Rates, ratios & proportions · Ages 13–14
Ratios and proportions
Multiply or divide both terms alike, and the ratio stays.
How to do it
Method
Multiply or divide both terms alike, and the ratio stays
x : 6 = 4 : 3, 3x = 6 × 4 = 24, x = 8
Another way
Spot the multiplier
6 ÷ 3 = 2, 4 × 2 = 8
In full
A ratio a : b can also be written as a fraction, ab. Multiplying or dividing both terms by the same number (not 0) leaves it the same. A ratio of fractions or decimals is made whole first: 12 : 23, both times 6, is 3 : 4; 0.6 : 1.5, both times 10, is 6 : 15, and both ÷ 3, 2 : 5. Amounts in different units go into the same unit before the ratio is written: 50 cm : 2 m = 50 : 200 = 1 : 4. Two equal ratios make a proportion, such as x : 6 = 4 : 3. One way: as fractions, then cross-multiply: x6 = 43, 3x = 24, x = 8. Another: spot the multiplier. The second term goes from 3 to 6, times 2, so the first goes from 4 to 8: x = 8. From 3x = 5y, dividing both sides by 3y gives xy = 53, so x : y = 5 : 3.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
Write 0.5 : 1.1 in its simplest whole-number form: :
Fill in each blank
Show the answer and working
Answer: 5, 11
Both terms times 10 make 5 : 11; both ÷ 1 make 5 : 11.
Example 2
x : 4 = 1 : 2, x =
Show the answer and working
Answer: 2
Cross-multiply: 2x = 4 × 1 = 4, so x = 2. Or spot the multiplier: 2 × 2 = 4, so x = 1 × 2 = 2.
Example 3
If x : y = 3 : 2, what is (x + y) : y?
- A. 2 : 5
- B. 3 : 5
- C. 5 : 2
- D. 5 : 3
Choose the right answer
Show the answer and working
Answer: C. 5 : 2
Let x = 3k and y = 2k: x + y = 5k, so (x + y) : y = 5 : 2.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.14 : 16 and 42 : 49 are equal ratios.
True or false?
Answer:
2.x : 24 = 5 : 3, x =
Answer:
3.Write 13 : 49 in its simplest whole-number form: :
Fill in each blank
Answer:
4.If 8x = 9y, then x : y = : (simplest whole numbers).
Fill in each blank
Answer:
5.x : 8 = (x + 4) : 9, x =
Answer:
All the answers
- 1. False
- 2. 40
- 3. 3, 4
- 4. 9, 8
- 5. 32
VIP practice notes and papers
Aimed at tests and exams: questions on this topic, new every time, to print or save as a PDF.
- Practice notes: The key point, worked examples step by step, then questions of the same kind to follow, hints fading
- Practice paper: Set out in parts like a school paper, marked, with answers and full working
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