The monthly incomes of Peter and Paul are in the ratio of 4 : 3. Their expenses are in the ratio of 3 : 2. If each saves ₹ 6,000 at the end of the month, their monthly incomes respectively are (in ₹)

(a) 24,000 and 18,000 ✓ UPSC's answer
(b) 28,000 and 21,000
(c) 32,000 and 24,000
(d) 34,000 and 26,000

Why the answer is (a)

• Let the monthly incomes of Peter and Paul be 4x and 3x respectively, based on the given ratio of 4:3.

• Let their monthly expenses be 3y and 2y respectively, based on the given ratio of 3:2.

• Since each saves ₹6,000, we form the equations: 4x - 3y = 6,000 and 3x - 2y = 6,000.

• To eliminate y, multiply the first equation by 2 and the second by 3, resulting in 8x - 6y = 12,000 and 9x - 6y = 18,000.

• Subtracting the first modified equation from the second gives x = 6,000.

• Substituting x = 6,000 back into the income expressions yields Peter's income as 4(6,000) = 24,000 and Paul's income as 3(6,000) = 18,000, which matches option (a).

Why the other options are wrong

(b) 28,000 and 21,000: The incomes 28,000 and 21,000 imply x = 7,000, which would result in savings of ₹8,400, not ₹6,000.
(c) 32,000 and 24,000: The incomes 32,000 and 24,000 imply x = 8,000, which would result in savings of ₹10,800, not ₹6,000.
(d) 34,000 and 26,000: The incomes 34,000 and 26,000 do not maintain the required 4:3 income ratio (34:26 simplifies to 17:13).

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2015, held on 23 August 2015. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.