Ratios & Proportions • Topic 4 of 4

Direct & Inverse Variation

Two quantities vary directly when one is a constant multiple of the other, y = kx: double x and y doubles, so the ratio y/x stays fixed. They vary inversely when their product is constant, y = k/x (equivalently xy = k): double x and y halves. The single decision that determines everything is which of these you are in — and the GRE deliberately blurs it. More workers finishing a job faster is inverse (workers × days is constant); more hours worked earning more pay is direct (pay/hours is constant). The reliable test: ask "if the first quantity goes up, does the second go up (direct) or down (inverse)?" For direct variation set the two ratios equal; for inverse set the two products equal. This is also the cleanest setup for classic "if 6 taps fill a tank in 8 hours, how long for 4 taps?" problems — fewer taps, more time, so it is inverse: 6 × 8 = 4 × t.

✅ Solved examples

1. y varies directly with x, and y = 18 when x = 4. Find y when x = 10.
k = y/x = 18/4 = 4.5. Then y = 4.5 × 10 = 45. (Or 18/4 = y/10 ⇒ y = 45.)
2. If 6 identical pumps fill a tank in 8 hours, how long would 4 pumps take?
Inverse: pumps × hours is constant. 6 × 8 = 48, so 4 × t = 48 ⇒ t = 12 hours.
3. Quantitative Comparison. y varies inversely with x. Quantity A: y when x = 2. Quantity B: y when x = 5. Which is greater?
Inverse variation means larger x gives smaller y, so y at x = 2 is greater than y at x = 5. Quantity A is greater.
4. The cost of fabric varies directly with length. 3.5 m costs $28. Find the cost of 6 m.
Direct: 28/3.5 = 8 per metre. Cost of 6 m = 6 × 8 = $48.

✏️ Practice — try these, take hints as needed

1. y varies directly with x; y = 30 when x = 12. Find y when x = 20.
Find k = y/x.
k = 30/12 = 2.5.
y = 2.5 × 20.
50
2. 8 workers build a wall in 15 days. How long would 10 workers take (same rate)?
More workers, less time — inverse.
workers × days is constant = 120.
10 × t = 120.
12 days
3. The pressure of a gas varies inversely with volume. At 4 L the pressure is 3 atm. Find the pressure at 6 L.
P × V is constant.
4 × 3 = 12.
6 × P = 12.
2 atm
4. A car travels a fixed distance. At 60 km/h it takes 5 hours. How long at 75 km/h?
Speed and time vary inversely for fixed distance.
speed × time = 300.
75 × t = 300.
4 hours
5. z varies directly with w. If z = 21 when w = 7, find w when z = 45.
k = z/w = 3.
z = 3w.
45 = 3w.
w = 15

📝 Topic test — 8 questions

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