Simple and Compound Interest

Amount = P * (1 + r/100)^2. Given Amount = 2.25 * P. (1 + r/100)^2 = 2.25. 1 + r/100 = 1.5. r/100 = 0.5, so r = 50%.

Principal = 24000, rate = 10% per annum, compounded semiannually means 5% per half-year. For 1.5 years (3 periods), the amount is 24000 * (1.05)^3 = 24000 * 1.157625 = 27783. The interest is 27783 - 24000 = 3783, which is between 3000 and 4000.

For compound interest, the formula is A = P(1 + r/100)^n. Here, 1210 = 1000(1 + 10/100)^n, which simplifies to 1.21 = (1.1)^n. Since 1.1^2 = 1.21, n = 2 years.

Compound interest = P * ((1 + r/100)^n - 1) = 25000 * ((1.12)^3 - 1) = 25000 * (1.404928 - 1) = 25000 * 0.404928 = 10123.20.

Amount = 10000 * (1.04) * (1.05) * (1.06) = 10000 * 1.15752 = 11575.20. Interest = 11575.20 - 10000 = 1575.20.

A = P(1+r)^n. 4624 = P(1+r)^2 and 4913 = P(1+r)^3. Dividing: 4913/4624 = 1+r. 1+r = 1.0625. P = 4624 / (1.0625)^2 = 4624 / 1.1289 = 4096.

Rate = 2.5% p.a. compounded half-yearly = 1.25% per half-year. Time = 18 months = 3 half-years. Amount = P(1 + 1.25/100)^3 = 9826. P = 9826 / (1.0125)^3 = 9826 / 1.03797 = 9466.54.

For 2 years, the difference between compound interest and simple interest is given by the formula D = P * (r/100)^2. Plugging in the values: 405 = 18000 * (r/100)^2, so (r/100)^2 = 405/18000 = 0.0225. Thus, r/100 = 0.15, meaning r = 15%.

For C.I., A = P(1+r/100)^n. 800 = P(1+r/100)^3 and 840 = P(1+r/100)^4. Dividing: 840/800 = 1+r/100. 1.05 = 1+r/100. r = 5%.

Principal = 60000, Rate = 12% p.a. compounded half-yearly (6% per 6 months), Time = 1 year (2 periods). Amount = 60000 * (1 + 6/100)^2 = 60000 * (1.06)^2 = 60000 * 1.1236 = 67416.

Principal = 4096. Rate = 12.5% per annum = 6.25% per half-year. Time = 18 months = 3 half-years. Amount = 4096 * (1 + 6.25/100)^3 = 4096 * (1 + 1/16)^3 = 4096 * (17/16)^3 = 4096 * 4913 / 4096 = 4913.

Interest during the third year is calculated on the balance after two years. That balance is 8000 x 1.05^2, so the third-year interest is 8000 x 1.05^2 x 0.05 = Rs. 441.

CI = P * ((1 + r/100)^n - 1). P = 800, r = 5, n = 2. CI = 800 * ((1.05)^2 - 1) = 800 * (1.1025 - 1) = 800 * 0.1025 = 82.

Using the compound interest formula A = P(1 + r/100)^n, A = 7500(1 + 0.04)^2 = 7500 * 1.04 * 1.04 = 7500 * 1.0816 = 8112.

The ratio of the amount after 3 years to the amount after 2 years is 4913/4624 = 1.0625. Since the amount grows by a factor of (1+r), the principal P satisfies P * (1.0625)^2 = 4624. Solving this gives P = 4624 / (1.0625)^2 = 4096.