Mixed Review: Choose the Relationship
These questions do not tell you which variation method to use. Read carefully and decide whether the relationship is direct, inverse, joint or partial.
Before calculating, ask:
- Is there a constant ratio?
- Is there a constant product?
- Are several variables multiplied?
- Is there a fixed amount plus a changing amount?
Part A: Identify the Type
- Cost is proportional to the number of identical items.
- Time for a fixed job decreases when the number of workers increases.
- Volume depends on length, width and height.
- A taxi fare contains a fixed charge and a charge per kilometre.
- y = 7x
- y = 50/x
- z = 3xy²
- y = 4x + 9
Part B: Calculations
- y ∝ x and y = 45 when x = 9. Find y when x = 14.
- y ∝ 1/x and y = 8 when x = 6. Find y when x = 15.
- z ∝ xy. If z = 60 when x = 4 and y = 5, find z when x = 6 and y = 7.
- y = kx + c. If (x, y) = (2, 9) and (7, 24), find the equation.
Part C: Exam-Style Applications
- The mass of identical metal rods varies directly as their total length. A 3.5 m length has mass 8.4 kg. Find the mass of 9 m.
- Fifteen workers complete a task in 28 days. After 7 days, five workers leave. Assuming equal rates, find the additional time required by the remaining workers.
- Q varies jointly as a² and b. Q = 144 when a = 3 and b = 4. Find Q when a = 5 and b = 2.
- An electricity bill contains a fixed charge and a charge per unit. A bill for 30 units is US$11 and a bill for 70 units is US$19. Find the bill for 100 units.
✅ Check Your Work
Show mixed review answers
Part A
- Direct
- Inverse
- Joint
- Partial
- Direct
- Inverse
- Joint
- Partial
Part B
- 70
- 3.2
- 126
- y = 3x + 3
Part C
-
21.6 kg.
k = 8.4/3.5 = 2.4 kg/m.
mass = 2.4(9) = 21.6. -
31.5 additional days.
Total work = 15×28 = 420 worker-days.
Work done in 7 days = 15×7 = 105.
Remaining work = 315 worker-days.
Remaining workers = 10.
Time = 315/10 = 31.5 days. -
200.
Q = ka²b
144 = k(9)(4), so k = 4.
Q = 4(25)(2) = 200. -
US$25.
11 = 30k + c
19 = 70k + c
8 = 40k, so k = 0.2.
c = 5.
Bill = 0.2(100) + 5 = 25.
Reflection
- Can you recognise all four variation types?
- Can you explain what k means in context?
- Can you distinguish direct from partial variation?
- Can you check whether your answer behaves as expected?
Revision Guidance and Readiness
- Attempt the questions without notes first.
- Use the answers or worked solutions to identify the exact step that needs attention.
- Return to the relevant lesson, then retry missed questions.
- You are ready to move on when you can explain your method and check your result independently.