Introduction to Ratios
Get started with ratios for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-05). A ratio such as \(4:3\) compares amounts of the same kind in a set order, and it ties in closely with fractions — in a drink mixed \(4:3\), the juice is \(\frac{4}{7}\) of the whole.
You will see how ratios and fractions connect, write ratios in simplest form (three-part ratios too), change amounts to the same unit first, and find the ratio between two amounts — everyday skills for recipes, mixing drinks and comparing groups.
Theory
A ratio compares quantities of the same kind, in a fixed order, such as \(4:3\). This Year 12 Standard 1 (NSW) guide shows how to write ratios in simplest form by dividing by the highest common factor, and how to turn a ratio into fractions of the whole.
The ratio \(a:b\) (read "\(a\) to \(b\)") means \(a\) parts of the first quantity for every \(b\) parts of the second. Order matters: write the parts in the order the question names them. A ratio itself has no units.
A ratio is in simplest form when the parts have no common factor. Divide every part by the highest common factor (HCF), for example \(15:25=3:5\) (HCF \(5\)). This also works for three-part ratios.
Before comparing, change both amounts to the same unit: \(1\text{ kg}=1000\text{ g}\), \(1\text{ L}=1000\text{ mL}\), \(1\text{ m}=100\text{ cm}\), \(1\text{ h}=60\text{ min}\), \(\$1=100\) cents. In \(a:b\) there are \(a+b\) parts altogether, so the first quantity is \(\dfrac{a}{a+b}\) of the whole. The ratio of the first quantity to the whole is \(a:(a+b)\), e.g. juice : whole drink \(=4:7\).
Simplest form: divide each part by the HCF.
Ratios to fractions: in \(a:b\) there are \(a+b\) parts altogether.
How to write a ratio in simplest form
- Order the quantities as the question names them.
- Convert to the same unit if needed, then drop the units.
- Divide every part by the highest common factor.
- Check the parts have no common factor left, and multiply back to see you get the original ratio.
The HCF of \(35\) and \(56\) is \(7\). Divide both parts by \(7\).
| \(35\div 7\) | \(=\) | \(5\) |
| \(56\div 7\) | \(=\) | \(8\) |
| \(35:56\) | \(=\) | \(5:8\) |
Check: \(5\times7:8\times7=35:56\). The ratio is \(5:8\).
Change both amounts to cents, then divide by the HCF (\(20\)).
| \(\$3\) | \(=\) | \(300\text{ cents}\) |
| \(80\text{c}:\$3\) | \(=\) | \(80:300\) |
| \(\) | \(=\) | \(4:15\) |
The ratio is \(4:15\).
(ii) In a class, \(\dfrac{2}{9}\) of the students walk to school and the rest catch the bus. Find the ratio walk : bus.
(i) Altos are \(3\) of the \(5+3=8\) parts. (ii) The rest is the fraction that catch the bus.
| \(\text{altos}\) | \(=\) | \(\dfrac{3}{8}\) |
| \(\text{bus}\) | \(=\) | \(1-\dfrac{2}{9}=\dfrac{7}{9}\) |
| \(\text{walk}:\text{bus}\) | \(=\) | \(\dfrac{2}{9}:\dfrac{7}{9}=2:7\) |
Multiply both fractions by \(9\) (same denominator) to get whole numbers.
(i) \(\dfrac{3}{8}\) of the choir (ii) \(2:7\)
| Fruit | Number |
|---|---|
| Mangoes | \(24\) |
| Pears | \(36\) |
| Apples | \(60\) |
(i) The HCF of \(24\), \(36\) and \(60\) is \(12\). (ii) Find apples out of all the fruit.
| \(24:36:60\) | \(=\) | \(2:3:5\) |
| \(\text{total}\) | \(=\) | \(24+36+60=120\) |
| \(\text{apples}\) | \(=\) | \(\dfrac{60}{120}=\dfrac{1}{2}\) |
(i) \(2:3:5\) (ii) \(\dfrac{1}{2}\) of the fruit
Common pitfalls
Frequently asked questions
What is a ratio in maths?
A ratio compares two or more quantities of the same kind in a fixed order. The ratio 4 to 3 means 4 parts of the first quantity for every 3 parts of the second.
How do you simplify a ratio?
Divide every part of the ratio by the highest common factor. For example, 35 to 56 has a highest common factor of 7, so it simplifies to 5 to 8.
Does the order of a ratio matter?
Yes. The parts must be written in the order the question names them. Boys to girls of 2 to 7 is not the same as girls to boys, which is 7 to 2.
How do you turn a ratio into a fraction?
Add the parts to get the total number of parts. Each quantity is its number of parts over the total. In the ratio 5 to 3 there are 8 parts, so the second quantity is 3 eighths of the whole, and the ratio of the second quantity to the whole is 3 to 8.
Do ratios need to be in the same units?
Yes. Convert both amounts to the same unit before you simplify. For example, 80 cents to 3 dollars becomes 80 to 300, which simplifies to 4 to 15.