Rates, ratios & proportions · Ages 13–14
Direct proportion
Direct proportion: y ÷ x stays the same, y = kx.
How to do it
Method
Direct proportion: y ÷ x stays the same, y = kx
60 ÷ 5 = 12, 12 × 8 = 96
Another way
A proportion
x : 8 = 60 : 5, 5x = 480, x = 96
In full
Two quantities that change by the same multiple together, one doubling as the other doubles, one halving as the other halves, are in direct proportion: y ÷ x is always the same, so y = kx for a constant k. 5 books cost $60: what do 8 of them cost? One way, the unitary method: one book first, 60 ÷ 5 = $12, so 8 cost 12 × 8 = $96. Another, a proportion: let 8 cost $x, then x : 8 = 60 : 5, 5x = 480, x = 96. To tell whether two quantities are in direct proportion, find y ÷ x for every pair: they must all be the same. y growing as x grows is not enough; y = x + 3 grows too, but isn't.
Worked examples
Have a go first, then open each one for the answer and the working.
Example 1
11 cakes take 1100 g of flour. How many of the same cakes do 2000 g of flour make?
Show the answer and working
Answer: 20 cakes
Each cake takes 1100 ÷ 11 = 100 g, and 2000 ÷ 100 = 20 cakes.
Example 2
y is in direct proportion to x. As x goes from 12 up to 14, y goes up by 1. When x = 12, y =
Show the answer and working
Answer: 6
The rise in y is in proportion to the rise in x too: k = 1 ÷ (14 − 12) = 12, so y = 12 × 12 = 6.
Example 3
y is in direct proportion to x. If x goes up 100%, what happens to y?
- A. It goes down 100%
- B. It goes up 100%
- C. It stays the same
- D. It goes down 50%
Choose the right answer
Show the answer and working
Answer: B. It goes up 100%
y = kx: x becomes 1 + 100% times as much, so y does too, 100% up.
Practice questions to try
Try these 5; the answers are at the end. For a whole worksheet, see VIP worksheets below.
1.In which set of values is y in direct proportion to x?
- A. (x, y) = (1, 6), (2, 10), (4, 18)
- B. (x, y) = (1, 16), (2, 8), (4, 4)
- C. (x, y) = (1, 4), (2, 8), (4, 16)
- D. (x, y) = (1, 3), (2, 4), (4, 6)
Choose the right answer
Answer:
2.2 storybooks cost $76, so 3 of them cost $
Answer:
3.y is in direct proportion to x: when x = 10, y = 4; when x = 55, y = ; when x = , y = 14.
Fill in each blank
Answer:
4.8 cakes take 880 g of flour. How many of the same cakes do 1870 g of flour make?
Answer:
5.y is in direct proportion to x. As x goes from 30 up to 45, y goes up by 21. When x = 30, y =
Answer:
All the answers
- 1. C. (x, y) = (1, 4), (2, 8), (4, 16)
- 2. 114
- 3. 22, 35
- 4. 17 cakes
- 5. 42
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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