Simple and Compound Interest

In compound interest, if it doubles in 1 year, it becomes 2^n times in n years. We want 8 times, which is 2^3. So it takes 3 years.

Under compound interest, the principal grows by the same factor over equal intervals of time. Since the sum doubles every 5 years, in 20 years it will undergo 4 doubling cycles (20 divided by 5). Therefore, the final amount will be 5,000 * 2^4 = 5,000 * 16 = Rs. 80,000.

Interest in 2nd year = P(1+r)^1 * r = 420. Interest in 3rd year = P(1+r)^2 * r = 462. The ratio (Interest 3rd / Interest 2nd) = (1+r) = 462/420 = 1.1. Thus, r = 0.1 or 10%.

Difference for 2 years = P * (r/100)^2. 25 = P * (5/100)^2 = P * (1/20)^2 = P / 400. P = 25 * 400 = 10,000.

Simple interest of 400 on 1000 in 4 years gives a rate of 10% per annum (400 = 1000 * r * 4 / 100). Compound interest on 1000 at 10% for 2 years is 1000 * (1.1^2 - 1) = 210.

SI = (P * R * T) / 100. SI = (P * 4 * 8) / 100 = 0.32P. Given P - SI = 340, so P - 0.32P = 340. 0.68P = 340. P = 340 / 0.68 = 500.

Simple interest formula: I = PRT/100. The difference in interest is (P * (R+4) * 3 / 100) - (P * R * 3 / 100) = 600. This simplifies to (P * 4 * 3) / 100 = 600, so 12P = 60000, P = 5000.

Using the compound interest formula A = P(1 + r/100)^n, we want A > 2P. (1 + 0.20)^n > 2, so (1.2)^n > 2. For n=3, 1.2^3 = 1.728. For n=4, 1.2^4 = 2.0736. Thus, 4 years are required.

For half-yearly compounding, the amount is 900 * (1 + 0.10)^2 = 1089, interest = 189. For quarterly compounding, the amount is 900 * (1 + 0.05)^4 = 1093.95, interest = 193.95. The difference is 193.95 - 189 = 4.95.

Joseph: SI = P, T = 4, R = r. P = (P*r*4)/100 => r = 25. George: SI = 2P, R = r/2 = 12.5, T = ?. 2P = (P*12.5*T)/100 => 2 = 0.125T => T = 16.

Year 1: 800 * 1.08 = 864. Remaining 7 years at 10%: 864 * (1.1)^7 = 864 * 1.9487 = 1683.69, which rounds to 1684.

The debt for Ram is 8000 + (8000 * 0.12 * t) and for Mohan is 9100 + (9100 * 0.10 * t). Setting these equal: 8000 + 960t = 9100 + 910t. Solving for t gives 50t = 1100, so t = 22 years.

For 2 years, the difference between CI and SI is P*(r/100)^2. Here, 10 = x*(10/100)^2 = x*(1/100) = x/100. So x = 1000.

Simple interest: Amount = P + SI. 3P = P + SI, so SI = 2P in 5 years. Rate R = (2P * 100) / (P * 5) = 40%. To become 6 times, SI = 5P. Time = (5P * 100) / (P * 40) = 12.5 years.