Equivalent Ratios and Ratio Tables | Grade 6 Math | LevelKite

A ratio compares two quantities. Suppose you make green paint with $2$ cups of blue paint for every $3$ cups of yellow paint. The ratio of blue to yellow is $2:3$ , which you read as 2 to 3. The two numbers are the terms of the ratio. Their order matters: here the first term counts cups of blue and the second term counts cups of yellow.

Now suppose you need more paint in exactly the same shade. How many cups of each color should you use? In this lesson, you will learn how to find amounts that keep the same comparison. Then you will check whether two ratios match, write a matching ratio of your own, and fill in a table of matching ratios.

Think of $2$ cups of blue and $3$ cups of yellow as one batch of green paint.

Two batches use $2 \times 2 = 4$ cups of blue and $3 \times 2 = 6$ cups of yellow.
Four batches use $2 \times 4 = 8$ cups of blue and $3 \times 4 = 12$ cups of yellow.

Every batch has the same mix inside it, so the shade stays the same. Only the total amount of paint changes. That is why $2:3$ , $4:6$ and $8:12$ all describe the same comparison: $2$ cups of blue for every $3$ cups of yellow. Ratios like these are called equivalent ratios. You can use an equals sign to show that two ratios are equivalent:

2 : 3 = (2 \times 4) : (3 \times 4) = 8 : 12

Key concept

Equivalent ratios keep the same comparison

Two ratios are equivalent when they describe the same comparison between two quantities.

Multiplying both terms of a ratio by the same nonzero number gives an equivalent ratio. Nonzero means not $0$ : multiplying by $0$ would give $0:0$ , which compares nothing. Dividing both terms by the same nonzero number also gives an equivalent ratio, because dividing undoes multiplying. If you multiply both terms of the new ratio by that number, you return to the ratio you started with.

In the formula, $a$ and $b$ are the two terms of a ratio and $n$ is the number you multiply both terms by. In this lesson, $a$ and $b$ are whole numbers greater than $0$ .

a : b = (a \times n) : (b \times n), \quad n \neq 0

Here is how to check whether two ratios are equivalent. Compare matching terms: the first term with the first term, and the second term with the second term.

Look at the two first terms. Find the number that one was multiplied or divided by to make the other. When you multiply, this number is called the multiplier.
Do the same thing, with the same number, to the second term of the ratio you started from.

If you get the second term of the other ratio, the two ratios are equivalent. If you get a different number, they are not equivalent. The check works because equivalent ratios use the same number on both terms. It applies when none of the terms is $0$ , which is true for every ratio in this lesson.

Worked example

Checking three ratios

Which of these ratios are equivalent to $6:8$ ? The choices are $18:24$ , $12:14$ and $3:4$ .

Test 18:24. Compare the first terms: 6 becomes 18 when you multiply by 3. Now use the same multiplier on the second term: 8 times 3 is 24, and 24 is the second term of 18:24. Both terms were multiplied by the same nonzero number, so 18:24 is equivalent to 6:8.
$6 \times 3 = 18 \qquad 8 \times 3 = 24$
Test 12:14. Compare the first terms: 6 becomes 12 when you multiply by 2. Use the same multiplier on the second term: 8 times 2 is 16. But the second term of 12:14 is 14, not 16. The two terms were not multiplied by the same number, so 12:14 is not equivalent to 6:8.
$6 \times 2 = 12 \qquad 8 \times 2 = 16 \neq 14$
Test 3:4. These terms are smaller than 6 and 8, so look for a division. Compare the first terms: 6 becomes 3 when you divide by 2. Divide the second term by 2 as well: 8 divided by 2 is 4, and 4 is the second term of 3:4. Both terms were divided by the same nonzero number, so 3:4 is equivalent to 6:8.
$6 \div 2 = 3 \qquad 8 \div 2 = 4$

Answer

The ratios $18:24$ and $3:4$ are equivalent to $6:8$ . The ratio $12:14$ is not equivalent to $6:8$ .

To write an equivalent ratio, pick a number and use it on both terms.

To get larger terms, multiply both terms by the same whole number greater than $1$ . For example, $5:8 = (5 \times 3):(8 \times 3) = 15:24$ .
To get smaller whole-number terms, divide both terms by a common factor greater than $1$ , if they share one. A common factor is a whole number that divides both terms with no remainder. For example, $10$ divides both $20$ and $30$ , so $20:30 = (20 \div 10):(30 \div 10) = 2:3$ .

Sometimes one term of the new ratio is already chosen for you. Then that term tells you which number to use, as the next example shows.

Worked example

Writing a ratio with a given term

Write a ratio that is equivalent to $3:7$ and has $28$ as its second term.

Find the multiplier from the terms you know. The second term changes from 7 to 28, so ask: 7 times what number is 28? Division undoes multiplication, so divide 28 by 7. The multiplier is 4.
$28 \div 7 = 4$
Use the same multiplier on the first term. An equivalent ratio needs both terms multiplied by the same nonzero number, so multiply 3 by 4.
$3 \times 4 = 12$
Write the new ratio. Keep the terms in the same order: the new first term is 12 and the new second term is 28.
$3 : 7 = (3 \times 4) : (7 \times 4) = 12 : 28$
Check by undoing the multiplication. Dividing both terms of 12:28 by 4 should bring back the ratio you started with, and it does.
$12 \div 4 = 3 \qquad 28 \div 4 = 7$

Answer

The ratio $12:28$ is equivalent to $3:7$ and has $28$ as its second term.

A ratio table is a table of pairs of values that all have the same ratio. In the ratio tables in this lesson, each column counts one of the two quantities and each row holds one pair of values. That makes every row a ratio that is equivalent to every other row.

The ratio table below is for the green paint. The first column counts cups of blue paint and the second column counts cups of yellow paint. Each row is one amount of green paint you could mix. From top to bottom, the rows show $1$ , $2$ , $3$ and $4$ batches.

Cups of blue and yellow paint in 1, 2, 3 and 4 batches of green paint

Cups of blue paint	Cups of yellow paint
$2$	$3$
$4$	$6$
$6$	$9$
$8$	$12$

Read across a row to see one ratio. Compare two rows to see a multiplier. From the first row to the fourth row, both columns are multiplied by $4$ : $2 \times 4 = 8$ cups of blue and $3 \times 4 = 12$ cups of yellow.

Any two rows are equivalent to each other, even when the multiplier between them is not a whole number. Look at the rows $4:6$ and $6:9$ . The multiplier is $1.5$ , because $4 \times 1.5 = 6$ and $6 \times 1.5 = 9$ , but that is hard to spot. An easier way is to divide each row down. Dividing both terms of $4:6$ by $2$ gives $2:3$ . Dividing both terms of $6:9$ by $3$ also gives $2:3$ . Both rows mean $2$ cups of blue for every $3$ cups of yellow, so they describe the same mix.

To complete a ratio table, start from a row that has both values. In the unfinished row, find the value you know and note its column. Work out what the complete row's value in that column was multiplied or divided by to make it. Then do the same thing to the complete row's value in the other column.

Any complete row works as a starting row, because all the rows are equivalent. Pick the one that makes the arithmetic easy.

Try the method on the table below. A craft club makes bead chains of different lengths. Every chain repeats the same pattern of red and white beads, so the ratio of red beads to white beads is the same for every chain.

Each row of the table is one chain. The first column counts the red beads in that chain and the second column counts the white beads. A question mark shows a missing value.

Red beads and white beads in four bead chains

Red beads	White beads
$6$	$15$
$18$	?
?	$5$
$10$	?

Worked example

Completing a ratio table

In the table just above, each row is one bead chain. The first column counts red beads and the second column counts white beads. Every chain has the same ratio of red beads to white beads. The rows are: $6$ red and $15$ white; $18$ red and a missing number of white; a missing number of red and $5$ white; $10$ red and a missing number of white. Find the three missing values.

Start from the complete row: 6 red and 15 white. In the second row you know the red beads, 18. In the red column, 6 becomes 18 when you multiply by 3. Every row keeps the same ratio, so multiply the white beads by 3 as well.
$6 \times 3 = 18 \qquad 15 \times 3 = 45$
In the third row you know the white beads, 5. In the white column, 15 becomes 5 when you divide by 3. Divide the red beads by 3 as well, because dividing both terms by the same nonzero number keeps the ratio equivalent.
$15 \div 3 = 5 \qquad 6 \div 3 = 2$
In the fourth row you know the red beads, 10. No whole number times 6 gives 10, so the first row is awkward here. Any complete row works, because all the rows are equivalent, so use the third row, which is now 2 red and 5 white. In the red column, 2 becomes 10 when you multiply by 5. Multiply the white beads by 5 as well.
$2 \times 5 = 10 \qquad 5 \times 5 = 25$
Check the second and fourth rows against the third row, 2 red and 5 white. Divide both terms of 18:45 by 9, and divide both terms of 10:25 by 5. Each one returns 2:5, so every row has the same ratio.
$18 \div 9 = 2, \quad 45 \div 9 = 5, \quad 10 \div 5 = 2, \quad 25 \div 5 = 5$

Answer

The second row has $45$ white beads, the third row has $2$ red beads and the fourth row has $25$ white beads.

One idea did all the work in this lesson: equivalent ratios describe the same comparison. To check one or to write one, use the same nonzero number on both terms, multiplying both or dividing both. A ratio table is a list of equivalent ratios, so one complete row lets you find a missing value in any other row that already has one value.

Next, you will use equivalent ratios to solve word problems and plot equivalent ratios on the coordinate plane.