Mathematics

Given: A and B together take 6 days, so their combined rate is 1/6 per day. B and C together take 12 days, so their combined rate is 1/12 per day. A and C together take 8 days, so their combined rate is 1/8 per day. Adding these rates: (A+B) + (B+C) + (A+C) = 1/6 + 1/12 + 1/8 = 9/24 = 3/8 per day. This sum counts each person twice, so the combined rate of A, B, and C is (3/8)/2 = 3/16 per day. Therefore, the time taken by A, B, and C together is the reciprocal of 3/16, which is 16/3 days or 5 1/3 days. Hence, the correct answer is option C (5 1/3 days).

The dividend is 9% on the face value of Rs. 25, so the dividend amount is 0.09 × 25 = Rs. 2.25. The man wants a 10% return on his investment, so the price he paid for the shares should be such that the dividend yield equals 10%. Using the formula: Price = Dividend / Desired Return = 2.25 / 0.10 = Rs. 22.50. Therefore, the correct price at which he bought the shares is Rs. 22.50, which corresponds to option B.

The total amount available for purchasing shares after deducting 2% brokerage from Rs. 12,750 is Rs. 12,750 - 2% of 12,750 = Rs. 12,750 - Rs. 255 = Rs. 12,495. Each share costs Rs. 25, so the number of shares that can be bought is 12,495 ÷ 25 = 499.8, which means 499 shares can be purchased. Since 499 shares is closest to option B (500 shares), and 500 shares cost Rs. 12,500 which is slightly more than Rs. 12,495 available, the practical maximum is 499 shares. However, given the options, 500 shares (option B) is the best answer as it is the closest to the calculated value and the question likely expects rounding to the nearest option.

The market value of a stock is calculated by dividing the dividend by the yield. Here, the dividend is 9% of the face value (assumed Rs. 100), so Rs. 9. Yield is 8%, so market value = 9 / 0.08 = Rs. 112.50.

The difference between compound interest (CI) and simple interest (SI) for 2 years at 8% per annum is given by the formula: Difference = P * (r/100)^2, where P is the principal and r is the rate of interest. Here, Difference = Rs.48 and r = 8%. So, 48 = P * (8/100)^2 = P * 0.0064. Solving for P gives P = 48 / 0.0064 = Rs.7500. Therefore, the sum invested is Rs.7500, which corresponds to option A. The initial question had the correct explanation but an incorrect answer choice marked; the correct answer is Rs.7500 (Option A).

Let the total work be W. Given A's work rate equals B and C's combined rate. A + B can do the work in 10 days, so their combined rate is W/10. C alone does it in 50 days, so C's rate is W/50. Since A = B + C, A's rate = B's rate + C's rate. Therefore, (A + B) = (B + C) + B = 2B + C = W/10. Substitute C = W/50, so 2B + W/50 = W/10. Solving for B gives B = W/20, meaning B alone can do the work in 20 days.

The question asks for the future value of an ordinary annuity where a man saves Rs. 200 at the end of each year and lends the money at 5% compound interest for 3 years. The correct formula for the future value of an ordinary annuity is: FV = P × [((1 + r)^n - 1) / r] where P = Rs. 200, r = 0.05, and n = 3. Calculating: FV = 200 × [((1.05)^3 - 1) / 0.05] = 200 × (1.157625 - 1) / 0.05 = 200 × 0.157625 / 0.05 = 200 × 3.1525 = Rs. 630.5 Among the options given, Rs. 565.25 (option A) is closest to the correct value if we consider rounding or slight differences in compounding assumptions. However, the exact calculated value is Rs. 630.5, which is not listed exactly. The other options are further off. Therefore, the correct answer should be option A (Rs. 565.25) if we consider the closest available option, but ideally, the question options should be corrected to include Rs. 630.5. Since the original options do not include the exact value, and option A is closest, it is the best choice. The original answer (option D) Rs. 666.50 is incorrect based on the standard formula and calculation. Hence, the question's correct answer is updated to A, with explanation reflecting the correct formula and calculation.

When interest is compounded half-yearly at 20% per annum, the effective rate per half year is 10%. Over 2 years (4 half years), the amount is higher than when compounded yearly at 20% per annum for 2 years. The difference in amounts is Rs.482, which leads to the principal sum being Rs.2000.

The market value of a stock is calculated by dividing the dividend by the yield. Given a 6% stock, the dividend is 6% of the face value (assumed Rs. 100), which is Rs. 6. The yield is 8%, so the market value = Dividend / Yield = 6 / 0.08 = Rs. 75. This matches option B. The original answer 'D' (Rs. 133.33) is incorrect based on this calculation.

The interest is compounded quarterly at an annual rate of 20%. The quarterly interest rate is 20% ÷ 4 = 5%. For six months, which is two quarters, the amount is calculated using the compound interest formula: A = P(1 + r/n)^(nt) = 2000 × (1 + 0.05)^2 = 2000 × 1.1025 = Rs. 2205. Hence, the amount after six months is Rs. 2205.