A borrowed a sum of ₹1,60,000 from B at 10% per annum simple interest. At the…

2022

A borrowed a sum of ₹1,60,000 from B at 10% per annum simple interest. At the same time he lent the same sum to C at the same rate on compound interest, compounded semi-annually for 2 years. Find the amount (in ₹) earned by A in the whole transaction.

  1. A.

    ₹4,281

  2. B.

    ₹4,280

  3. C.

    ₹2,481

  4. D.

    ₹2,840

Attempted by 2 students.

Show answer & explanation

Correct answer: C

Simple interest is SI = P x R x T / 100. When a sum is compounded semi-annually at an annual rate R% for T years, the amount becomes P x (1 + R/200)2T, so the compound interest earned is CI = Amount - P. Since the same principal is borrowed from B and lent to C, the principal cancels out, so A's overall gain in the transaction is simply the compound interest earned from C minus the simple interest paid to B.

  1. Interest A pays to B (simple interest): SI = ₹1,60,000 x 10 x 2 / 100 = ₹32,000.

  2. Terms for the sum lent to C, compounded semi-annually: half-yearly rate = 10% / 2 = 5%, and the number of half-year periods over 2 years = 2 x 2 = 4.

  3. Amount C owes A: Amount = ₹1,60,000 x (1.05)4 = ₹1,60,000 x 1.21550625 = ₹1,94,481.

  4. Compound interest A earns from C: CI = ₹1,94,481 - ₹1,60,000 = ₹34,481.

  5. Net amount A earns in the whole transaction: CI earned - SI paid = ₹34,481 - ₹32,000 = ₹2,481.

(1.05)4 = (21/20)4 = 194481/160000 exactly, since 212 = 441 and 4412 = 1,94,481, so there is no rounding error in the amount. A period-by-period build-up (₹1,60,000 to ₹1,68,000 to ₹1,76,400 to ₹1,85,220 to ₹1,94,481 after each half-year) confirms the same compound interest of ₹34,481, giving the same net gain of ₹2,481.

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