If ₹ R are divided between ‘A’ and ‘B’ in the ratio of a/b : c/d, then what is…

Concept: A ratio does not change when both of its terms are multiplied by the same non-zero quantity — a : b and ka : kb describe exactly the same division, because both shares are scaled by the identical factor k. That invariance is what allows a ratio written with fractions to be replaced by an equivalent whole-number ratio. And once a total is to be split in a whole-number ratio m : n, the total is read as m + n equal parts, so the first party receives R × m/(m + n).

1. The two shares stand in the ratio A : B = a/b : c/d. As written, both terms are fractions, so they cannot yet be read as a count of parts — the denominators must be cleared first.

2. Choose the multiplier. The invariance above permits any non-zero multiplier, so choose one that clears both denominators at the same time: bd, the product of the two denominators, because bd is divisible by b and also by d.

3. Multiply the first term by bd: (a/b) × bd = a × (bd ÷ b) = a × d = ad. The b in the denominator cancels the b inside bd, so only d is left standing beside a.

4. Multiply the second term by bd: (c/d) × bd = c × (bd ÷ d) = c × b = bc. This time the d in the denominator cancels the d inside bd, so only b is left standing beside c.

5. Both terms were multiplied by the same quantity bd, so by the invariance the ratio is unchanged: A : B = a/b : c/d = ad : bc. This is the step the rest of the working rests on — A corresponds to ad parts and B corresponds to bc parts.

6. Total number of parts = ad + bc.

7. A's share = R × [ad/(ad + bc)] = adR/(ad + bc).

The clearing step at a glance:

Cross-check with numbers: take a = 1, b = 2, c = 1, d = 3, so the ratio is 1/2 : 1/3. Multiplying both terms by bd = 6 gives 3 : 2, that is 5 parts in all, so A should get (3/5)R. Substituting the same values into adR/(ad + bc) gives ad = 1 × 3 = 3 and bc = 2 × 1 = 2, so the share is 3R/(3 + 2) = 3R/5. The direct split and the formula agree, so the amount A receives is adR/(ad + bc).