Ratios & Dividing Quantities
Master ratios and dividing quantities for NSW Year 12 Mathematics Standard 2. A ratio compares quantities of the same kind, and you learn to write one in simplest form, compare two quantities in the same units, and use equivalent ratios to scale amounts up or down.
The key skill is to divide a quantity in a given ratio with the one-part method: add the parts, find the value of one part, then multiply. You will apply it to sharing money, scaling recipes and mixing materials — a practical Standard 2 problem-solving skill.
Theory
Ratios compare quantities of the same kind and are a core skill in Year 12 Standard 2 (NSW). This guide shows how to write a ratio in simplest form, compare two quantities in the same units, use equivalent ratios, and divide a quantity in a given ratio using the one-part method for sharing money, recipes and mixtures.
A ratio compares two or more quantities of the same kind, written with a colon such as \(3:5\). Order matters: \(3:5\) is not the same as \(5:3\). Before you write or simplify a ratio, put the quantities in the same unit.
A ratio is in simplest form when its parts are whole numbers with no common factor left — divide every part by their highest common factor (HCF). Ratios that simplify to the same thing, like \(2:8\) and \(1:4\), are equivalent ratios.
To divide a quantity in a given ratio you use the one-part method: add the ratio numbers to get the total parts, divide the quantity by the total parts to find one part, then multiply. This Year 12 Standard 2 (NSW) skill is used for sharing money, scaling recipes and mixing materials.
For a quantity \(Q\) divided in the ratio \(a:b:c\):
Each share is that ratio number times one part, for example the first share is:
How to divide a quantity in a given ratio
- Same units. Make sure every quantity is in the same unit, and keep the ratio parts in the order given.
- Total parts. Add the ratio numbers together.
- One part. Divide the quantity by the total number of parts.
- Each share. Multiply one part by each ratio number, then check the shares add back to the original quantity.
Convert to the same unit, then divide both parts by their HCF.
| \(2\text{ h}\) | \(=\) | \(120\text{ min}\) |
| \(45:120\) | \(=\) | \(\dfrac{45}{15}:\dfrac{120}{15}\) |
| \(\) | \(=\) | \(3:8\) |
The ratio in simplest form is \(3:8\).
Add the parts, find one part, then multiply each ratio number by it.
| \(\text{total parts}\) | \(=\) | \(2+3+5 = 10\) |
| \(\text{one part}\) | \(=\) | \(\dfrac{\$600}{10} = \$60\) |
| \(\text{shares}\) | \(=\) | \(2\times\$60,\ 3\times\$60,\ 5\times\$60\) |
| \(\) | \(=\) | \(\$120,\ \$180,\ \$300\) |
The shares are \(\$120\), \(\$180\) and \(\$300\) (they add to \(\$600\)).
Juice is \(2\) parts, so find one part first.
| \(2\text{ parts}\) | \(=\) | \(500\text{ mL}\) |
| \(\text{one part}\) | \(=\) | \(\dfrac{500}{2} = 250\text{ mL}\) |
| \(\text{soda water}\) | \(=\) | \(3\times250 = 750\text{ mL}\) |
| \(\text{total}\) | \(=\) | \(500+750 = 1250\text{ mL}\) |
\(750\) mL of soda water, for a total of \(1250\) mL (\(1.25\) L) of punch.
Gravel is \(4\) parts, so work out one part, then the rest.
| \(4\text{ parts}\) | \(=\) | \(15\text{ kg}\) |
| \(\text{one part}\) | \(=\) | \(\dfrac{15}{4} = 3.75\text{ kg}\) |
| \(\text{cement}\) | \(=\) | \(1\times3.75 = 3.75\text{ kg}\) |
| \(\text{total mass}\) | \(=\) | \(7\times3.75 = 26.25\text{ kg}\) |
Cement is \(3.75\) kg and the total mass is \(26.25\) kg.
Common pitfalls
Frequently asked questions
How do you divide a quantity in a given ratio?
Add the ratio numbers to get the total number of parts, divide the quantity by that total to find the value of one part, then multiply one part by each ratio number. The shares should add back to the original quantity.
How do you simplify a ratio?
Divide every part of the ratio by their highest common factor (HCF). For example, 24:36 both divide by 12 to give 2:3. Keep dividing until the parts share no common factor other than 1.
Does the order of a ratio matter?
Yes. A ratio of 3:5 is different from 5:3 because each number is tied to a particular quantity. Always write the parts in the same order as the quantities are named in the question.
What are equivalent ratios?
Equivalent ratios are ratios that simplify to the same thing, such as 1:4, 2:8 and 3:12. You get an equivalent ratio by multiplying or dividing every part by the same number, which is how you scale a recipe up or down.
How do you write two quantities as a ratio when the units differ?
Convert both quantities to the same unit first, then write and simplify the ratio. For example, 45 minutes to 2 hours becomes 45:120 (since 2 hours is 120 minutes), which simplifies to 3:8.
Is a ratio the same as a fraction?
They are related but not identical. In the ratio 3:5 there are 8 parts in total, so the first quantity is 3 out of 8 (the fraction 3/8) of the whole, not 3/5. A ratio compares the parts to each other, while a fraction compares a part to the whole.