In a business, A, B and C had invested in the ratio of 2 : 3 : 5, but they all earned the same amount of profit at the end. If the profit is proportional to the amount as well as the duration of the investment, what will be the ratio of duration of their investments ?
- (a)5 : 3 : 2
- (b)2 : 3 : 5
- (c)10 : 15 : 6
- (d)15 : 10 : 6
Answer
Why
Correct — D, (d) 15 : 10 : 6.
The governing relation. The stem states it: profit is proportional to the amount invested AND to the duration of the investment. So for each partner, profit is proportional to (capital × time). This product is the partner's 'capital-months', and it is the only quantity that matters in a partnership problem.
Setting up. Let the capitals be 2k, 3k and 5k and the durations t1, t2 and t3. All three earned the SAME profit, so their products are equal: 2k·t1 = 3k·t2 = 5k·t3. The k cancels, leaving 2t1 = 3t2 = 5t3.
Solving. Three quantities whose multiples are equal are in the ratio of the RECIPROCALS of those multipliers: t1 : t2 : t3 = 1/2 : 1/3 : 1/5. Multiply through by the lowest common multiple of 2, 3 and 5, which is 30: t1 : t2 : t3 = 15 : 10 : 6.
Check it. Take the durations as 15, 10 and 6 months and the capitals as 2, 3 and 5. The products are 2 × 15 = 30, 3 × 10 = 30 and 5 × 6 = 30 — equal, so the profits are equal, as required.
Sense-check the shape of the answer before writing it down. A partner who put in the LEAST money must leave it in the LONGEST to earn as much as the others, so the durations must run in the opposite order to the capitals: the 2-share has the largest duration and the 5-share the smallest. Options (b) and (c) fail that test at a glance, and (a), though it does decrease, is the wrong kind of decrease — it is a simple reversal rather than an inversion.
A concrete illustration makes the result obvious. Suppose A puts in ₹ 2,000 for 15 months, B ₹ 3,000 for 10 months and C ₹ 5,000 for 6 months. Each has 30,000 rupee-months to his credit, so each takes an equal share of the profit — which is what the question stipulates. The durations in that example are 15, 10 and 6, exactly the keyed ratio.
Why the others are wrong
- (a)5 : 3 : 2 — This REVERSES the capital ratio instead of INVERTING it, and the two operations are different. Reversing gives 5 : 3 : 2; inverting gives 1/2 : 1/3 : 1/5, which is 15 : 10 : 6. Test it: capitals 2, 3, 5 with durations 5, 3, 2 give products 10, 9 and 10 — not equal, so the profits would not be equal. Reversal only coincides with inversion in the special case of two terms.
- (b)2 : 3 : 5 — The capital ratio copied unchanged. It would mean the partner who invested most also left it in longest, so the profits would be wildly unequal: products 4, 9 and 25. This option catches a candidate who reads the question quickly and answers the ratio that is printed in it, which is why examiners include the given ratio among the choices in almost every problem of this type.
- (c)10 : 15 : 6 — The right three numbers in the wrong order — the first two are interchanged. It is what results from computing the reciprocals correctly and then attaching them to the partners carelessly, or from writing 1/3 before 1/2. The order is fixed by the stem, which names A, B and C with capitals in the ratio 2 : 3 : 5, so the largest duration must stand first, against the smallest capital.
Concept
In a partnership, profit is shared in the ratio of each partner's capital multiplied by the time for which it was invested. Where all partners invest for the same period, the time factor is common and the profit ratio is simply the capital ratio — a SIMPLE partnership. Where the periods differ, the products must be used — a COMPOUND partnership. The same product appears in reverse in questions like this one, where the profits are given as equal and the durations are asked for. The general principle behind the arithmetic is that when the products of paired quantities are equal, the quantities in each pair are inversely proportional: doubling one halves the other. The same product also governs the reverse question — given the profits and the capitals, find the times — and the mixed case, where a partner joins or withdraws part-way and his capital-months are computed in two pieces and added. In every version the working rule is the same: reduce each partner to a single number, capital multiplied by the months for which it stood, and compare those numbers.
Partnership questions are a fixture of the quantitative block because they test proportional reasoning in a form that has an obvious business meaning — and an Accounts Officer meets the same computation when profits are apportioned between partners in a firm. This particular item inverts the usual direction of the question, giving the profits and asking for the time, which is exactly the twist a candidate must be ready for.
Partnership questions come in two directions and this is the less common one: the profits are given as equal and the durations are asked for, so the reasoning runs backwards from the usual drill. That reversal is where marks are lost, because a candidate who has practised only the forward form reaches for the capital ratio. Guard against it with a sense check before writing the answer — the partner with the least capital must have the longest duration, so the answer must run in the opposite order to the ratio given. Two of the four options fail that check on sight.
Key facts
- Profit share is proportional to capital multiplied by time — the capital-months of each partner.
- Equal profits mean equal products of capital and time.
- If 2t1 = 3t2 = 5t3, then t1 : t2 : t3 = 1/2 : 1/3 : 1/5.
- Multiplying a ratio of reciprocals by the lowest common multiple of the denominators clears the fractions: here 30 gives 15 : 10 : 6.
- Inverting a ratio is not the same as reversing it, except in the two-term case.
- Check: capitals 2, 3, 5 with times 15, 10, 6 give equal products of 30.
- Where all partners invest for the same period, the profit ratio reduces to the capital ratio.
- A working partner's salary, where one is provided for, is deducted before the remaining profit is divided in this ratio.
Study next
Common traps
- Reversing the ratio instead of inverting it.
- Answering with the ratio given in the question.
- Attaching the correct reciprocals to the wrong partners.
- Forgetting that profit depends on capital AND time when the periods are unequal.
Partnership items appear in most papers in this family, sometimes forward — find the profit share — and sometimes inverted, as here. Both are handled by writing capital × time for each partner first; every question in the family is then a matter of comparing those products.
Related PYQs
EPFO_APFC_2023_Q76The ages of Mr. Kumar and his son are in the ratio 5 : 3. Fifteen years back this ratio was 2 : 1. What was the age (in years) of Mr. Kumar when his son was born?
- (a) 30
- (b) 35
- (c) 40
- (d) 45
Answer(a) 30
A ratio problem from the earlier paper worked with the same discipline — ages in a given ratio now and a different ratio fifteen years earlier, solved by writing each quantity as a multiple of one unknown.
Practice
- practice — not a real PYQ
P and Q invest in a business in the ratio 4 : 5 and receive equal profits. What is the ratio of the durations of their investments ?
- (a)4 : 5
- (b)5 : 4
- (c)16 : 25
- (d)25 : 16
Answer(b) 5 : 4
- practice — not a real PYQ
X invests ₹ 12,000 for 6 months and Y invests ₹ 9,000 for 8 months in the same business. In what ratio should the profit be divided ?
- (a)1 : 1
- (b)4 : 3
- (c)3 : 4
- (d)2 : 3
Answer(a) 1 : 1