Three students A, B and C of a school receive cash prize in the ratio 3 : 4 : 5 in a competition. Then the school principal gives ₹ 4,000 to each student. As a result now the cash prize of A, B and C becomes in the ratio 5 : 6 : 7. How much did B get in the competition ?
- (1)₹ 12,000
- (2)₹ 10,000
- (3)₹ 8,000
- (4)₹ 6,000
Answer
Why
Correct — option (3), ₹ 8,000.
Step 1 — write the prizes in the ratio 3 : 4 : 5.
A = 3x, B = 4x, C = 5x
Step 2 — add ₹ 4,000 to each.
A = 3x + 4,000, B = 4x + 4,000, C = 5x + 4,000
Step 3 — use the new ratio A : B = 5 : 6.
6 × (3x + 4,000) = 5 × (4x + 4,000)
18x + 24,000 = 20x + 20,000
2x = 4,000
x = 2,000
Step 4 — B's prize in the competition.
B = 4x = 4 × 2,000 = ₹ 8,000
Step 5 — check all three.
Prizes: ₹ 6,000, ₹ 8,000, ₹ 10,000
After ₹ 4,000 each: ₹ 10,000, ₹ 12,000, ₹ 14,000
10,000 : 12,000 : 14,000 = 5 : 6 : 7 ✓
Quicker route: an equal gift leaves the gaps between prizes unchanged. The gap between neighbours is one part in 3 : 4 : 5 and one part in 5 : 6 : 7, so both ratios have the same unit x. Then A's new amount is 5x: 3x + 4,000 = 5x, and x = 2,000.
The idea to remember: when an equal amount is added to every share, the ratio of unequal shares changes but the gaps between shares do not.
Why the others are wrong
- (1)₹ 12,000 — Option (1) is B's total after the principal's gift: ₹ 8,000 + ₹ 4,000 = ₹ 12,000.
The stem asks how much B got in the competition, before the ₹ 4,000 was added. That is 4 × 2,000 = ₹ 8,000.
- (2)₹ 10,000 — Option (2) is C's prize in the competition, 5 × 2,000, and also A's amount after the gift, 6,000 + 4,000.
B's share is 4 parts of the 3 : 4 : 5 ratio, not 5. With one part worth ₹ 2,000, B got ₹ 8,000.
- (4)₹ 6,000 — Option (4) is A's prize in the competition, 3 × 2,000 = ₹ 6,000.
It is the right value for 3 parts, but B holds 4 parts of the 3 : 4 : 5 ratio, so B got 4 × 2,000 = ₹ 8,000.
Concept
A ratio a : b : c says the shares are a, b and c parts of one common unit. Writing the shares as ax, bx and cx turns the ratio into amounts once x is known.
Adding the same amount to unequal shares changes the ratio but not the differences between shares. That gives two routes: an equation from the new ratio, or the unchanged gaps.
Subtracting an equal amount behaves the same way. Multiplying every share by the same number leaves the ratio itself unchanged.
RPSC's 2023 syllabus lists "Ratio, Proportion and Partnership" under Basic Numeracy in Reasoning & Mental Ability.
Ages are a case of equal additions: everyone grows older by the same number of years, so the gap between two people's ages stays fixed while the ratio of their ages, if they differ, changes.
Partnership sums use the same idea of parts: profit is divided in the ratio of each partner's capital multiplied by the time for which it was invested, unless the sum states another agreement.
Key facts
- Shares in the ratio 3 : 4 : 5 are 3x, 4x and 5x for a common unit x.
- Adding an equal amount to each share leaves the differences between shares unchanged.
- Here x = ₹ 2,000, so the prizes were ₹ 6,000, ₹ 8,000 and ₹ 10,000.
- After ₹ 4,000 each, the amounts ₹ 10,000, ₹ 12,000 and ₹ 14,000 are in the ratio 5 : 6 : 7.
- In a partnership sum, profit is shared in the ratio of capital × time invested, unless another agreement is stated.
The gap of ₹ 2,000 between neighbours is the same before and after the gift.
Study next
Common traps
- Answering with the amount after the gift. The stem asks what B got in the competition, ₹ 8,000, not B's later total of ₹ 12,000.
- Treating ratio numbers as rupees. 3 : 4 : 5 turning into 5 : 6 : 7 does not mean ₹ 2 was added to each; the parts are multiples of an unknown x.
- Solving from one pair and not checking the third. x = 2,000 from A and B must also give C : A = 7 : 5; ₹ 14,000 : ₹ 10,000 confirms it.
A question can add or subtract an equal amount from every share and give the new ratio, as this one does, or give the amounts and ask for the new ratio.
A question can set the same idea in ages, mixtures, salaries or partnership profits.
Related PYQs
UnlockIAS will link similar questions from RAS Pre 2016 and 2013 here once those papers are published on this site.
Practice
- practice — not a real PYQ
The present ages of two brothers are in the ratio 2 : 3. After 6 years, the ratio of their ages will be 3 : 4. What is the present age of the elder brother ?
- (a)12 years
- (b)18 years
- (c)24 years
- (d)15 years
Answer(2) — Ages 2x and 3x keep their gap x. After 6 years the ratio 3 : 4 also has a gap of one part, so the younger brother's age then is 3x: 2x + 6 = 3x, x = 6. The elder is 3 × 6 = 18 years.Option (1) is the younger brother's present age, option (3) the elder's age after 6 years, and option (4) gives ages 10 and 15, which become 16 and 21, not 3 : 4.
- practice — not a real PYQ
Two numbers are in the ratio 4 : 7. If 6 is subtracted from each, the ratio becomes 1 : 2. What is the larger number ?
- (a)24
- (b)36
- (c)42
- (d)18
Answer(3) — Numbers 4x and 7x: 2 × (4x − 6) = 7x − 6, so 8x − 12 = 7x − 6 and x = 6. The numbers are 24 and 42, and 18 : 36 = 1 : 2 ✓.Option (1) is the smaller number, option (2) the larger number after subtracting 6, and option (4) the smaller number after subtracting 6.