Mathematics

The compound interest on Rs 16,000 for 9 months at 20% per annum compounded quarterly is calculated as follows: The quarterly interest rate is 20%/4 = 5%. The number of quarters in 9 months is 3. Using the compound interest formula A = P(1 + r)^n, the amount A = 16000 * (1 + 0.05)^3 = 16000 * 1.157625 = Rs 18,522 (approximately). The compound interest is the amount minus the principal, i.e., 18522 - 16000 = Rs 2522. This matches option C exactly, so the correct answer is C, not B. The original explanation incorrectly chose the closest option instead of the exact match.

Let the speed of the stream be x km/h. The man's speed in still water is 4.5 km/h. Downstream speed = 4.5 + x, upstream speed = 4.5 - x. Given time upstream is twice time downstream, so (distance / upstream speed) = 2 × (distance / downstream speed). Simplifying, downstream speed = 2 × upstream speed, so 4.5 + x = 2(4.5 - x). Solving gives x = 1.5 km/h.

The principal can be found using the simple interest formula: SI = P × R × T / 100, so P = (SI × 100) / (R × T) = (50 × 100) / (5 × 2) = Rs. 500. Compound interest for 2 years at 5% is CI = P[(1 + R/100)^T - 1] = 500[(1 + 0.05)^2 - 1] = 500(1.1025 - 1) = Rs. 51.25, which matches option A.

The compound interest (CI) on Rs. 4000 for 2 years at 10% per annum compounded annually is calculated as: CI = P[(1 + R/100)^T - 1] = 4000[(1 + 10/100)^2 - 1] = 4000[1.21 - 1] = 4000 × 0.21 = Rs. 840. The simple interest (SI) on the unknown sum for 3 years at 8% per annum is half of Rs. 840, which is Rs. 420. Using SI = (P × R × T)/100, we get: 420 = (P × 8 × 3)/100 => P = (420 × 100)/(8 × 3) = Rs. 1750. Among the options, Rs. 1750 (option C) is the correct sum placed on simple interest. The previous answer (option B) was incorrect because the calculated principal (Rs. 1708.33) is closer to Rs. 1750 than Rs. 1650, so option C is the best choice.

The principal is calculated from the simple interest formula: SI = (P × R × T)/100, so P = (SI × 100)/(R × T) = (80 × 100)/(4 × 2) = Rs. 1000. Compound interest for 2 years at 4% per annum is CI = P[(1 + R/100)^T - 1] = 1000[(1 + 0.04)^2 - 1] = 1000(1.0816 - 1) = Rs. 81.60, which matches option A.

If the sum triples in 3 years under compound interest, the growth factor for 3 years is 3. The annual growth factor is therefore the cube root of 3, i.e., $3^{1/3}$. To become 9 times the original amount, the total growth factor must be 9, which is $3^2$. This means it will take two periods of 3 years each, i.e., 6 years. Mathematically, using the compound interest formula $(1+r{)}^t=9$ and $1+r=3^{1/3}$, we get $t=\frac{\ln 9}{\ln (3^{1/3})}=6$ years. Hence, the amount will be 9 times itself in 6 years, corresponding to option D.

Let the speed of the stream be x kmph. The upstream speed is (42 - x) kmph and downstream speed is (42 + x) kmph. Given time upstream is twice time downstream, so distance/(42 - x) = 2 × distance/(42 + x). Simplifying gives (42 + x) = 2(42 - x), leading to x = 14 kmph.

The man rows downstream and upstream to the place and back in a total time of 1 hour. The downstream speed is 6 + 1.2 = 7.2 km/h, and the upstream speed is 6 - 1.2 = 4.8 km/h. Let the distance to the place be x km. The total time taken is (x/7.2) + (x/4.8) = 1 hour. Solving this equation gives x = 2.88 km. Therefore, the distance to the place is 2.88 km, which corresponds to option B.

Given the compound interest (C.I.) for 2 years at 10% per annum is Rs. 525, the principal can be calculated. Then, the simple interest (S.I.) for 4 years (double the time) at 5% per annum (half the rate) on the same principal is Rs. 400. Hence, option A is correct.

The compound interest for the second year is calculated on the amount at the beginning of the second year. Given the interest rate of 10% per annum and compound interest of Rs.132 for the second year, the amount at the start of the second year must be Rs.1320 (since 10% of Rs.1320 is Rs.132). The principal amount P is such that after the first year, the amount becomes P × (1 + 0.10) = 1320, so P = 1320 / 1.10 = Rs.1200. Therefore, the original principal amount is Rs.1200, which earns Rs.132 as compound interest in the second year at 10% per annum.