EQN-02-02

Direct & Inverse Proportion

Tell the two kinds of proportion apart and solve speed, work, and price problems without setting up heavy algebra.

~5 min read•Difficulty 3/5

Direct proportion

Two quantities are directly proportional when one grows, the other grows by the same factor (their ratio stays constant).

y = k \cdot x y / x = constant

More items $\to$ more cost. If 3 pens cost 12, then 5 pens cost $(12/3) \times 5 = 20.$

Inverse proportion

Two quantities are inversely proportional when one grows, the other shrinks by the same factor (their product stays constant).

x \cdot y = constant

More workers $\to$ less time. If 6 workers finish a job in 10 days, then 4 workers take $(6 \times 10) /4 = 15$ days.

How to tell them apart

Ask: "as the first quantity goes up, does the second go up or down?"

Up together $\to$ direct $\to$ keep the ratio.
One up, one down $\to$ inverse $\to$ keep the product.

Speed–distance–time

distance $=$ speed $\times$ time. With distance fixed, speed and time are inversely proportional; with time fixed, distance and speed are directly proportional.

Worked examples

Example 1 (difficulty 2/5)

If 8 machines produce a batch in 6 hours, how long do 12 machines take (same rate)?

◎ Predict the answer first:

Example 2 (difficulty 2/5)

A car travels 150 km on 10 litres. How far on 14 litres at the same rate?

◎ Predict the answer first:

Tips

Decide direct vs inverse before touching numbers — it sets whether you keep the ratio or the product.
Inverse-proportion answers move opposite to the input: more workers should give a smaller time.
The "constant product" of an inverse problem (e.g. machine-hours, worker-days) is the total work.

Common mistakes

Using direct proportion for a "more workers, less time" problem

Why: Setting up a single ratio is the default habit.

Fix: When one quantity falls as the other rises, hold the product constant, not the ratio.

Scaling distance and time as direct when distance is fixed

Why: Speed and time both feel like they should rise together.

Fix: With a fixed distance, faster means less time — they are inversely related.

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