Simplifying Ratios
A ratio is in simplest form when its terms are whole numbers with no common factor other than \(1\), so \(45:75\) simplifies to \(3:5\). In NSW Year 8 maths this subtopic takes simplifying further than Year 7, with larger numbers and more terms (NESA outcome MA4‑RAT‑C‑01).
You will divide every term by the highest common factor, simplify ratios with three or four terms, clear decimals, fractions and mixed numerals first, and change amounts to the same unit before writing the ratio, so \(40\text{c} : \$2 = 1:5\). Ratios with pronumerals, money, drink recipes and canteen sales all appear, along with comparing one part to the whole. It is part of Year 8 Rates and Ratios.
Theory
To simplify a ratio, make every term a whole number in the same unit, then divide every term by their HCF. \(18:42:48\) has HCF \(6\), so it simplifies to \(3:7:8\). This Year 8 maths page shows how, with decimals, fractions and units.
A ratio is in simplest form when every term is a whole number and the only factor they all share is \(1\). It works for any number of terms.
Finding a large HCF. Write each term as the HCF times something: \(102 = 34 \times 3\) and \(170 = 34 \times 5\), so \(102:170 = 3:5\). Two smaller steps reach the same answer: \(\div 17\) then \(\div 2\).
Decimals. The term with the most decimal places decides the multiplier: one place needs \(\times 10\), two need \(\times 100\). Multiply every term, including whole-number terms, then divide by the HCF.
Fractions. Write mixed numerals as improper fractions, multiply every term by the lowest common denominator, then divide by the HCF.
Different units. Change to the same unit first, then write the ratio without units: \(1\) km \(= 1000\) m, \(1\) m \(= 1000\) mm, \(1\) kg \(= 1000\) g, \(1\) L \(= 1000\) mL, \(1\) h \(= 60\) min, and \(\$1\) is \(100\)c.
Pronumerals. A pronumeral in every term is part of the HCF and divides out; one in only some terms stays: \(10ab:15a = 2b:3\).
Part to whole. The whole is the sum of the parts, so first add them.
The term with the most decimal places decides the multiplier:
| Ratio | Multiply by | Simplest |
|---|---|---|
| \(0.4:1.6\) | 10 | \(1:4\) |
| \(0.24:1.6\) | 100 | \(3:20\) |
| \(2.5:1:0.75\) | 100 | \(10:4:3\) |
Simplifying a ratio
- Units: change every quantity to the same unit.
- Whole numbers: multiply every term by \(10\), \(100\) or the lowest common denominator.
- Divide every term by the HCF.
- Check that the terms share no factor other than \(1\).
| \(2.1 \times 1000\) | \(=\) | \(2100 \text{ mm}\) |
| \(70:2100\) | \(=\) | \(1:30\) |
\(70\) mm : \(2.1\) m \(= 1:30\) (dividing both terms by \(70\)).
\(18 = 3 \times 6\), \(42 = 7 \times 6\), \(48 = 8 \times 6\), so the HCF is \(6\).
| \(18:42:48\) | \(=\) | \(3:7:8\) |
\(3:7:8\)
(i) \(91:117:143\)
(ii) \(46:69:115:92\).
(i) \(91 = 13 \times 7\), \(117 = 13 \times 9\), \(143 = 13 \times 11\): the HCF is \(13\).
(ii) Every term is a multiple of \(23\).
| \(91:117:143\) | \(=\) | \(7:9:11\) |
| \(46:69:115:92\) | \(=\) | \(2:3:5:4\) |
(i) \(7:9:11\)
(ii) \(2:3:5:4\)
(i) \(0.36:0.96\)
(ii) \(2.5:1:0.75\).
Both have two decimal places somewhere, so multiply every term by \(100\), including the \(1\).
| \(0.36:0.96\) | \(=\) | \(36:96\) |
| \(\) | \(=\) | \(3:8\) |
| \(2.5:1:0.75\) | \(=\) | \(250:100:75\) |
| \(\) | \(=\) | \(10:4:3\) |
(i) \(3:8\) \((\div 12)\)
(ii) \(10:4:3\) \((\div 25)\)
(i) \(\dfrac{3}{8}:\dfrac{5}{6}\)
(ii) \(2\dfrac{1}{2}:1\dfrac{1}{4}\).
(i) The lowest common denominator of \(8\) and \(6\) is \(24\).
(ii) \(2\dfrac{1}{2} = \dfrac{5}{2}\) and \(1\dfrac{1}{4} = \dfrac{5}{4}\); multiply both by \(4\).
| \(\dfrac{3}{8} \times 24\) | \(=\) | \(9\) |
| \(\dfrac{5}{6} \times 24\) | \(=\) | \(20\) |
| \(\dfrac{5}{2} \times 4\) | \(=\) | \(10\) |
| \(\dfrac{5}{4} \times 4\) | \(=\) | \(5\) |
(i) \(9:20\)
(ii) \(10:5 = 2:1\)
(i) Write maths : reading : music in simplest form.
(ii) Write music : total time in simplest form.
Use minutes: \(1\) h \(= 60\) min. The HCF of \(45\), \(60\) and \(30\) is \(15\).
| \(45:60:30\) | \(=\) | \(3:4:2\) |
| \(45 + 60 + 30\) | \(=\) | \(135\) |
| \(30:135\) | \(=\) | \(2:9\) |
(i) \(3:4:2\)
(ii) \(2:9\)
(i) \(90\)c : \(\$2.70\)
(ii) \(10ab:15a\)
(iii) length : width : perimeter for a rectangle \(1.2\) m long and \(40\) cm wide.
(i) \(\$2.70\) is \(270\)c.
(ii) The HCF is \(5a\).
(iii) Use cm: \(120\) and \(40\); perimeter \(= 2 \times (120 + 40) = 320\).
| \(90:270\) | \(=\) | \(1:3\) |
| \(10ab:15a\) | \(=\) | \(2b:3\) |
| \(120:40:320\) | \(=\) | \(3:1:8\) |
(i) \(1:3\)
(ii) \(2b:3\)
(iii) \(3:1:8\) \((\div 40)\)
Common pitfalls
Frequently asked questions
What does simplest form mean for a ratio?
Every term is a whole number and the terms have no common factor other than 1.
How do I simplify a ratio with decimals?
Multiply every term by 10 or 100 to make whole numbers, then divide by the HCF.
How do I simplify a ratio of fractions?
Multiply every term by the lowest common denominator, then divide by the HCF.
Why must the units be the same?
A ratio compares amounts of the same kind. 500 g : 2 kg is 500 : 2000 = 1 : 4, not 500 : 2.
Can a ratio have more than two terms?
Yes. Divide every term by the HCF of all of them, such as 18:42:48 = 3:7:8.