Ratios
A ratio compares two or more amounts of the same kind, such as 5 red counters to 2 blue counters, written \(5:2\). This first ratio subtopic in NSW Year 7 maths shows why the order of the terms matters and how a ratio splits a whole into equal parts (NESA outcome MA4‑RAT‑C‑01).
You will write ratios from pictures, tables and words, tell part-to-part from part-to-whole, turn a ratio into a fraction of the total, and find missing terms in equivalent ratios by multiplying every term by the same number. Mixing cordial so every glass tastes the same is a ratio at work. This subtopic is part of Year 7 Ratios, Rates and Time.
Every question with a fully worked solution.
- Ratios - Video - Equivalent Ratios Watch
Theory
A ratio compares two or more quantities measured in the same units. The ratio \(3:5\) is read “\(3\) to \(5\)”. This Year 7 maths page shows how to write a ratio in the right order, turn it into a fraction of the whole, and find missing terms in equivalent ratios.
The numbers in a ratio are its terms. Order matters: the order of the words gives the order of the numbers. With \(5\) red and \(2\) blue counters, red : blue \(= 5:2\), but blue : red \(= 2:5\).
Part : part and part : whole. Red : blue compares a part with a part. Red : all counters \(= 5:7\) compares a part with the whole.
A ratio as a fraction. Each term is a number of equal parts. Add the terms to find the total number of parts. With cats : dogs \(= 4:3\) there are \(4 + 3 = 7\) parts, so \(\dfrac{4}{7}\) of the animals are cats and \(\dfrac{3}{7}\) are dogs.
Equivalent ratios. Multiply or divide every term by the same number: \(2:3 = 4:6 = 6:9\). To find a missing term, find the number that turns one term into the other, then do the same to the other term.
Not a whole number? Simplify first, or allow a fraction or decimal. For \(6:9 = 8:\,?\), \(6:9 = 2:3\), and \(2 \times 4 = 8\), so the missing term is \(3 \times 4 = 12\). Terms can also become decimals: \(4:3:7 = 2:1.5:3.5\) (each term halved).
Equivalent ratios and fractions of the whole:
How to write a ratio
- Count each quantity (same units).
- Write the numbers in the same order as the words, with a colon between them.
- For a fraction of the whole, add the terms to get the total parts.
How to find a missing term
- Find the pair of matching terms you know (for example \(5\) and \(20\)).
- Find the multiplier: \(20 \div 5 = 4\).
- Multiply the other term by the same number, then check.
(i) tomatoes to lettuces
(ii) lettuces to tomatoes
(iii) tomatoes to all the seedlings
| \(5 + 8\) | \(=\) | \(13\) |
(i) \(5:8\)
(ii) \(8:5\)
(iii) \(5:13\)
\(3\) has been multiplied by \(3\) to make \(9\). Do the same to \(4\). (As fractions: \(\dfrac{4}{3} = \dfrac{x}{9}\).)
| \(9 \div 3\) | \(=\) | \(3\) |
| \(4 \times 3\) | \(=\) | \(12\) |
\(x = 12\), so \(4:3 = 12:9\).
(i) \(5:6 = 20:\,?\)
(ii) \(7:2 = \,?:10\)
| \(20 \div 5\) | \(=\) | \(4\) |
| \(6 \times 4\) | \(=\) | \(24\) |
| \(10 \div 2\) | \(=\) | \(5\) |
| \(7 \times 5\) | \(=\) | \(35\) |
(i) \(24\)
(ii) \(35\)
\(4\) has been divided by \(2\) to make \(2\). Divide every term by \(2\).
| \(3 \div 2\) | \(=\) | \(1.5\) |
| \(7 \div 2\) | \(=\) | \(3.5\) |
\(x = 1.5\) and \(y = 3.5\), so \(4:3:7 = 2:1.5:3.5\).
(i) What fraction of the punch is lemonade?
(ii) A big batch uses \(8\) cups of orange juice. How many cups of pineapple juice and lemonade does it need?
| \(2 + 3 + 4\) | \(=\) | \(9\) |
| \(8 \div 2\) | \(=\) | \(4\) |
| \(3 \times 4\) | \(=\) | \(12\) |
| \(4 \times 4\) | \(=\) | \(16\) |
(i) \(\dfrac{4}{9}\)
(ii) \(2:3:4 = 8:12:16\), so \(12\) cups of pineapple juice and \(16\) cups of lemonade.
(i) \(9\) L red and \(12\) L white
(ii) \(5\) L red and \(6\) L white
| \(9 \div 3\) | \(=\) | \(3\) |
| \(12 \div 4\) | \(=\) | \(3\) |
| \(5 \div 3\) | \(=\) | \(1\dfrac{2}{3}\) |
| \(6 \div 4\) | \(=\) | \(1\dfrac{1}{2}\) |
(i) Yes: both terms are multiplied by \(3\), so \(9:12 = 3:4\).
(ii) No: \(2\) L was added to each term, which does not keep the ratio.
\(6\) does not divide into \(8\). Simplify first: \(6:9 = 2:3\) (divide by \(3\)).
| \(8 \div 2\) | \(=\) | \(4\) |
| \(3 \times 4\) | \(=\) | \(12\) |
\(6:9 = 8:12\)
Common pitfalls
Frequently asked questions
What is a ratio?
A comparison of quantities in the same units, written with a colon, such as \(3:5\).
Does the order of a ratio matter?
Yes. The order of the numbers follows the order of the words: red to blue \(5:2\) is not blue to red \(2:5\).
How do you find an equivalent ratio?
Multiply or divide every term by the same number: \(2:3 = 6:9\).
How do you change a ratio into a fraction?
Add the terms to find the total parts. In cats : dogs \(= 4:3\), cats are \(\dfrac{4}{7}\) of the whole.
How do you find a missing term in a ratio?
Find what one term was multiplied by, then multiply the other term by the same number. \(5:6 = 20:24\).