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Direct and inverse proportion

2026-10-21Percentage, Rate & Ratio Ms Lam

Decide whether y varies directly or inversely with x, then solve with k — TSA-style classification plus calculation.

Key formulas

Direct: y = kx

Inverse: xy = k

y ∝ 1/x \text{means inverse}

Recognising the type

If doubling x doubles y, the relation is direct. If doubling x halves y (product constant), it is inverse. Word clues: “shared equally”, “speed and time for fixed distance” often signal inverse proportion.

Finding k

Use one known pair to find k, then substitute the new value. Keep units consistent — convert hours and minutes before forming rates.

Tables and graphs

Direct proportion graphs are straight lines through the origin. Inverse proportion graphs are rectangular hyperbolas in the first quadrant; the product of coordinates stays constant.

Exam habit

First write “direct” or “inverse” and the matching formula, then find k.

Worked example

Inverse proportion

Problem. y is inversely proportional to x. When x = 6, y = 10. Find y when x = 4.

Solution. k = xy = 60. When x = 4, y = 60/4 = 15.

Review quiz

1 / 4

Direct: y=20 when x=8. y when x=14?

Practice questions

y is directly proportional to x. When x = 8, y = 20. Find y when x = 14.
y is inversely proportional to x. When x = 6, y = 10. Find y when x = 4.
12 workers finish a job in 10 days (same rate). How many days for 8 workers?
State whether buying x identical pens for total cost y is direct or inverse.

Study notes

Is “more workers, fewer days” inverse?

Yes, if the total work is fixed and workers work at the same rate.

Does y = kx + 3 count as direct proportion?

No — direct proportion requires y = kx with no extra constant.