Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.
Level 1 · Warm-up
1 Which rule fits this table?
Show worked solution
7 ÷ 2 = 21 ÷ 6 = 35 ÷ 10 = 3.5, so y = 3.5x. y = x + 5 and y = 3x + 1 both fit the first column only (6 + 5 = 11 and 3 × 6 + 1 = 19, not 21); y = \(\dfrac{2}{7}\)x uses x ÷ y, k upside down.
2 y ∝ x, and y = 18 when x = 4. Find k, and then y when x = 14.
Show worked solution
k = 18 ÷ 4 = 4.5 (or 9/2), so y = 4.5 × 14 = 63.
3 w varies directly with t, and w = 36 when t = 4. Find t when w = 99.
Show worked solution
k = 36 ÷ 4 = 9, so w = 9t. 99 = 9t gives t = 11.
Level 2 · Core
4 y varies directly with the square of x. Which set of points (x, y) could fit?
Show worked solution
From (1, 4), k = 4 ÷ 12 = 4, so y = 4x2: 4 × 4 = 16 and 4 × 9 = 36. The set is (1, 4), (2, 16), (3, 36). (1, 4), (2, 8), (3, 12) is y = 4x, direct with x not x2; (1, 4), (2, 16), (3, 64) multiplies by 4 each time; (1, 4), (2, 6), (3, 8) is y = 2x + 2.
5 The line shows y varying directly with x. Find k, and then y when x = 20.
Show worked solution
k = 3 ÷ 5 = 0.6 (or 3/5), so y = 0.6 × 20 = 12.
6 The stretch of a spring varies directly with the mass hung on it. A 300 g mass stretches it 4.5 cm. What mass, in grams, stretches it 7.5 cm?
Show worked solution
7.5 ÷ 4.5 = \(\dfrac{5}{3}\), so the mass is 300 × \(\dfrac{5}{3}\) = 500 g. Or: 1.5 cm for every 100 g, and 7.5 cm is 5 lots of 1.5 cm.
Level 3 · Challenge
7 Three statements:
1 If y ∝ x2 and x is halved, then y is halved.
2 The points (2, 5) and (6, 15) could both lie on the graph of a direct variation.
3 y = 2x + 1 shows direct variation, because its graph is a straight line.
Which statements are correct?
1 If y ∝ x2 and x is halved, then y is halved.
2 The points (2, 5) and (6, 15) could both lie on the graph of a direct variation.
3 y = 2x + 1 shows direct variation, because its graph is a straight line.
Which statements are correct?
Show worked solution
Halving x multiplies y by (1/2)2 = 1/4, so y is quartered, not halved (1 ✗). 5 ÷ 2 = 15 ÷ 6 = 2.5, so both lie on y = 2.5x (2 ✓). y = 2x + 1 passes through (0, 1), not the origin, so it is not direct variation (3 ✗). 2 only.
8 y varies directly with x2, and y = 80 when x = 4. Find k, and the positive value of x for which y = 180.
Show worked solution
k = 80 ÷ 16 = 5, so y = 5x2. 180 = 5x2 gives x2 = 36, so x = 6.
9 P(2, 11) lies on the graph of a direct variation. The point Q(a, 66) lies on the same graph. Find a.
Show worked solution
k = 11 ÷ 2 = 5.5, so 66 = 5.5a and a = 12. Check: 66 is 6 times 11, so a is 6 times 2 = 12.