Two quantities are in direct proportion if one is a constant multiple of the other: y = kx, where k is the constant of proportionality.
When x doubles, y doubles. The graph of y against x is a straight line through the origin.
Find k from a known pair, then use it: if y = 15 when x = 3, k = 5, so y = 5x and y = 40 when x = 8.
Direct proportion through the origin
Inverse proportion
For positive quantities in these applications, inverse proportion means xy = k > 0, or y = k/x with x > 0. Increasing x decreases y while keeping their product fixed.
When x doubles, y halves. The graph is a reciprocal curve (hyperbola).
For a fixed job and equal independent constant work rates, 6 workers × 10 days = 60 worker-days, so 4 workers take 15 days. Real work may not scale this way if rates or overhead change.
Inverse proportion for positive values
Other forms of proportion
y \propto x2 means y = kx2; y \propto \√{x} means y = k\√{x}; y \propto \frac{1}{x2} means y = \frac{k}{x2}.
Method: write the equation with k, substitute known values to find k, then use the formula.
Recognising proportion
Check by looking at a table: for direct proportion \frac{y}{x} is constant; for inverse proportion xy is constant.
Unit rates
Unitary method: find the value of one item, then scale. If 5 pens cost £4, one pen costs £0.80.
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1.If y is directly proportional to x, what happens to y when x doubles?
Easy
Ait halves
Bit doubles
Cit stays the same
Dit increases by 2
2.If y is inversely proportional to x, what happens to y when x doubles?
Easy
Ait doubles
Bit halves
Cit stays the same
Dit quadruples
3.5 pens cost £4. How much do 8 pens cost?
Easy
A£6.40
B£5.00
C£7.50
D£32
4.y = kx and y = 12 when x = 4. What is k?
Easy
A3
B8
C48
D\frac{1}{3}
5.y is directly proportional to x and y = 20 when x = 4. Find y when x = 7.
Easy
A35
B28
C5
D140
6.At a constant fuel-use rate, a car uses 8 litres for 100 km. How many litres for 350 km?
Easy
A28
B35
C12.5
D24
7.y is inversely proportional to x and y = 6 when x = 4. What is k?
Easy
A24
B1.5
C10
D2
8.Which table shows direct proportion?
Easy
Ax = 1, 2, 3; y = 4, 8, 12
Bx = 1, 2, 3; y = 12, 6, 4
Cx = 1, 2, 3; y = 5, 6, 7
Dx = 1, 2, 3; y = 1, 4, 9
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9.Which table shows inverse proportion?
Easy
Ax = 1, 2, 3; y = 4, 8, 12
Bx = 1, 2, 4; y = 12, 6, 3
Cx = 1, 2, 3; y = 5, 6, 7
Dx = 1, 2, 3; y = 1, 4, 9
10.y is inversely proportional to x. When x = 5, y = 8. Find y when x = 10.
Medium
A4
B16
C2
D40
11.Four identical machines working independently at constant equal rates make a batch in 9 hours. With no setup delay, how long would six such machines take?
Medium
A6 hours
B13.5 hours
C36 hours
D4 hours
12.A recipe for 6 people needs 450 g of flour. How much for 10 people?
Medium
A750 g
B540 g
C900 g
D270 g
13.y \propto x2 and y = 18 when x = 3. What is k?
Medium
A2
B6
C54
D9
14.y \propto x2, k = 2. What is y when x = 5?
Medium
A50
B10
C25
D20
15.y \propto \√{x} and y = 6 when x = 9. What is y when x = 25?
Medium
A10
B15
C8
D6
16.y is inversely proportional to x2 and y = 4 when x = 3. Find y when x = 6.
Medium
A1
B2
C8
D16
17.The area of a circle is directly proportional to the square of the radius. A circle of radius 2 cm has area 4\π. What is the area for radius 5 cm?
Medium
A25\π
B10\π
C20\π
D100\π
18.If y is directly proportional to x and x is increased by 20%, y is increased by:
Medium
A20%
B40%
C44%
D120%
19.For nonzero k and nonzero real x, what shape does y = k/x have?
Easy
Aa straight line through the origin
Ba hyperbola
Ca parabola
Da circle
20.What is the graph of direct proportion?
Easy
Aa straight line through the origin
Ba curve
Ca straight line not through the origin
Da circle
21.Which formula gives inverse proportion?
Medium
Ay = 3x
By = \frac{12}{x}
Cy = x + 2
Dy = x2
22.Which is the unit cost of an item if 6 cost £15?
Easy
A£2.50
B£0.40
C£90
D£9
23.In direct proportion, \frac{y}{x} is constant.
Easy
True or false?
24.In inverse proportion, xy is constant.
Easy
True or false?
25.The graph of inverse proportion is a straight line.
Easy
True or false?
26.If y \propto x then doubling x doubles y.
Easy
True or false?
27.If y \propto x2 and x doubles, y doubles.
Medium
True or false?
28.Assume fixed unit prices and equal constant work rates, as appropriate. Which quantity pairs are directly proportional?
Medium
ATotal cost and number of items at £3 per item
Bdistance at constant speed over time
Cspeed and time for a fixed journey
Darea of a square and its side
Enumber of workers and days to finish a job
29.For fixed jobs or distances, assume equal independent constant working rates. Which quantity pairs are inversely proportional?
Medium
Anumber of workers and time to finish a job
Bspeed and time for a fixed distance
Ccost and number of items at a fixed price
Dheight and weight of children
Eside and area of a square
30.3 kg of apples cost £6. How much do 5 kg cost?
Easy
A£10
B£9
C£12
D£15
31.y is directly proportional to x. If x is halved, y is:
Easy
Ahalved
Bdoubled
Cunchanged
Dquartered
32.It takes 8 workers 6 days to build a wall. How long would it take 12 workers working at the same rate?
