Ratio, proportion & rates of change
Ratio: Simplifying, Sharing, and Fractions
Year 7 · Year 8 · Year 9
- ✓Understand what a ratio is and how to write it correctly.
- ✓Simplify ratios to their simplest form.
- ✓Share a given quantity into a ratio.
- ✓Convert between ratios and fractions, and vice versa.
Key concepts
A ratio compares two or more quantities of the same type. It shows how much of one thing there is compared to another. Ratios are written using a colon (:) to separate the quantities. For example, if there are 3 red apples and 2 green apples, the ratio of red to green apples is 3:2. The order of the numbers in a ratio is important.
To simplify a ratio, you divide all parts of the ratio by their highest common factor (HCF). This is similar to simplifying fractions. A ratio is in its simplest form when the numbers in the ratio have no common factors other than 1. You must divide all parts of the ratio by the same number.
To share a quantity in a given ratio, follow these steps: 1. Add together all the parts of the ratio to find the total number of parts. 2. Divide the total quantity by the total number of parts to find the value of one part. 3. Multiply the value of one part by each number in the ratio to find the amount for each share.
Ratios can be directly related to fractions. If a ratio compares two quantities, A and B, as A:B, then the total number of parts is A + B. The fraction of the total that A represents is A/(A+B), and the fraction of the total that B represents is B/(A+B). This concept extends to ratios with more than two parts.
Key facts to remember
- 1A ratio compares quantities of the same type.
- 2The order of numbers in a ratio is crucial; 2:3 is different from 3:2.
- 3Ratios can be simplified by dividing all parts by their highest common factor (HCF).
- 4Ratios can compare more than two quantities, e.g., 2:3:5.
- 5To share a quantity in a ratio, first find the total number of parts.
- 6If a ratio is A:B, the total number of parts is A+B.
- 7The fraction of the total represented by 'A' in the ratio A:B is A/(A+B).
- 8Ratios do not have units if the quantities being compared have the same units.
Worked examples
Example 1
Simplify the ratio 24:36.
Answer
2:3
Always ensure the ratio is fully simplified by checking for any remaining common factors.
Example 2
Share £70 in the ratio 3:4.
Answer
£30 and £40
It's good practice to check your answer by adding the shares together to ensure they sum to the original total.
Example 3
A fruit bowl contains apples and bananas in the ratio 5:3. If there are 15 apples, how many bananas are there? What fraction of the fruit are bananas?
Answer
There are 9 bananas. The fraction of the fruit that are bananas is 3/8.
Remember to simplify fractions to their lowest terms.
Common mistakes
- ✗Not simplifying ratios fully to their simplest form.
- ✗Mixing up the order of the ratio, especially when the question specifies 'A to B' but the student writes 'B to A'.
- ✗Forgetting to add all parts of the ratio together before dividing the total quantity when sharing.
- ✗Incorrectly converting a ratio A:B to a fraction as A/B instead of A/(A+B).
- ✗Only dividing some parts of a ratio when simplifying, rather than all parts.
Exam tips
- ★Always read the question carefully to ensure you understand which quantities are being compared and in what order.
- ★Show all your working, especially when sharing a quantity, as method marks are often awarded even if the final answer is incorrect.
- ★Check your simplified ratios by ensuring there are no common factors left between the numbers.
- ★When sharing a quantity, check that your final amounts add up to the original total quantity given in the problem.
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