2/3 of a principal amount is deposited in the bank at compound interest at the rate of 10% per annum and rest of the principal amount is deposited in the post office at the simple interest rate of 15% per annum. If the difference between compound interest and simple interest for two years be ₹ 480, then total principal amount is equal to

Correct — option (2), ₹ 12,000.

Let the total principal be P. Two different parts earn interest under two different rules, so each interest is worked out on its own part.

Step 1 — split the principal.
Bank part = 2P/3
Post office part = P − 2P/3 = P/3

Step 2 — compound interest on the bank part, 10% a year for 2 years.
Amount factor = 1.1 × 1.1 = 1.21
CI = 2P/3 × (1.21 − 1) = 2P/3 × 0.21 = 0.14P

Step 3 — simple interest on the post office part, 15% a year for 2 years.
SI = P/3 × 15 × 2 ÷ 100 = P/3 × 0.30 = 0.10P

Step 4 — subtract.
CI − SI = 0.14P − 0.10P = 0.04P

Step 5 — equate to ₹ 480.
0.04P = 480
P = 480 ÷ 0.04 = ₹ 12,000

Check: the bank gets ₹ 8,000 and earns 8,000 × 0.21 = ₹ 1,680. The post office gets ₹ 4,000 and earns 4,000 × 0.15 × 2 = ₹ 1,200. The difference is 1,680 − 1,200 = ₹ 480.

The idea to remember: when two parts earn interest at different rates, find each interest as a fraction of P first, then subtract.

Why the others are wrong

• (1)
  ₹ 16,000 — With P = ₹ 16,000, the bank part is ₹ 32,000/3 and earns 0.14 × 16,000 = ₹ 2,240. The post office part earns 0.10 × 16,000 = ₹ 1,600.

  The difference is ₹ 640, not ₹ 480. Since the gap is always 0.04P, a gap of ₹ 480 fixes P at ₹ 12,000.

• (3)
  ₹ 10,000 — With P = ₹ 10,000, the compound interest is 0.14 × 10,000 = ₹ 1,400 and the simple interest is 0.10 × 10,000 = ₹ 1,000.

  The difference is ₹ 400, which falls short of ₹ 480. The gap 0.04P equals ₹ 480 only when P = ₹ 12,000.

• (4)
  ₹ 8,000 — ₹ 8,000 is the amount placed in the bank when the total is ₹ 12,000 (two-thirds of ₹ 12,000), not the total principal.

  Taken as the total, ₹ 8,000 gives CI = ₹ 1,120 and SI = ₹ 800, a difference of ₹ 320, not ₹ 480.

Simple interest is charged on the original principal only: SI = P × R × T ÷ 100. Over 2 years at 15%, it adds 30% of the principal.

Compound interest adds each year's interest to the principal before the next year's interest is worked out. Over T years, Amount = P × (1 + R/100)ᵀ and CI = Amount − P.

At 10% for 2 years, 1.1 × 1.1 = 1.21, so compound interest is 21% of the principal. The extra 1% over simple interest is 10% interest on the first year's interest.

Key facts

• Simple interest: SI = P × R × T ÷ 100.
• Compound interest, compounded yearly: Amount = P × (1 + R/100)ᵀ; CI = Amount − P.
• At 10% a year compounded yearly for 2 years, CI = 21% of the principal.
• For one principal at one rate over 2 years, CI − SI = P × (R/100)².
• Here the gap is 0.14P − 0.10P = 0.04P = ₹ 480, so P = ₹ 12,000 (₹ 8,000 in the bank, ₹ 4,000 in the post office).

Common traps

• Applying CI − SI = P(R/100)² here. That formula needs one principal at one rate; this stem splits the money and uses 10% compound against 15% simple.
• Taking compound interest for 2 years at 10% as 20%. Compounding gives 21%, because the second year also earns interest on the first year's ₹ 0.10 per rupee.
• Stopping at a part instead of the total. ₹ 8,000 is the bank deposit and ₹ 4,000 the post office deposit; the stem asks for the whole principal.

UnlockIAS will link similar questions from RAS Pre 2018 and 2016 here once those papers are published on this site.

• practice — not a real PYQ

  The simple interest on ₹ 4,200 for 2 years at 10% per annum equals the compound interest (compounded annually) on another sum for 2 years at 10% per annum. The other sum is

  • (a)₹ 4,000
  • (b)₹ 4,200
  • (c)₹ 3,800
  • (d)₹ 4,410
  Answer(1) — SI = 4,200 × 10 × 2 ÷ 100 = ₹ 840. CI on x for 2 years at 10% = 0.21x, so 0.21x = 840 and x = ₹ 4,000. Option (2) would earn CI of ₹ 882, option (3) ₹ 798 and option (4) ₹ 926.10.
• practice — not a real PYQ

  A sum of ₹ 20,000 is split into two equal parts. One part earns 10% per annum compound interest (compounded annually) and the other 10% per annum simple interest, both for 2 years. What is the difference between the two interests?

  • (a)₹ 200
  • (b)₹ 100
  • (c)₹ 210
  • (d)₹ 50
  Answer(2) — Each part is ₹ 10,000. CI = 10,000 × 0.21 = ₹ 2,100; SI = 10,000 × 0.20 = ₹ 2,000; difference = ₹ 100. Option (1) applies P(R/100)² to the whole ₹ 20,000 though only half earns compound interest; options (3) and (4) do not equal 2,100 − 2,000.