Simple and Compound Interest

Let the amounts be x at 12.5% and y at 11.11% (which equals 1/9). The equations are: 0.125x + (1/9)y = 120 and (1/9)x + 0.125y = 128. Testing option A (2000, 3000): 0.125(2000) + (1/9)(3000) = 250 + 333.33 ≠ 120. Testing option C (1000, 1975): 0.125(1000) + (1/9)(1975) ≈ 125 + 219.44 ≠ 120. None of the given options satisfy both conditions simultaneously. The correct solution requires solving the system of equations, which yields different values.

For simple interest: SI = P×R×T/100. Given: Principal = Rs. 4800, Amount = Rs. 6480, Time = 7 years. Interest earned = 6480 - 4800 = Rs. 1680. Using the formula: 1680 = 4800×R×7/100. Solving: R = (1680×100)/(4800×7) = 168000/33600 = 5%. Therefore, the rate of simple interest is 5% per annum.

Principal = 19500, rate = 7%, time = 3 years. SI = P × R × T/100 = 19500 × 7 × 3/100 = 4095. Amount = 19500 + 4095 = 23595. After penalty of 840: received amount = 23595 - 840 = 22755.

Using compound interest formula: A = P(1 + r/100)^t. CI for 2 years = P[(1 + 24/100)^2 - 1] = P[(1.24)^2 - 1] = P[1.5376 - 1] = 0.5376P = 10080. Therefore P = 10080/0.5376 = Rs. 18750. Option B is correct.

Let x be each installment. Present value of all installments equals principal: x/(1.2) + x/(1.2²) + x/(1.2³) = 150000. This gives x(0.8333 + 0.6944 + 0.5787) = 150000, so x(2.1064) = 150000, x ≈ 71209.

SI for 2 years = Rs.550 at 5%, so SI for 1 year = Rs.275. Principal = 275/0.05 = Rs.5500. CI for 2 years = 5500 × (1 + 0.05)² - 5500 = 5500 × 1.1025 - 5500 = 6063.75 - 5500 = Rs.563.75. Difference = CI - SI = 563.75 - 550 = Rs.13.75. Option B is correct.

From Statement I: Principal becomes 5 times in 12 years at simple interest. Using P(1 + 12r/100) = 5P, we get r = 400/12 = 33.33% per annum. Statement II: At 3/4 of this rate (25% CI), principal earns Rs. 1350 in 2 years, giving P((1.25)^2 - 1) = 1350, so P = 2400. Statement III: Difference between CI and SI for 3 years is Rs. 487.5, which can verify the rate. With P=2400 and r=33.33%, we can calculate amount at 3.5 years using compound interest.

Simple Interest formula: SI = P × R × T / 100. Here, SI = 270, R = 8%, T = 9 months = 9/12 years = 0.75 years. So 270 = P × 8 × 0.75 / 100 = P × 6/100. Therefore P = 270 × 100/6 = Rs. 4500. Remember to convert time to years when rate is per annum.

Simple Interest = Principal × Rate × Time. 2550 = P × 0.17 × 3. So P = 2550/(0.51) = 5000. The formula gives direct calculation when interest, rate, and time are known.

Simple Interest = (P×R×T)/100. Given SI = 7P/8 for T=5 years. So (7P/8) = (P×R×5)/100. Solving gives R = (7×100)/(8×5) = 17.5%.

For semi-annual compounding at 8% per annum, the rate per half-year is 4%. In 1 year, there are 2 half-year periods. Using A = P(1 + r)^n, we get 6760 = P(1.04)^2 = P(1.0816). Therefore, P = 6760/1.0816 = Rs. 6250. Answer C is correct.

Simple interest increases principal by 32% in 4 years, so rate = 32% ÷ 4 = 8% per year. Compound interest on 24,000 for 3 years at 8%: 24,000 × (1.08³ - 1) = 24,000 × (1.259712 - 1) = 24,000 × 0.259712 = 6,233.09 ≈ 6,233.

Using compound interest formula: A = P(1 + r/100)^n. Substituting values: 1210 = 1000(1 + r/100)^2, giving 1.21 = (1 + r/100)^2. Taking square root: 1.1 = 1 + r/100, so r = 10%. The amount grows by 21% in 2 years at compound interest, which corresponds to 10% annual rate. Options B, C, and D are incorrect rates.

Amount doubles in 5 years at simple interest, so P + SI = 2P. This means SI = P in 5 years. Annual interest = P/5 = 0.2P. For amount to become 8P: P + n × 0.2P = 8P. Therefore n × 0.2P = 7P, giving n = 7/0.2 = 35 years.

Interest for 3 years (from year 2 to year 5) = Rs. 2250 - Rs. 2100 = Rs. 150. Annual interest = Rs. 150 ÷ 3 = Rs. 50. Principal = Amount after 2 years - Interest for 2 years = Rs. 2100 - (Rs. 50 × 2) = Rs. 2000. Rate = (Annual Interest ÷ Principal) × 100 = (50 ÷ 2000) × 100 = 2.5%.