Ratios Revision Notes for AQA GCSE Maths

Ratios revision note

Introduction to ratios

A ratio is a way of comparing quantities or amounts. Ratios show the relationship between different parts of a whole or between different quantities. Understanding ratios is essential for solving many mathematical problems, and there are several key techniques you need to master.

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Ratios are fundamental to many areas of mathematics, from geometry and algebra to statistics and probability. Mastering ratio techniques will help you solve complex problems more efficiently.

Writing ratios as fractions

The most straightforward way to express a ratio is by converting it into a fraction. When you have a ratio like $2:9$ , you can write this as a fraction by placing one number over the other.

For example, if apples and oranges are in the ratio $2:9$ , this tells us there are $\frac{2}{9}$ as many apples as oranges, or alternatively, $\frac{9}{2}$ times as many oranges as apples. This fraction form makes it easier to work with ratios in calculations.

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Worked Example: Converting Ratios to Fractions

If books and magazines are in the ratio $3:7$ :

There are $\frac{3}{7}$ as many books as magazines
There are $\frac{7}{3}$ times as many magazines as books
For every 3 books, there are 7 magazines

Reducing ratios to their simplest form

Just like fractions, ratios can be simplified by finding common factors. To reduce a ratio to its simplest form, you need to divide all the numbers in the ratio by the same value - specifically, by their highest common factor.

A ratio is in its simplest form when there's no number (other than 1) that divides evenly into all parts of the ratio. For example, to simplify $15:18$ , you would notice that both numbers can be divided by 3. So $15:18$ becomes $5:6$ , which cannot be simplified further.

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Worked Example: Simplifying Ratios

Simplify the ratio $24:36$ :

Step 1: Find the highest common factor of 24 and 36

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24
Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36
HCF = 12

Step 2: Divide both numbers by the HCF $24:36 = (24 ÷ 12):(36 ÷ 12) = 2:3$

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Calculator tip: If you're allowed to use a calculator, you can enter a ratio as a fraction (like $\frac{8}{12}$ ) and use the fraction button to automatically reduce it. Then convert the result back to ratio form.

The more awkward cases

Some ratios require special handling before they can be simplified. These more challenging scenarios include ratios with decimals, mixed units, and cases where you need unit ratios.

Ratios with decimals or fractions

When dealing with ratios that contain decimal numbers, you need to eliminate the decimal places first. Do this by multiplying both parts of the ratio by an appropriate power of 10.

For instance, if you have the ratio $2.4:3.6$ , multiply both sides by 10 to get $24:36$ . Then you can simplify this by dividing both numbers by their highest common factor (12) to get $2:3$ .

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Worked Example: Ratios with Decimals

Simplify the ratio $1.5:2.4$ :

Step 1: Eliminate decimals by multiplying by 10 $1.5:2.4 = 15:24$

Step 2: Find the HCF of 15 and 24 HCF = 3

Step 3: Divide both by the HCF $15:24 = (15 ÷ 3):(24 ÷ 3) = 5:8$

Ratios with mixed units

When a ratio contains different units, you must convert everything to the same unit before simplifying. Always convert to the smaller unit to avoid decimals.

For example, with $24$ mm: $7.2$ cm, first convert $7.2$ cm to millimetres (72 mm). This gives you $24$ mm: $72$ mm. Now you can simplify by dividing both by 24 to get $1:3$ .

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Always convert to the smaller unit when dealing with mixed units. This prevents you from introducing decimals, which would make the problem more complex.

Creating unit ratios

Sometimes you need to express a ratio in the form $1:n$ or $n:1$ . This is particularly useful when you want to see how many times bigger one quantity is than another.

To create a unit ratio, simply divide both parts of the ratio by one of the numbers. For example, to convert $3:56$ to the form $1:n$ , divide both parts by 3. This gives you $1:18\frac{2}{3}$ .

Scaling up ratios

When you know the ratio between parts and the actual size of one part, you can find the size of the other parts by scaling up the ratio proportionally.

The key principle is that the two parts of a ratio are always in direct proportion. This means if one part doubles, the other part doubles too. If one part triples, the other part triples as well.

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Worked Example: Scaling Up Ratios

Mortar is made from sand and cement in the ratio $7:2$ . If you use 21 buckets of sand, how much cement do you need?

Step 1: Find the scale factor $21 ÷ 7 = 3$ (sand has been multiplied by 3)

Step 2: Apply the same scale factor to cement $2 × 3 = 6$ buckets of cement needed

Part
ratios

Part

ratios occur when the left-hand side of a ratio is included in the right-hand side. These ratios compare a part to the total amount.

For example, if Mrs Miggins has tabby cats and ginger cats in the ratio $3:5$ (tabby to total), this means that out of every 5 cats, 3 are tabby cats and 2 are ginger cats (since $5 - 3 = 2$ ).

When working with part

ratios, remember that:

The ratio tells you what fraction of the whole each part represents
You can find the other parts by subtracting from the whole
You can scale up using the same proportional principles

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In part

ratios, be careful to identify which number represents the part and which represents the whole. The whole will always be larger than any individual part.

Proportional division

Proportional division involves splitting a total amount into parts according to a given ratio. This is a three-step process that's essential for many ratio problems.

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The Three-Step Process for Proportional Division:

Step 1: ADD UP THE PARTS If you have a ratio like $2:4:7$ , add these numbers together to find the total number of parts: $2 + 4 + 7 = 13$ parts.

Step 2: DIVIDE TO FIND ONE "PART" Divide the total amount by the number of parts to find the value of one part. If £9100 is being divided in the ratio $2:4:7$ , then one part = £9100 ÷ 13 = £700.

Step 3: MULTIPLY TO FIND THE AMOUNTS Multiply the value of one part by each number in the ratio to find the individual amounts. So the person with 4 parts gets $4 × £700 = £2800$ .

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Worked Example: Proportional Division

Three friends share £240 in the ratio $3:5:8$ . How much does each person get?

Step 1: Add up the parts $3 + 5 + 8 = 16$ parts

Step 2: Find the value of one part One part = £240 ÷ 16 = £15

Step 3: Calculate each person's share

Person 1: $3 × £15 = £45$
Person 2: $5 × £15 = £75$
Person 3: $8 × £15 = £120$

Check: £45 + £75 + £120 = £240 ✓

Changing ratios

Some problems involve ratios that change when quantities are added or removed. These require careful step-by-step analysis.

The key approach is to:

Find the original numbers of each quantity
Work out how the quantities change
Write the new ratio and simplify if necessary

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For more complex changing ratio problems, you might need to:

Write ratios as equations using variables
Turn ratios into fractions
Solve simultaneous equations

For example, if you know the ratio of male to female pupils is $5:3$ , and then teachers are added in the ratio $4:9$ (male to female), you can use algebraic methods to find the new overall ratio.

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Key Points to Remember:

Simplify ratios by dividing all parts by their highest common factor
For decimals and fractions: multiply to eliminate decimal places first
For mixed units: convert to the smaller unit before simplifying
For proportional division: add up the parts, find one part, then multiply
Part
ratios include one part within the total - subtract to find the other parts
Unit ratios show direct comparison by making one part equal to 1
Scaling up uses direct proportion - if one part changes by a factor, all parts change by the same factor

Ratios (AQA GCSE Maths): Revision Notes