Simple and Compound Interest

For 3 years, the difference between CI and SI is P[(1+r/100)³-1] - P×r×3/100. At 20%, this equals P[(1.2)³-1-0.6] = P[1.728-1-0.6] = P(0.128). Setting 0.128P = 48 gives P = 48/0.128 = 375. The formula for 3 years is CI-SI = P×r²×(300+r)/(100)³.

Using SI = P * R * T / 100, where R = T = r years. So 432 = 1200 * r * r / 100, giving 12r² = 432, so r² = 36, r = 6%. The key insight is rate and time are numerically equal.

18 months = 3 half-year periods at 10% each. A = 7200 * 1.1³ = 7200 * 1.331 = 9583.20. CI = 9583.20 - 7200 = 2383.20, rounding to Rs 2383. Remember to adjust rate for compounding period.

Let each installment be x. Rate = 11 1/9% = 100/9% = 1/9. PV of first installment (year 1): x/(1 + 1/9) = x/(10/9) = 0.9x. PV of second installment (year 2): x/(10/9)^2 = 0.81x. PV of third installment (year 3): x/(10/9)^3 = 0.729x. Sum = 2.439x = 7317. Solving: x = 7317/2.439 ≈ ₹3000. Verification: 3000 × 2.439 = ₹7317.

Calculate SI separately and sum: For ₹500 at 7% for 3 years: SI = 500 × 7 × 3/100 = ₹105. For ₹7000 at 10% for 3 years: SI = 7000 × 10 × 3/100 = ₹2100. For ₹1000 at 5% for 3 years: SI = 1000 × 5 × 3/100 = ₹150. Total SI = 105 + 2100 + 150 = ₹2355. The approach is to apply the SI formula separately to each principal amount and then add.

Annual interest paid = 20000 × 6% = Rs 1200. Annual interest received = 20000 × 8.5% = Rs 1700. Gain per year = 1700 - 1200 = Rs 500. The difference in interest rates (8.5% - 6% = 2.5%) applied to the principal gives the annual profit.

Simple Interest = Amount - Principal = Rs. 31500 - Rs. 26500 = Rs. 5000. Using SI = P×R×T/100: 5000 = 26500 × R × 5 / 100. Solving: R = 5000 × 100 / (26500 × 5) = 500000 / 132500 ≈ 3.77%, which rounds to 3.8%. Always find interest amount first.

Let principal = P and rate = r%. Amount after 2 years: P(1 + r/100)² = 3600. After 4 years: P(1 + r/100)⁴ = 6480. Dividing: (1 + r/100)² = 6480/3600 = 1.8. So P × 1.8 = 3600, giving P = ₹2000. Option D (₹23600) appears to be a typo for ₹2,360, which is incorrect. The compound interest doubles between years 2 and 4, confirming the calculation.

The difference between simple interest for 5 years and 3 years at the same rate equals exactly 2 years' interest. At 14% p.a., 2 years' interest is 28% of the principal. Setting 0.28P = 4200 gives P = 4200/0.28 = 15000. The answer is Rs 15,000.

For 6 years at 25%, simple interest equals 1.5 times the principal (25% × 6 = 150%). Since this interest is Rs. 360 more than the principal, we have 1.5P = P + 360, giving 0.5P = 360, so P = Rs. 720. Option A (Rs. 600) would give interest = 1.5 × 600 = 900, which is Rs. 300 more than principal.

Let x be invested at 4% and (5000-x) at 5%. Interest: 0.04x + 0.05(5000-x) = 223. Solving: 0.04x + 250 - 0.05x = 223, so -0.01x = -27, giving x = 2700. Verification: ₹108 from 4% + ₹115 from 5% = ₹223.

Let principal be P and rate be R%. Difference in interest = P×3×(R+2)/100 - P×3×R/100 = 6P/100 = 450. Solving: 6P = 45000, P = 7500. The additional 2% for 3 years gives 6% extra interest, which equals ₹450.

Let P be principal and R be rate. Amount after 2 years = P + 2P*R/100 = 890. Amount after 7 more years (total 9) = P + 7P*R/100 = 1420. Solving: 5P*R/100 = 530, so P*R/100 = 106. Then P + 2*106 = 890, so P = 678. Thus R = 106*100/678 = 15.63%.

Using the simple interest formula SI = P × R × T / 100, we substitute the known values: 9600 = P × 16 × 5 / 100. This gives P = 9600 × 100 / (16 × 5) = 960000 / 80 = ₹12,000. The interest earned over 5 years at 16% was ₹9,600 on a principal of ₹12,000.

For half-yearly compounding at 20% p.a. for 2 years: rate per half-year = 10%, periods = 4. Amount = 15750 × (1.1)⁴ = 15750 × 1.4641 = ₹23,059.575. Compound Interest = 23059.575 - 15750 = ₹7,309.575. Option D is correct.