 2 = 9$	$12 \div 2 = 6$	$9:6$	$9 \div 6 = 1.5$
$3$	$18 \div 3 = 6$	$12 \div 3 = 4$	$6:4$	$6 \div 4 = 1.5$
$6$	$18 \div 6 = 3$	$12 \div 6 = 2$	$3:2$	$3 \div 2 = 1.5$

Every row of the table describes the same playlist. Splitting it into equal groups divides both terms by the same nonzero number, the number of groups. Each group has the same mix as the whole playlist, so the comparison does not change: in every row, the number of pop songs is $1.5$ times the number of rock songs. Ratios that describe the same comparison are equivalent ratios.

That is the rule behind simplifying: dividing both terms of a ratio by the same nonzero number gives an equivalent ratio. It is the same move you use to simplify a fraction. The divisor, the number you divide by, must not be zero, because division by zero is undefined. In this lesson every quantity is a positive amount, so every divisor is positive.

A common factor of two whole numbers is a whole number that divides both of them with no remainder. To simplify a ratio of two positive whole numbers, divide both terms by a common factor greater than $1$ . The terms get smaller, but the ratio stays equivalent.

The ratio is in simplest form when the only common factor of its terms is $1$ . To get there in one step, divide both terms by their greatest common factor (GCF), the largest of their common factors. The GCF of two positive whole numbers is at least $1$ , so it is never zero.

For $18:12$ the common factors are $1$ , $2$ , $3$ and $6$ , so the GCF is $6$ . Dividing both terms by $6$ gives $3:2$ , the last row of the table. The playlist has $3$ pop songs for every $2$ rock songs.

Worked example

Write a ratio in simplest form

A bag holds $24$ red tiles and $36$ blue tiles, and no other tiles. Write the ratio of red tiles to all the tiles in the bag in simplest form.

Find the two quantities being compared. A ratio can compare a part to another part or a part to the whole. This one compares a part, the red tiles, to the whole bag, so you need the total. The bag holds only red and blue tiles, so add the two counts.
$24 + 36 = 60$
Write the terms in the order of the wording. In the ratio of red tiles to all the tiles, red tiles come first, so 24 is the first term and the total, 60, is the second term.
$24 : 60$
Find the greatest common factor of the two terms. List the factors of 24, the whole numbers that divide 24 with no remainder: 1, 2, 3, 4, 6, 8, 12 and 24. Start from the largest: 60 divided by 24 leaves a remainder, but 60 divided by 12 is exactly 5, so the GCF of 24 and 60 is 12.
$60 \div 12 = 5, \qquad \text{GCF}(24, 60) = 12$
Divide both terms by 12. The divisor is the same for both terms and is not zero, so the new ratio is equivalent to the first one.
$(24 \div 12) : (60 \div 12) = 2 : 5$
Check the result. The only common factor of 2 and 5 is 1, so the ratio is in simplest form. Multiplying both terms by 12 gives back the original terms.
$2 \times 12 = 24, \qquad 5 \times 12 = 60$

Answer

In simplest form, the ratio of red tiles to all the tiles is $2:5$ , so $2$ out of every $5$ tiles are red.

Some ratios compare quantities measured in different units, such as miles and hours. A ratio like this is a rate. For example, a car that travels $150$ miles in $3$ hours at a steady speed has a rate of $150$ miles to $3$ hours.

A unit rate is a rate whose second quantity is exactly $1$ unit. It tells you how much of the first quantity goes with one unit of the second. The word per means for each one, so $50$ miles per hour means $50$ miles for each $1$ hour.

You already know the move that produces a unit rate: divide both terms by the second term. As long as the second term is not zero, this makes the second term $1$ and keeps the rate equivalent. For the car, dividing both terms by $3$ gives a unit rate of $50$ miles per hour:

\frac{150\ \text{miles}}{3\ \text{hours}} = \frac{(150 \div 3)\ \text{miles}}{(3 \div 3)\ \text{hours}} = \frac{50\ \text{miles}}{1\ \text{hour}}

Key concept

One move, two uses

Dividing both terms of a ratio by the same nonzero number gives an equivalent ratio: both terms change in the same way, so the comparison between them stays the same.

In the formula, $a$ and $b$ are the amounts of the first and second quantities, and $c$ is the number you divide by. Division by zero is undefined, so neither $b$ nor $c$ may be zero.

To simplify: when $a$ and $b$ are positive whole numbers, let $c$ be their greatest common factor.
To find a unit rate: let $c = b$ . The second term becomes $1$ , and the first term, $a \div b$ , is the amount of the first quantity per $1$ unit of the second.

\frac{a}{b} = \frac{a \div c}{b \div c} \quad \text{for } b \neq 0 \text{ and } c \neq 0

Worked example

Find a unit rate

A printer prints $135$ pages in $6$ minutes at a steady pace. Find the unit rate in pages per minute.

Read the units you want. Pages per minute means pages for each 1 minute, so pages are the first term and minutes are the second term.
$\frac{135\ \text{pages}}{6\ \text{minutes}}$
Divide both terms by the second term, 6. The divisor is the same for both terms and is not zero, so the rate stays equivalent. A nonzero number divided by itself is 1, so the second term becomes 1 minute.
$\frac{(135 \div 6)\ \text{pages}}{(6 \div 6)\ \text{minutes}} = \frac{(135 \div 6)\ \text{pages}}{1\ \text{minute}}$
Work out 135 divided by 6. Twenty-two 6s make 132, which leaves 3, and 3 divided by 6 is 0.5. The division ends, so 22.5 is an exact value, not a rounded one.
$135 \div 6 = 22 + \frac{3}{6} = 22.5$
Check by multiplying the unit rate by the number of minutes. The result should be the original number of pages.
$22.5 \times 6 = 135$

Answer

The unit rate is exactly $22.5$ pages per minute, so the printer prints $22.5$ pages for each $1$ minute.

Watch the order. A tap running steadily takes $8$ minutes to fill a $20$ -liter bucket, and you want the rate in liters per minute. A common mistake is to divide the numbers in the order you read them: $8 \div 20 = 0.4$ , reported as $0.4$ liters per minute.

Test it. At $0.4$ liters per minute, $8$ minutes would give only $0.4 \times 8 = 3.2$ liters, not $20$ .

It fails because $8 \div 20$ divides minutes by liters. The result is $0.4$ minute per liter, the time needed to add $1$ liter. That is a true rate, but it answers a different question. Liters per minute puts liters first and minutes second: $20 \div 8 = 2.5$ liters per minute. Both divisors are nonzero, so both rates exist, but only the second has the units you want.

Worked example

Find a unit rate with fractions

A hiker walks $\frac{9}{4}$ miles in $\frac{3}{5}$ of an hour at a steady pace. Find the unit rate in miles per hour.

Miles per hour means miles for each 1 hour, so miles are the first term and hours are the second term.
$\frac{9}{4}\ \text{miles} : \frac{3}{5}\ \text{hour}$
Divide both terms by the second term, three fifths, which is not zero. The second term becomes 1 hour, and the first term becomes the unit rate.
$\left(\frac{9}{4} \div \frac{3}{5}\right)\ \text{miles} : 1\ \text{hour}$
Dividing by a nonzero fraction is the same as multiplying by its reciprocal, the fraction with the numerator and denominator switched. The reciprocal of three fifths is five thirds, and its denominator, 3, is not zero.
$\frac{9}{4} \div \frac{3}{5} = \frac{9}{4} \times \frac{5}{3} = \frac{9 \times 5}{4 \times 3} = \frac{45}{12}$
Simplify. Divide the numerator and the denominator by their greatest common factor, 3. It is the same nonzero divisor for both, so the value does not change.
$\frac{45 \div 3}{12 \div 3} = \frac{15}{4}$
Write the exact value as a decimal. Three 4s make 12, which leaves 3, and 3 divided by 4 is 0.75.
$\frac{15}{4} = 3 + \frac{3}{4} = 3.75$
Check by multiplying the unit rate by the time. After you divide the numerator and the denominator by 5, the result should be the original distance.
$\frac{15}{4} \times \frac{3}{5} = \frac{15 \times 3}{4 \times 5} = \frac{45}{20} = \frac{9}{4}$

Answer

The hiker's unit rate is $\frac{15}{4}$ miles per hour, which is exactly $3.75$ miles per hour.

Takeaway. A ratio compares two quantities in a set order. Dividing both terms by the same nonzero number gives an equivalent ratio, and that one move does two jobs:

Divide by the greatest common factor to write a ratio of positive whole numbers in simplest form.
Divide by the second term, which must not be zero, to get a unit rate: the amount of the first quantity for each $1$ unit of the second.

Let the word per tell you which quantity goes second, and check a unit rate by multiplying it by the second quantity to get back the first.

Next, you will write an equation that says two ratios are equal, called a proportion, and solve it for a missing value.