Simple and Compound Interest

Compound Interest formula: A = P(1 + r/100)^n. Here P=24000, r=20%, n=3. Amount after 3 years = 24000 * (1.2)^3 = 24000 * 1.728 = 41472. Compound Interest = Amount - Principal = 41472 - 24000 = 17472. Option D matches.

First find rate: SI = P×R×T/100, so 5580 = 15500×R×3/100, giving R = 12%. Now calculate CI for 3 years at 12%: Amount = 15500×(1.12)³ = 15500×1.404928 = 21776.384. CI = Amount - Principal = 21776.384 - 15500 = 6276.384. This matches option B.

Compound Interest = P[(1 + r/100)² - 1] = 3500[(1 + 12/100)² - 1] = 3500[(1.12)² - 1] = 3500[1.2544 - 1] = 3500 × 0.2544 = Rs. 890.40. Option D matches this calculation exactly.

Simple Interest = (P × R × T)/100 = (72000 × 17 × 3)/100 = Rs. 36720. Total amount = Principal + Interest = 72000 + 36720 = Rs. 108720. Option A is correct.

Compound Interest formula: A = P(1 + r/100)^n. Here P = 36000, r = 8%, n = 2 years. A = 36000 × (1.08)^2 = 36000 × 1.1664 = 41990.40. CI = A - P = 41990.40 - 36000 = 5990.40. Option B matches this calculation.

Using the compound interest formula: A = P(1 + r/100)ⁿ = 4000(1 + 12/100)² = 4000(1.12)² = 4000 × 1.2544 = 5017.60. Compound Interest = 5017.60 - 4000 = 1017.60.

Simple Interest = Principal × Rate × Time. Given SI = (1/8) × 24000 = 3000 over 2 years. So 3000 = 24000 × R × 2, which gives R = 3000/(24000×2) = 6.25%. The key is recognizing that the simple interest is one-eighth of the principal, not the amount.

Compound Interest = P(1+r/100)³ - P = 3000(1.12)³ - 3000 = 3000(1.404928) - 3000 = 4214.78 - 3000 ≈ 1215. Option D matches.

From statement II: Rate = 5%. From statement I: CI for 2 years = 484. Using CI formula A = P(1 + r/100)^t, we get P(1.05)^2 - P = 484, giving P = 4400. SI on 4400 at 5% = (4400 × 5 × 1)/100 = 220, not 500. From statement III: CI - SI = 41 for 2 years at 5%. The difference formula is P(r/100)^2 = 41, giving P = 16400. SI on 16400 at 5% for 1 year = (16400 × 5 × 1)/100 = 820, not 500. The question asks for SI = 500, but all statements lead to different principals. With all three statements, we can solve but none give SI = 500, so the answer is 'None of these'.

In compound interest, amount doubles in 15 years: 2P = P(1+r)¹⁵, so (1+r)¹⁵ = 2. For 8 times: 8P = P(1+r)ⁿ, so (1+r)ⁿ = 8 = 2³. Since (1+r)¹⁵ = 2, we need (1+r)ⁿ = [(1+r)¹⁵]³ = (1+r)⁴⁵, so n = 45 years.

For 3 years simple interest = 4000 × R × 3/100 = 120R. For 2 years compound interest = 4000[(1+R/100)² - 1] = 80R + 0.4R². The difference |120R - (80R + 0.4R²)| = 190 gives 0.4R² - 40R + 190 = 0. Solving: R = [100 ± √8100]/2 = [100 ± 90]/2. R = 5% or 95%. Only R=5% is practically valid, which is not in options A-D, making E correct.

Simple Interest = (Principal × Rate × Time)/100 = (10200 × 13 × 7)/100 = 10200 × 91/100 = 9282. None of the given options (9172, 9256, 9318, 9035) match Rs. 9282, so option E (None of these) is correct.

Compound Interest = P[(1 + r/100)^n - 1]. CI = 1800[(1 + 6/100)^2 - 1] = 1800[(1.06)^2 - 1] = 1800[1.1236 - 1] = 1800 × 0.1236 = Rs. 222.48. The amount after 2 years would be 1800 × 1.1236 = Rs. 2022.48, so interest earned is Rs. 222.48.

Simple Interest = P × R × T / 100. For same P and T, difference in SI = P × T × (R1 - R2) / 100. Given difference = Rs. 2.50, P = Rs. 500, T = 2 years. So 2.50 = 500 × 2 × (R1 - R2) / 100, which gives 2.50 = 10 × (R1 - R2), so (R1 - R2) = 0.25%. The rate difference is 0.25%.

Principal = 4000, time = 3 years, rate = 5%. Simple Interest = PRT/100 = 4000×5×3/100 = 600. Compound Interest = P[(1+R/100)^T - 1] = 4000[(1.05)³ - 1] = 4000[1.157625 - 1] = 4000×0.157625 = 630.50. Difference = 630.50 - 600 = 30.50.