Ratio, Proportion & Percentage

Snapshot

• Ratio, Proportion & Percentage is the engine room of the General Test. It is asked directly and it powers almost every other quant topic — profit & loss, averages, data interpretation, even time–speed–work. Get fluent here and a third of the quant paper becomes mental maths.
• Three skills carry the whole topic: the fraction ↔ percent conversions, dividing a quantity in a ratio, and the iron rule that percentage change is always taken on the base (old) value.
• This guide covers percentage (including successive and reverse change), ratio and proportion (direct, inverse, compound), partnership, and mixtures / alligation — with worked examples for each.
• Exam reality: +5 / −1. These questions are short; speed and a clear head about which base a percentage sits on win the marks.

Part 1 — Percentage

"Percent" means "per hundred". What percent is A of B = (A ÷ B) × 100. X percent of N = (X ÷ 100) × N. Memorise these pairs so you never long-divide:

Percentage change = (new − old) ÷ OLD × 100 — always divide by the old value. Rise from 80 to 100 is 20 ÷ 80 = 25%, not 20%.

Successive change. Two changes of a% and b% combine to a + b + (ab ÷ 100) (use minus signs for decreases). A 20% rise then a 20% fall = 20 − 20 − 4 = a net 4% fall — never zero.

Reverse percentage. If a price after a 20% discount is ₹240, the original was 240 ÷ 0.8 = ₹300. If A is 25% more than B, then B is 20% less than A (not 25%) — because the base changes.

Percentage point vs percent. A share rising from 20% to 25% is 5 percentage points, but a 25% relative increase (5 ÷ 20). CUET offers both as options on purpose.

Part 2 — Ratio & Proportion

A ratio A : B compares two amounts, reduced to lowest terms. To divide a quantity in a ratio, add the parts to get total units, find one unit, then scale — the single most common ratio question:

Split ₹3,600 in 2 : 3 : 4: total = 9 units, one unit = 3,600 ÷ 9 = ₹400 → shares ₹800, ₹1,200, ₹1,600.

A proportion says two ratios are equal: a : b = c : d means a × d = b × c (cross-multiply). In a continued proportion a : b = b : c, the middle term is the mean proportional, b = √(a × c) — mean proportional of 4 and 9 is √36 = 6.

Direct proportion: more of one ⇒ more of the other (workers ↔ output); the ratio stays constant. Inverse proportion: more of one ⇒ less of the other (workers ↔ time); the product stays constant. Compound proportion chains several together (men, days and hours in the classic "work" formula M₁D₁H₁ ÷ W₁ = M₂D₂H₂ ÷ W₂).

Partnership. Profit is shared in the ratio of (capital × time) for each partner. If A invests ₹x for 12 months and B invests ₹y for 6 months, their profit ratio is 12x : 6y.

Part 3 — Mixtures & Alligation

When two ingredients at different "prices" (or strengths) are mixed, the rule of alligation gives the mixing ratio in one line:

(quantity of cheaper) : (quantity of dearer) = (dearer − mean) : (mean − cheaper).

So to get ₹40 "milk" by mixing ₹60 milk with free water, milk : water = (40 − 0) : (60 − 40) = 2 : 1. Alligation also answers "what fraction is water" and "replacement" sums (the repeated replacement result: after removing and replacing x from a vessel of V, n times, the pure quantity left = V(1 − x/V)ⁿ).

Part 4 — Speed techniques

1. Convert every percentage to a fraction before multiplying: 37.5% of 800 = 3/8 × 800 = 300.
2. Flip the percentage when one number is friendly: 16% of 25 = 25% of 16 = 4.
3. Anchor to 10% and 1%: 23% of 350 = (10% × 2) + (1% × 3) = 70 + 10.5 = 80.5.
4. Use unit value for ratios — find one part, then every share is one multiplication.
5. Use complements: if 35% is spent, compute the 65% that remains if it is friendlier.
6. For "x% more / less", remember the base flips — +25% one way is −20% the other.
7. Sanity-check direction — an increase answer must exceed the original; a discount answer must be smaller.

Part 5 — Worked examples

1. Successive change. A number is increased 25% then decreased 20%. Net? 25 − 20 − (25×20÷100) = 0% — no change.

2. Reverse percentage. After a 12% rise, a salary is ₹28,000. Original? 28,000 ÷ 1.12 = ₹25,000.

3. Divide in a ratio. Split ₹6,300 among A, B, C in 2 : 3 : 4. 9 units = 6,300 → unit = 700 → ₹1,400, ₹2,100, ₹2,800.

4. Inverse proportion. 15 workers finish a job in 8 days; how long for 20 workers? 15×8 = 20×t → t = 6 days.

5. Mean proportional. Mean proportional of 8 and 32 = √(8×32) = √256 = 16.

6. Partnership. A puts ₹12,000 for 8 months, B ₹9,000 for 12 months. Profit ratio? (12000×8) : (9000×12) = 96 : 108 = 8 : 9.

7. Alligation. In what ratio mix rice at ₹30/kg and ₹40/kg to sell the mix at ₹36/kg? cheaper : dearer = (40−36):(36−30) = 4 : 6 = 2 : 3.

8. Replacement. From 40 L of pure milk, 4 L is removed and replaced with water, twice. Milk left? 40(1 − 4/40)² = 40 × (0.9)² = 32.4 L.

Part 6 — Common traps

• Base trap — percentage change is on the old value, every time.
• Successive ≠ sum — +20% then −20% is a net loss, not zero.
• "x% more" flips the base — A 25% more than B does not make B 25% less than A.
• Ratio direction — "A to B" is A : B; read the order before reducing.
• Direct vs inverse — workers↔output is direct; workers↔time is inverse. Decide first, then set up.
• Percentage point vs percent — read which one the question wants.

Part 7 — How to use this page

Memorise the fraction–percent table and the alligation rule, re-solve the eight examples with the page hidden, then attempt the practice set below — for each miss, name the rule that caught you — and finish with the timed test.

One-line revision: convert percentages to fractions, divide percentage change by the old value, split a quantity by finding one unit first, keep the product constant for inverse proportion, and reach for the alligation cross whenever two things are mixed.