Solve proportion problems with the unitary method and with formulae y = kx and y = k/x, recognise proportion in tables and graphs, and (Higher) work with y ∝ x², y ∝ 1/x² and y ∝ √x.
What you will learn
- Use the unitary method for direct proportion
- Find and use y = kx and y = k/x
- Solve workers-and-time problems with inverse proportion
- Recognise proportion from a table or a graph
- Use y ∝ x², y ∝ 1/x² and y ∝ √x (Higher)
Direct and inverse
Direct proportion: y = kx.
Double x and y doubles; y ÷ x is always k. The graph is a straight line through (0, 0).
The letter k is the constant of proportionality: a fixed number you find from one pair of values.
- y is directly proportional to x, and y = 21 when x = 3
- k = y ÷ x = 21 ÷ 3 = 7, so y = 7x
- when x = 5, y = 7 × 5 = 35
- when y = 56, x = 56 ÷ 7 = 8
Inverse proportion: y = k/x. Double x and y halves; x × y is always k. The graph is a curve that gets closer to both axes.
- y is inversely proportional to x, and y = 6 when x = 4
- k = 4 × 6 = 24, so y = 24/x
- when x = 3, y = 24 ÷ 3 = 8
- when y = 12, x = 24 ÷ 12 = 2 (x = k/y)
Without a formula, the unitary method also works for direct proportion: if 5 kg of apples cost £7.50, 1 kg costs £1.50, so 8 kg cost £12.
If 4 builders take 6 days to build a wall, the total work is 4 × 6 = 24 builder-days. So 3 builders take 24 ÷ 3 = 8 days.
More builders, fewer days: that is inverse proportion. Turn it round the same way: to finish in 2 days you need 24 ÷ 2 = 12 builders.
To tell which kind a table shows, test every pair. If y ÷ x is the same for every pair, y is directly proportional to x.
If x × y is the same for every pair, y is inversely proportional to x. If y ÷ x² is the same, y is directly proportional to x².
If none of these stays the same, it is none of them: a table such as x = 1, 2, 3 with y = 5, 7, 9 goes up in equal steps, but y ÷ x changes, so it is a straight line that misses (0, 0) (here y = 2x + 3), not direct proportion.
To fill in a table, find k from a complete pair, then use the formula for each gap.
The rest of this lesson and its 7 practice questions are locked.
In this lesson
- Direct and inverse
- Other powers (Higher)
- Try it: See the shapes
7 practice questions, marked as you go; about 25 minutes in all.
In this unit: Ratio and proportion
Convert units and use scales, write and simplify ratios, share amounts in a ratio, compare best buys and exchange rates, and solve direct and inverse proportion problems.