Ratio & Proportion

A ratio a : b means the quantities are ak and bk for some common multiplier k. Introduce that k and every question becomes a one-line equation.

The four types:

1. Divide a sum in a given ratio. Total parts = a + b + c; one part = total ÷ total parts; each share = its parts × one part.
2. Add or subtract the same amount from each. Take the numbers as ak and bk, form the new ratio, cross-multiply, solve for k.
3. Fourth proportional. If a : b :: c : x then ax = bc, so x = bc/a. In any proportion, product of the extremes = product of the means.
4. Compounding two ratios. Given A : B = 2 : 3 and B : C = 4 : 5, make B common by multiplying the first by 4 and the second by 3 → A : B : C = 8 : 12 : 15.

The single most common mistake is reading the wrong pair. If a vessel has milk and water in the ratio 3 : 2, then milk to water is 3 : 2 but milk to mixture is 3 : 5. Before dividing, ask yourself whether the denominator asked for is one part or the whole.

Speed tip: never convert to decimals. Keeping everything in "parts" means the arithmetic stays whole-number all the way to the answer.

a : b → take ak, bk · share = total × own parts / total parts · a : b :: c : x → x = bc/a · milk : mixture = a : (a+b)

₹6,300 is divided among A, B and C in the ratio 2 : 3 : 4. Find B's share. → Total parts = 2 + 3 + 4 = 9, so one part = 6300 ÷ 9 = ₹700. B has 3 parts = 3 × 700 = ₹2,100. Sanity check: the three shares are ₹1,400, ₹2,100 and ₹2,800, which add back to ₹6,300 and are in the ratio 2 : 3 : 4.