These eight original questions follow the variation section. Write the equation with k first, find k, then answer. Open the answer only after your own attempt.
Questions
1. y varies directly as x. y = 15 when x = 5. Find y when x = 8.
Answer
y = kx, so 15 = 5k and k = 3. Then y = 3 × 8 = 24.
2. y varies inversely as x. y = 6 when x = 4. Find y when x = 3, and find x when y = 2.
Answer
y = k/x, so k = 6 × 4 = 24. When x = 3, y = 24 ÷ 3 = 8. When y = 2, x = 24 ÷ 2 = 12.
3. y varies directly as x². y = 50 when x = 5. Find y when x = 7.
Answer
y = kx², so 50 = 25k and k = 2. Then y = 2 × 49 = 98.
4. y varies directly as √x. y = 6 when x = 9. Find y when x = 16.
Answer
y = k√x, so 6 = 3k and k = 2. Then y = 2 × 4 = 8.
5. z varies jointly as x and y. z = 24 when x = 2 and y = 3. Find z when x = 5 and y = 6.
Answer
z = kxy, so 24 = 6k and k = 4. Then z = 4 × 5 × 6 = 120.
6. p varies directly as q and inversely as r. p = 6 when q = 9 and r = 3. Find p when q = 12 and r = 4.
Answer
p = kq/r, so 6 = k × 9 ÷ 3 = 3k and k = 2. Then p = 2 × 12 ÷ 4 = 6.
7. The time t to finish a job varies inversely as the number of workers n. Six workers take 10 hours. How long will 15 workers take?
Answer
t = k/n, so k = 6 × 10 = 60. For n = 15, t = 60 ÷ 15 = 4 hours.
The answer is smaller than 10, as expected when workers are added.
8. y varies inversely as x, and y = 5 when x = 2. A student writes y = 10x. Explain the mistake and find x when y = 4.
Answer
The student used the direct form. The correct equation is y = k/x with k = 5 × 2 = 10, so y = 10/x.
When y = 4, x = 10 ÷ 4 = 2.5. Check: 4 × 2.5 = 10.
If you got these wrong
If questions 1 to 4 went wrong, revisit solving direct variation problems and solving inverse variation problems.
For question 5, see working with joint variation. For question 6, see translating combined variation into equations.
Log each slip in the mistake log and paper-error review, then build a timed set with the timed original practice session builder.
For a teacher to go through your working, see online one-to-one Mathematics tuition.