Direct and inverse proportion — identification and problem solving

Question

If 5 workers can complete a job in 12 days, how many days will 8 workers take? Is this direct or inverse proportion? Also, if 3 metres of cloth costs Rs 210, what will 7 metres cost?

(CBSE Class 8 — direct and inverse proportion)

Solution — Step by Step

Workers and days: More workers → fewer days needed. This is inverse proportion (as one increases, the other decreases).

Cloth length and cost: More cloth → more cost. This is direct proportion (both increase or decrease together).

In inverse proportion: $x_1 \times y_1 = x_2 \times y_2$

5 \times 12 = 8 \times y_2

y_2 = \frac{60}{8} = \mathbf{7.5 \text{ days}}

In direct proportion: $\dfrac{x_1}{y_1} = \dfrac{x_2}{y_2}$

\frac{3}{210} = \frac{7}{y_2}

y_2 = \frac{7 \times 210}{3} = \mathbf{\text{Rs } 490}

Why This Works

graph TD
    A["Two quantities changing"] --> B{"How do they relate?"}
    B -->|"Both increase together<br/>OR both decrease"| C["Direct Proportion<br/>x/y = constant"]
    B -->|"One increases,<br/>other decreases"| D["Inverse Proportion<br/>x × y = constant"]
    C --> E["Formula: x₁/y₁ = x₂/y₂"]
    D --> F["Formula: x₁ × y₁ = x₂ × y₂"]
    A --> G{"How to identify?"}
    G --> H["Ask: if I double x,<br/>does y double or halve?"]
    H -->|"Doubles"| C
    H -->|"Halves"| D

Direct proportion means the ratio stays constant: $y = kx$ . If you double one quantity, the other doubles too. Inverse proportion means the product stays constant: $xy = k$ . If you double one quantity, the other halves.

Alternative Method — Unitary Method

Find the value for 1 unit first:

Cloth problem: 3 m costs Rs 210, so 1 m costs $210/3 = 70$ . Therefore 7 m costs $70 \times 7 = \text{Rs } 490$ .

Workers problem: 5 workers take 12 days, so 1 worker takes $5 \times 12 = 60$ days. Therefore 8 workers take $60/8 = 7.5$ days.

The unitary method always works and avoids formula confusion.

Common real-world examples to remember: speed-time (inverse), distance-time at constant speed (direct), workers-days (inverse), items-cost (direct), speed-fuel consumption (roughly direct). Having these associations helps you quickly identify the type in word problems.

Common Mistake

Students set up the equation for direct proportion when it should be inverse, or vice versa. The giveaway: if more of one thing means LESS of the other, it’s inverse. Many students see “more workers” and write $5/12 = 8/x$ , getting $x = 19.2$ days — which says 8 workers take MORE time than 5, which makes no sense. Always do a quick sanity check on your answer.

Question

Solution — Step by Step

Why This Works

Alternative Method — Unitary Method

Common Mistake

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