Simplifying Ratios
Simplifying a ratio means rewriting it with the smallest whole numbers that keep the same comparison, so \(18:24\) becomes \(3:4\). In NSW Year 7 maths this builds directly on finding common factors (NESA outcome MA4‑RAT‑C‑01).
You will divide every term by the highest common factor, simplify ratios with three terms, clear decimals and fractions first, and change measurements to the same unit before comparing, so \(30\text{ cm} : 1\text{ m} = 3:10\). Recipes, paint mixes and concrete all rely on simplified ratios. This subtopic is part of Year 7 Ratios, Rates and Time.
Every question with a fully worked solution.
- Simplifying Ratios - Video - What are Ratios & How to Simplify Them Watch
Theory
A ratio is in simplest form when its terms are whole numbers whose only common factor is \(1\). To simplify, divide every term by the highest common factor (HCF): \(18:24 = 3:4\). This Year 7 maths page also shows ratios with decimals, fractions and different units.
Simplify with the HCF. Find the highest common factor of the terms and divide every term by it, just like simplifying a fraction. \(18:24\): the HCF is \(6\), so \(18:24 = 3:4\).
Check: the new terms have no common factor other than \(1\).
Three terms. Use the HCF of all three terms: \(12:20:32 = 3:5:8\) (divide by \(4\)).
Decimals and fractions. Make whole numbers first. For decimals, multiply every term by \(10\) (or \(100\)): \(0.4:1.2 = 4:12 = 1:3\). For fractions, multiply every term by the lowest common denominator.
Different units. Change to the same unit first, then write the ratio without units: \(30\) cm : \(1\) m \(= 30:100 = 3:10\).
Unit changes you need before simplifying a ratio of measurements:
| Unit change | Unit change |
|---|---|
| \(1\) cm \(= 10\) mm | \(1\) m \(= 100\) cm |
| \(1\) m \(= 1000\) mm | \(1\) km \(= 1000\) m |
| \(1\) kg \(= 1000\) g | \(1\) L \(= 1000\) mL |
| \(1\) h \(= 60\) min | \(1\) week \(= 7\) days |
| \(1\) year \(= 12\) months | \(1\) day \(= 24\) h |
How to simplify a ratio
- Change to the same units, and to whole numbers (multiply decimals by \(10\) or \(100\), fractions by the lowest common denominator).
- Find the HCF of all the terms.
- Divide every term by the HCF.
- Check that the only common factor left is \(1\).
(ii) Write \(35:56\) in simplest form.
Factors of \(35\): \(1, 5, 7, 35\). Factors of \(56\): \(1, 2, 4, 7, 8, 14, 28, 56\).
| \(35 \div 7\) | \(=\) | \(5\) |
| \(56 \div 7\) | \(=\) | \(8\) |
(i) \(7\)
(ii) \(5:8\)
(i) \(32:24\)
(ii) \(18:30:42\)
(iii) \(18:42:48\)
| \(32\) | \(=\) | \(8 \times 4\) |
| \(24\) | \(=\) | \(8 \times 3\) |
| \(18\) | \(=\) | \(6 \times 3\) |
| \(30\) | \(=\) | \(6 \times 5\) |
| \(42\) | \(=\) | \(6 \times 7\) |
| \(48\) | \(=\) | \(6 \times 8\) |
(i) \(4:3\) (HCF \(8\))
(ii) \(3:5:7\) (HCF \(6\))
(iii) \(3:7:8\) (HCF \(6\))
(i) \(0.9:2.4\)
(ii) \(\dfrac{3}{4}:\dfrac{5}{8}\)
(i) Multiply both terms by \(10\), then divide by the HCF \(3\).
(ii) The lowest common denominator is \(8\). Multiply both terms by \(8\).
| \(0.9 \times 10\) | \(=\) | \(9\) |
| \(2.4 \times 10\) | \(=\) | \(24\) |
| \(\dfrac{3}{4} \times 8\) | \(=\) | \(6\) |
| \(\dfrac{5}{8} \times 8\) | \(=\) | \(5\) |
(i) \(9:24 = 3:8\)
(ii) \(6:5\) (already in simplest form)
Change to the same unit: \(1\) m \(= 1000\) mm.
| \(2.1 \times 1000\) | \(=\) | \(2100\) |
| \(70 \div 70\) | \(=\) | \(1\) |
| \(2100 \div 70\) | \(=\) | \(30\) |
\(70:2100 = 1:30\) (the HCF is \(70\)).
(i) Write walk : bus in simplest form.
(ii) Write walk : whole trip in simplest form.
| \(6 \times 1000\) | \(=\) | \(6000\) |
| \(800 \div 400\) | \(=\) | \(2\) |
| \(6000 \div 400\) | \(=\) | \(15\) |
| \(800 + 6000\) | \(=\) | \(6800\) |
| \(6800 \div 400\) | \(=\) | \(17\) |
(i) \(2:15\)
(ii) \(2:17\)
| \(1.25 \times 60\) | \(=\) | \(75\) |
| \(2 \times 60\) | \(=\) | \(120\) |
| \(30 \div 15\) | \(=\) | \(2\) |
| \(75 \div 15\) | \(=\) | \(5\) |
| \(120 \div 15\) | \(=\) | \(8\) |
\(30:75:120 = 2:5:8\) (the HCF is \(15\)).
Common pitfalls
Frequently asked questions
How do you simplify a ratio?
Divide every term by the highest common factor. \(35:56 = 5:8\) (divide by \(7\)).
How do you simplify a ratio with decimals?
Multiply every term by \(10\) or \(100\) to make whole numbers, then divide by the HCF. \(0.9:2.4 = 9:24 = 3:8\).
How do you simplify a ratio with fractions?
Multiply every term by the lowest common denominator. \(\dfrac{3}{4}:\dfrac{5}{8} = 6:5\).
What if the units are different?
Change to the same unit first. \(70\) mm : \(2.1\) m \(= 70:2100 = 1:30\).
How do I know a ratio is in simplest form?
The terms are whole numbers and their only common factor is 1.