Ratio

A ratio compares amounts by counting equal-sized parts. A recipe, a team, or a box of counters can all use the same idea: the numbers tell you how many parts of each kind there are, while the actual amount can change.

Parents: let your child predict before they reveal each step. The tappable terms and method boxes name the exact ratio language to use in a P6 solution.

By the end you'll be able to interpret, form and simplify ratios, find missing equivalent terms, share an amount, and connect a ratio with a fraction of the whole.

Read a ratio as equal parts

A ratio of 2:3 means 2 equal parts of the first kind for every 3 equal parts of the second kind. It does not say that the actual amounts must be 2 and 3. Add the terms to find the total number of equal parts, then use each term over that total to describe a fraction of the whole.

Show the worked solution

So the whole has 5 equal parts: the first amount is $\frac{2}{5}$ of the total and the second is $\frac{3}{5}$.

Equivalent ratios and a missing term

Two ratios are equivalent when they describe the same comparison. You can multiply or divide every term by one common factor while keeping the terms whole. For example, 4:6 and 6:9 are equivalent because both simplify to 2:3, even though the scale factor between those two written ratios is 3/2.

The same scale factor solves a missing-term question. Find the factor from a known pair, then use it on every term. Keep every ratio term as a whole number and scale all terms, never just one side.

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Build an equivalent ratio

🤔 Predict first: If 3 parts become 12, what must 5 parts become?

82015

Find both equivalent terms

$$3\times4=12$$

$$5\times4=20$$

$$3:5=12:20$$

Simplest form

A ratio is in its simplest form when all its terms have no common factor greater than 1. Divide every term by the greatest common factor. For example, 18:30 becomes 3:5 after dividing both terms by 6.

For three quantities, simplify all three together. The ratio 6:9:15 becomes 2:3:5 after division by 3. The order stays fixed, so the first number still names the first kind of object.

Form and interpret two-part ratios

To form a ratio, put the quantities in the order asked and simplify if possible. If a box has 6 red counters and 10 blue counters, red:blue is 6:10 = 3:5. Reversing the order gives blue:red = 5:3, which is a different statement.

To interpret a two-part ratio, add the terms to find the number of equal parts in the whole. A ratio of 3:2 has 5 parts, so the first amount is 3 out of 5 parts and the second is 2 out of 5 parts.

Form a ratio from three quantities

There are 6 red objects, 9 blue objects and 12 green objects. The question asks for the ratio in red:blue:green order, so write the three quantities in exactly that order before simplifying.

Form and simplify the three-part ratio

$$\text{red:blue:green}=6:9:12$$

$$6:9:12\div3=2:3:4$$

The formed ratio is 6:9:12, and its simplest form is 2:3:4 because every term divides by 3.

Work with three-part ratios

A three-part ratio compares three kinds of amount at once. Use one common scale factor across all three terms to complete a missing term. Keep the order fixed while scaling.

Show the worked solution

For this case, the common scale factor is 4, so 2:3:5 becomes 8:12:20.

Divide an amount in a ratio

To share an amount, first add the ratio terms to find the total units. Then divide the amount by that number to find one unit, and multiply by each share. For $72 in the ratio 4:5, there are 9 units, so one unit is $8. The shares are 4 × $8 = $32 and 5 × $8 = $40.

The same unitary method works for three parts. For $100 in the ratio 2:3:5, there are 10 units, so one unit is $10 and the shares are $20, $30 and $50.

Share $72 in the ratio 4:5

$$4+5=9\text{ units}$$

$$\text{1 unit}=72\div9=8$$

$$\text{Smaller share}=4\times8=32$$

$$\text{Larger share}=5\times8=40$$

Ratio and fraction relationship

The total of the ratio terms becomes the denominator of each fraction of the whole. If boys:girls is 2:3, there are 5 parts, so boys are $\frac{2}{5}$ and girls are $\frac{3}{5}$ of the class. If $\frac{1}{3}$ of a class are boys, the remaining $\frac{2}{3}$ are girls, so boys:girls is 1:2.

Watch out — these are easily mixed up

Three practice problems

1. Complete 3:5 = ?:20.

Use one scale factor

$$5\times4=20$$

$$3\times4=12$$

$$3:5=12:20$$

2. $72 is shared in the ratio 4:5. Find the smaller share.

Find one unit

$$4+5=9\text{ units}$$

$$72\div9=8$$

$$4\times8=32$$

$$\text{Smaller share}=\$32$$

3. Complete 2:3:5 = 8:12:?.

Scale all three terms

$$2\times4=8$$

$$3\times4=12$$

$$5\times4=20$$

$$2:3:5=8:12:20$$

Quick recap

🎯 Mastery check

Answer all 8 — your progress is saved on this device.

1. Which ratio is equivalent to 3:4?

   6:83:84:3

2. What is 18:30 in its simplest form?

   6:103:52:3

3. In the ratio 2:3, what fraction of the whole is the first part?

   $\frac{2}{3}$$\frac{2}{5}$$\frac{3}{5}$

4. Complete 3:5 = ?:20.

   81215

5. Complete 2:3:5 = 8:12:?.

   162024

6. There are 6 red counters and 10 blue counters. What is red:blue in simplest form?

   3:55:36:16

7. $72 is shared in the ratio 4:5. What is the smaller share?

   $32$40$8

8. The ratio of boys to girls is 2:3. What fraction of the class are girls?

   $\frac{2}{5}$$\frac{3}{5}$$\frac{3}{2}$

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