Ratio and rate · Usually met around P6
Equivalent ratios and simplest form
Divide every term by the HCF.
How to do it
Method
Divide every term by the HCF
12 : 18 → HCF 6 → 2 : 3
Another way
Divide step by step
12 : 18 → 6 : 9 → 2 : 3
In full
Multiply or divide every term of a ratio by the same number (not 0) and the new ratio is equal to the old one, an equivalent ratio: 2 : 3, 4 : 6 and 6 : 9 are all equal. Adding the same number does not work: 2 : 3 and 3 : 4 are not equal. To write a ratio in its simplest form, divide every term by their highest common factor (HCF), until no common factor is left: the HCF of 12 and 18 is 6, 12 ÷ 6 = 2 and 18 ÷ 6 = 3, so the simplest form is 2 : 3. Or divide a step at a time: 12 : 18 → 6 : 9 → 2 : 3. When the units differ, change them to the same unit first: 50 cm : 2 m is 50 : 200, which is 1 : 4. A ratio of three amounts works the same way, all three terms at once: 6 : 9 : 12 becomes 2 : 3 : 4.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
7 : 3 = 63 : □
Show the answer and working
Answer: 27
7 × 9 = 63, so 3 must be multiplied by 9 too: 3 × 9 = 27.
Example 2
8 : 24 in its simplest form is 1 : 3.
True or false?
Show the answer and working
Answer: True
The HCF of 8 and 24 is 8; dividing by 8 gives 1 : 3. Dividing by a smaller common factor doesn't reach the simplest form.
Example 3
1 m = 100 cm. Write 75 cm : 3 m in its simplest form: □ : □
Fill in each box
Show the answer and working
Answer: 1，4
Same units first: 3 m = 300 cm. The HCF of 75 and 300 is 75: 1 : 4.
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