Three persons A, B and C run a business together and their shares are 17%, 37% and 46% respectively. Any profit they earn is distributed according to the proportion of their shares. If the difference of the profits of B and A on a given date is ₹ 1,000, what is the profit of C on that day?

Correct — A, (a) ₹ 2,300. Because the profit is divided in proportion to the shares, a difference in shares translates directly into a difference in rupees. B holds 37 per cent and A holds 17 per cent, so B's profit exceeds A's by 37 − 17 = 20 percentage points of the day's profit, and the stem says that difference is ₹ 1,000. One percentage point of the profit is therefore ₹ 1,000 ÷ 20 = ₹ 50. That single number answers everything else.

C holds 46 per cent, which is 46 such points, so C's profit is 46 × ₹ 50 = ₹ 2,300 — option (a).

The whole picture is worth writing out, because it confirms the working and takes only a moment. If one point is ₹ 50, the day's profit is 100 × ₹ 50 = ₹ 5,000. A receives 17 per cent of that, ₹ 850; B receives 37 per cent, ₹ 1,850; C receives 46 per cent, ₹ 2,300. The three add to ₹ 5,000, as they must, and B's ₹ 1,850 exceeds A's ₹ 850 by exactly the ₹ 1,000 the stem gives. Notice also that the three shares themselves add to 100 per cent, which is the first thing to check in any question of this kind: if the stated shares did not exhaust the whole, some part of the profit would be going elsewhere and the proportional reasoning would need adjusting.

The method generalises. Whenever a quantity is divided in a fixed proportion, find the value of one unit of the proportion — one percentage point here, one part where a ratio is given — from whatever piece of information the question supplies, and then multiply. It does not matter whether the information given is one person's share, the total, or the difference between two shares.

Why the others are wrong

• (b)₹ 2,350 — This is ₹ 2,350, which is fifty rupees more than the correct figure and corresponds to a share of 47 per cent rather than 46. It is a reading slip rather than a method error, and the check that catches it is to add the three stated shares: 17 + 37 + 46 comes to exactly 100, so no partner holds 47 per cent and nothing in the question can produce that figure. Tested the other way, the option fails too. If C's profit were ₹ 2,350, then one percentage point would be worth 2,350 ÷ 46 = ₹ 51·09, and the difference between B's and A's profits would come to 20 × ₹ 51·09, about ₹ 1,022, rather than the ₹ 1,000 the stem states. On proportional-division items the shares should always be added first; the check takes two seconds and it confirms that the percentages have been read correctly.
• (c)₹ 2,450 — ₹ 2,450 corresponds to a share of 49 per cent, which again is not one of the three figures given. The option is placed to catch a candidate who has arrived at the right method but has taken the wrong partner's percentage, or who has added a difference to a share instead of multiplying. Back-substitution settles it: if C received ₹ 2,450 then a percentage point would be worth 2,450 ÷ 46 = ₹ 53·26, and the gap between B and A would be 20 × ₹ 53·26, about ₹ 1,065, which contradicts the stem. There is a further structural check available here. C's share is 46 points against the 20-point gap between B and A, so C's profit must be 46 ÷ 20 = 2·3 times the stated ₹ 1,000, that is ₹ 2,300 exactly. Any option that is not 2·3 times ₹ 1,000 is wrong before any arithmetic is done.
• (d)₹ 4,600 — ₹ 4,600 is exactly twice the correct answer, and it comes from valuing a percentage point at ₹ 100 instead of ₹ 50 — that is, from dividing the ₹ 1,000 by ten points rather than by the twenty that actually separate B from A. A candidate who subtracts 46 − 37 = 9 and then rounds, or who halves the difference somewhere in the working, lands here. The consequences are easy to test and plainly wrong: at ₹ 100 a point the whole day's profit would be ₹ 10,000, and B's share of ₹ 3,700 would exceed A's of ₹ 1,700 by ₹ 2,000, not by the ₹ 1,000 given. An answer that is a round multiple of the correct one is nearly always produced by a factor error of this kind, and running the answer back through the sentence that generated it exposes it at once.

Proportional division is the arithmetic of sharing a quantity among parties in stated proportions, and it is the same whether the proportions are given as percentages, as a ratio or as fractions. The technique that never fails is the unitary one: express every party's entitlement in the same units — percentage points here, parts where a ratio such as 2 : 3 : 5 is given — and then find what one unit is worth. Any single piece of numerical information will do it. The total profit divided by 100 gives the value of a point; one partner's amount divided by that partner's points gives it; and, as in this question, the difference between two partners' amounts divided by the difference in their points gives it too. Once one unit is valued, every other quantity in the problem follows by multiplication. Two checks come with the method. The first is that the stated proportions must exhaust the whole — the three percentages here add to 100 — since otherwise part of the amount is unaccounted for. The second is that the answers must add back to the total. In the vocabulary of the topic it is worth keeping percentage points distinct from percentages: the gap between 37 per cent and 17 per cent is twenty percentage points of the profit, and it is that gap, not any percentage of a percentage, which the ₹ 1,000 measures.

Key facts

• Profit divided in proportion to shares means that any difference in shares corresponds to the same difference in profit.
• B's share of 37 per cent exceeds A's of 17 per cent by 20 percentage points, and that gap is worth ₹ 1,000.
• One percentage point of the profit is therefore ₹ 1,000 ÷ 20 = ₹ 50, and the whole profit is ₹ 5,000.
• C's 46 per cent gives 46 × ₹ 50 = ₹ 2,300, which is the answer.
• The three partners receive ₹ 850, ₹ 1,850 and ₹ 2,300, which add back to the ₹ 5,000 total.
• The three shares add to exactly 100 per cent, so no part of the profit is unaccounted for.
• C's profit must be 46 ÷ 20 = 2·3 times the stated difference of ₹ 1,000, which fixes the answer without finding the total.
• The unitary method works from any single datum — a total, one partner's amount, or the difference between two amounts.

Common traps

• Looking for the total profit, which the question never gives, instead of working from the difference between two shares.
• Dividing the stated difference by the wrong number of percentage points, which scales every answer by a factor.
• Using a partner's share other than C's, which produces the two near-miss options in this set.
• Failing to check that the three shares add to a hundred, the quickest confirmation that the percentages have been read correctly.
• Confusing a difference in percentage points with a percentage of a percentage, which are different quantities.

• practice — not a real PYQ

  Three partners hold 20 per cent, 30 per cent and 50 per cent of a firm, and profits are shared in proportion to their holdings. If the profit of the second partner exceeds that of the first by ₹ 2,000, what is the profit of the third partner?

  • (a)₹ 5,000
  • (b)₹ 8,000
  • (c)₹ 10,000
  • (d)₹ 12,000

  Answer(c) ₹ 10,000

• practice — not a real PYQ

  A sum of ₹ 7,500 is divided among X, Y and Z in the ratio 2 : 3 : 5. What is Z's share?

  • (a)₹ 1,500
  • (b)₹ 2,250
  • (c)₹ 3,000
  • (d)₹ 3,750

  Answer(d) ₹ 3,750