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Ratio & Proportion

Ratio is how you compare quantities and share things out fairly, from recipes to scale drawings. Proportion is what lets you scale a recipe up or down without spoiling it.

MYP 4Standard MathsNumberCriteria A · D~9 min read

Written by Fayek Jawad Haider · MYP student

Simplifying ratios

A ratio compares two or more quantities of the same kind, written with a colon, such as \(3:2\). Just like a fraction, a ratio simplifies: divide every part by the same number until no common factor is left. Dividing by the highest common factor gets you there in one move.

Ratio
A comparison of quantities in the same units, written \(a:b\). It shows relative size, not actual amounts.
Worked example

Simplify the ratio \(12:18\).

1
Find the highest number that divides both 12 and 18: that is 6.
2
Divide each part by 6: \(12 \div 6 = 2\) and \(18 \div 6 = 3\).
\(12:18 = 2:3\)

Match the units first

A ratio only makes sense when both parts are in the same unit. Before simplifying \(50\text{ cm} : 2\text{ m}\), convert the metres: \(50 : 200 = 1:4\).

Sharing in a ratio

To split an amount in a given ratio, add the parts to find the total number of parts, work out what one part is worth, then multiply back up. That first step, adding the parts, is the one people forget.

Worked example

Share \(\pounds 60\) between two people in the ratio \(2:3\).

1
Add the parts: \(2 + 3 = 5\) parts in total.
2
Find one part: \(\pounds 60 \div 5 = \pounds 12\).
3
Multiply back up: \(2 \times \pounds 12 = \pounds 24\) and \(3 \times \pounds 12 = \pounds 36\).
\(\pounds 24\) and \(\pounds 36\) (which check back to \(\pounds 60\))

Always check the total

Add your shares back together at the end. If they do not come to the original amount, something has slipped, so it is a quick, free way to catch an error.

Direct proportion

Two quantities are in direct proportion when they grow at the same rate: double one and the other doubles too. The neatest way to solve these is the unitary method, find the value of one item first, then scale up to however many you need.

Worked example

If 5 pens cost \(\pounds 2.00\), how much do 8 pens cost?

1
Find the cost of one pen: \(\pounds 2.00 \div 5 = \pounds 0.40\).
2
Scale up to 8 pens: \(8 \times \pounds 0.40 = \pounds 3.20\).
8 pens cost \(\pounds 3.20\)

Where this is assessed

Ratio and proportion span Criterion A (the method) and Criterion D (applying it to real recipes, maps and money). The unitary method is worth showing in full, since the "cost of one" line earns method marks even if the final arithmetic slips.

Check yourself

1. Simplify the ratio \(20:35\). +

The highest common factor of 20 and 35 is 5. Divide both parts: \(20\div5 = 4\) and \(35\div5 = 7\), giving \(\mathbf{4:7}\).

2. Share 48 sweets in the ratio \(3:5\). +

Parts: \(3+5 = 8\). One part: \(48 \div 8 = 6\) sweets. So the shares are \(3\times6 = 18\) and \(5\times6 = 30\), that is \(\mathbf{18}\) and \(\mathbf{30}\) sweets.

3. If 4 apples cost 96p, how much do 7 apples cost? +

One apple: \(96 \div 4 = 24\)p. Seven apples: \(7 \times 24 = 168\)p, which is \(\mathbf{\pounds 1.68}\).

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