Ratio and Proportion — The Word-Problem Method That Never Fails
"Divide ₹4,500 between A and B in the ratio 5:4." "6 workers finish a job in 8 days; how many days for 12 workers?" Both are ratio-and-proportion problems, and both are solved by the same small set of moves — yet students lose marks by applying the wrong move to the wrong type. This guide gives the method, the direct-vs-inverse distinction, and four fully worked examples.
The core relationship
In the proportion a : b = c : d, b and c are called the means, and a and d are called the extremes. The rule "product of extremes = product of means" is exactly the cross-multiplication a × d = b × c above — the same statement, two names.
Direct vs inverse proportion
Two quantities are in direct proportion if one increases as the other increases, at a fixed ratio (more workers hired for a fixed time = more total wages paid). They are in inverse proportion if one increases as the other decreases, so that their product stays constant (more workers on a fixed job = fewer days needed). Choosing the wrong one is the single biggest source of error in this topic — always ask in words, before writing any equation, whether the second quantity should logically go up or down.
Worked example 1 — sharing a sum in a given ratio
Question: ₹4,500 is divided between A and B in the ratio 5 : 4. Find each share.
Total parts = 5 + 4 = 9
A's share = (5 ÷ 9) × 4500 = ₹2,500
B's share = (4 ÷ 9) × 4500 = ₹2,000
Check: 2500 + 2000 = 4500 ✓
Worked example 2 — finding an unknown term
Question: Solve for x: 3 : 5 = x : 20
Cross-multiply: 5 × x = 3 × 20
5x = 60
x = 12
Check: 3 : 5 = 12 : 20 → 3 ÷ 5 = 0.6 and 12 ÷ 20 = 0.6 ✓
Worked example 3 — inverse proportion (a work-rate problem)
Question: 6 workers, all working at the same rate, finish a job in 8 days. How many days would 12 workers take?
More workers means fewer days needed — this is inverse proportion, so workers × days = constant, not workers ÷ days.
6 × 8 = 48 (the constant — total worker-days needed for the job)
12 × days = 48
days = 4
Doubling the workers halved the time — the hallmark of an inverse relationship.
Worked example 4 — combining two ratios
Question: If A : B = 2 : 3 and B : C = 4 : 5, find A : B : C.
The value of B must match in both ratios before they can be combined. B is 3 in the first ratio and 4 in the second; the LCM of 3 and 4 is 12.
A : B = 2 : 3 = (2 × 4) : (3 × 4) = 8 : 12
B : C = 4 : 5 = (4 × 3) : (5 × 3) = 12 : 15
A : B : C = 8 : 12 : 15
Check: 8 ÷ 12 = 2 ÷ 3 ✓ and 12 ÷ 15 = 4 ÷ 5 ✓
Common mistakes that cost marks
- Cross-multiplying in the wrong order. a : b = c : d gives a × d = b × c — mixing up which pair multiplies together produces the reciprocal of the correct answer.
- Applying the direct-proportion method to an inverse-proportion problem (and vice versa). Always decide in words first: should the second quantity rise or fall as the first one rises?
- Not simplifying the ratio to lowest terms before working with it, which makes later arithmetic needlessly heavy.
- Forming a ratio from quantities in different units. "3 hours : 40 minutes" must become "180 minutes : 40 minutes" (or both in hours) before it is a valid ratio.
- Combining two ratios without matching the shared term first — exactly the step skipped when a student writes A : B : C = 2 : 3 : 5 in example 4 by simply lining the numbers up, which is wrong because B is not the same size in both original ratios.
Where this appears in exams
| Context | Typical use |
|---|---|
| Class 6–8 Maths | Ratio, proportion and unitary method chapters |
| Class 10 Maths | Similar triangles and scale-factor problems use the same ratio logic |
| Commerce/Accounts | Sharing partnership profit in the ratio of capital invested |
| Competitive exams (SSC/Bank quantitative sections) | Direct and inverse proportion word problems, mixture-and-alligation questions |
| Chemistry stoichiometry | Mole ratios from a balanced equation are proportion problems in disguise |
Check your own arithmetic. Set up the ratio by hand using the method above, then verify the final numbers on the Scientific Calculator before moving to the next question.
Open the ABC Chemistry Calculator Suite →Ratio and proportion is foundational to Class 11–12 chemistry numericals as much as to maths. ABC Chemistry teaches both together in its Class 11–12 batches, at the Gurugram centre and online across India: abcchemistry.in.