Simple and Compound Interest

Continuous compounding formula is A = P × e^(rt). Here, P=60000, r=0.12, t=3 years. A = 60000 × e^(0.36) = 60000 × 1.433329 = 85999.74 (approximately). The closest match is Rs. 85,999.97, which accounts for precise calculation.

The difference in interest earned is 2320 - 2200 = 120. This difference is due to the difference in interest rates: 8% - 5% = 3%. So 3% of principal for 1 year equals 120. Therefore: P × 3/100 × T = 120. Since the amounts are at simple interest, we can write: P × 5/100 × T = 220 (interest at 5%) and P × 8/100 × T = 320 (interest at 8%). Subtracting: P × 3/100 × T = 100. The principal can be found as 2200/(1 + 5T/100) or we can directly solve: T = 4 years.

Half-yearly compounding at 10% p.a. means 5% per half-year. For 15 months = 5 half-years (30 months) - no, 15 months has only 2.5 half-years. Actually 15 months = 1.25 years, with half-yearly compounding we have 3 half-years (18 months) would be 3 periods, but 15 months is exactly 2.5 half-years, which means 2 full half-years plus 3 months at simple interest. Let me recalculate: P = 20000, rate = 5% per half-year. After 2 half-years: 20000 × 1.05 × 1.05 = 22050. For remaining 3 months at 10% p.a. simple interest on 22050: 22050 × (10/100) × (3/12) = 551.25. Total = 22050 + 551.25 = 22601.25. Gain = 2601.25, gain% = 2601.25/20000 × 100 = 13.0%. Option C is correct.

For compound interest with fractional time, calculate full periods first: A = 60000(1.08)^2 = 69984. For the remaining 6 months, use simple interest on this amount: SI = 69984 × 8/100 × 0.5 = 2799.36. Total amount = 69984 + 2799.36 = 72783.36. Compound Interest = 72783.36 - 60000 = 12783.36.

Monthly rates form GP: 6%, 12%, 24%, 48%, ... doubling each month. For 12 months at simple interest on Rs.100: Total interest = sum of (100 × rate × 1/12 × 1/100) = (1/12) × sum of rates in %. Sum = 6 + 12 + 24 + 48 + 96 + 192 + 384 + 768 + 1536 + 3072 + 6144 + 12288 = 24570. Interest = 24570/12 = 2047.5. The GP sum is a(r^n-1)/(r-1) = 6(2^12-1)/(2-1) = 6(4095) = 24570 paise = Rs. 2047.5.

After year 1 at 12.5%: 240000 × 1.125 = 270000. Year 2 at 12%: 270000 × 1.12 = 302400. Year 3 at 12%: 302400 × 1.12 = 338688. The key is applying different rates for different years.

Using the compound interest formula: CI = P[(1 + r/100)ⁿ - 1]. Here, CI = 81, r = 2.5%, n = 2 years. So: 81 = P[(1 + 2.5/100)² - 1] = P[(1.025)² - 1] = P[1.050625 - 1] = P × 0.050625. Therefore: P = 81/0.050625 = 1600. The principal sum is Rs. 1600. This is a straightforward application of the CI formula - solve for P by dividing the compound interest by the interest factor.

Let P be the principal. Compound interest formula for 2 years: A = P(1 + 0.125)² = P(1.125)² = P × 1.265625. CI = P(1.265625 - 1) = 0.265625P = 510, so P = 1920. Simple interest for 2 years at 12.5%: SI = 1920 × 0.125 × 2 = 480.

For compound interest: CI = P(1+r/100)^n - P. Given CI=624, r=8%, n=2. So 624 = P(1.08)^2 - P = P(1.1664 - 1) = P(0.1664). Therefore P = 624/0.1664 = 3750. The principal sum is Rs 3750. Verification: 3750 at 8% for 2 years gives CI = 3750×0.1664 = 624.

Use compound interest formula: A = P(1 + r)^t. 13824 = 8000(1.2)^t, so (1.2)^t = 1.728. Recognizing 1.2³ = 1.728, we get t = 3 years. The key is checking powers of 1.2 or using logarithms.

For SI: P×12×3/100 = 2340, so P = 6500. For CI at 10% for 2 years on (6500+Y): (6500+Y)×[(1+0.1)² - 1] = 1890. Solving gives Y = 2500. This uses SI and CI formulas sequentially.

For CI at 12.5%: P=24576. Year 1: Amount=24576×1.125=27648, CI=3072. Year 2: Amount=27648×1.125=31104, CI=3456. Year 3: Amount=31104×1.125=34992, CI=3888. Difference between CI of 3rd and 2nd year: 3888-3456=432.

The difference between compound interest and simple interest for 2 years is P × r² ÷ 10000. Here: 659.1 = 39000 × r² ÷ 10000. So r² = 659.1 × 10000 ÷ 39000 = 169. So r = 13%. The rate of interest is 13% per annum.

Promised simple interest at 20% p.a., but interest is added every 6 months (compounding semi-annually). Period 1 (0-6 months): Principal = 15000, Interest = 15000 × 20% × 1/2 = 1500. Period 2 (6-12 months): New principal = 16500, Interest = 16500 × 20% × 1/2 = 1650. Period 3 (12-18 months): New principal = 18150, Interest = 18150 × 20% × 1/2 = 1815. Total interest = 1500 + 1650 + 1815 = 4965. Option C matches.

The difference between CI and SI for 2 years at 8.5% is ₹28.90. Using formula CI - SI = P × (r/100)², we get P × (8.5/100)² = 28.90, so P = 28.90 × 10000/72.25 = ₹4000. CI on ₹4000 for 2 years at 8.5% = 4000 × [(1.085)² - 1] = 4000 × 0.177225 = ₹708.90, matching option C.