Every unitary-method question starts by finding the value of just ONE unit — but the direction you multiply or divide in depends entirely on whether the quantities move together or in opposite directions.
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The first question in any unitary-method problem is always: do these two quantities increase TOGETHER, or does one increase while the other decreases?
📌 Telling direct from inverse
Direct variation: more of one thing needs MORE of the other. More pens cost more money. Multiply.
Inverse variation: more of one thing needs LESS of the other. More workers finish a job in FEWER days. Cross-multiply and divide.
Find the value of ONE unit first, then scale up or down as needed.
✨ Worked examples
Direct: 5 pens cost ₹250
cost of 8 pens?
250 ÷ 5 × 8 = ₹400
Inverse: 12 workers, 15 days
20 workers, how many days?
(12 × 15) ÷ 20 = 9 days
When THREE or more quantities are linked together, the chain rule keeps their combined product constant.
The chain rule formula: for work problems, (workers₁ × hours₁ × days₁) = (workers₂ × hours₂ × days₂) — because the total amount of work stays the same.
📊 Worked example
| Given | Value |
|---|---|
| 15 workers, 6 hours/day, 10 days | Total work units = 15 × 6 × 10 = 900 |
| New: 12 workers, 5 hours/day | Days needed = 900 ÷ (12 × 5) = 15 days |
Work-days problems are inverse variation in disguise — more workers always means fewer days for the same job.
Trap: if some workers leave partway through, split the problem into two stages — work done before the change, and work remaining after — rather than applying one formula to the whole job.
12 questions on direct variation, inverse variation and the chain rule.
