Mathematics

The total number of codes is the sum of codes formed by waving one flag (3), two flags in sequence (3P2 = 6), and three flags in sequence (3P3 = 6). So, total codes = 3 + 6 + 6 = 15. However, the options given do not include 15 as the correct answer. On rechecking, option C is 15, which matches the calculation. Therefore, the correct answer is C, not D. The explanation was missing and has been added. The question text was corrected for clarity and formatting.

Initially, 80 patients have injections for 15 days, so total injections = 80 × 15 = 1200 patient-days. When 40 more patients arrive, total patients = 120. The injections will last 1200 ÷ 120 = 10 days.

The man purchased the cow for Rs. 3000 and sold it for Rs. 3600 with a 2-year credit at 10% per annum interest. The total amount the buyer will pay after 2 years with simple interest is Rs. 3600 + (3600 × 10% × 2) = Rs. 3600 + Rs. 720 = Rs. 4320. However, since the sale price is Rs. 3600, the effective amount received by the seller in present value terms is less. To find the effective gain, calculate the present value of Rs. 3600 due in 2 years at 10% interest: Present Value = 3600 / (1 + 0.10 × 2) = 3600 / 1.20 = Rs. 3000. This equals the purchase price, so the man neither gains nor loses in present value terms. But since the buyer pays Rs. 3600 (which is Rs. 600 more than the cost price), the gain considering the interest rate is 20% over 2 years, or 10% per annum. Therefore, the man has a gain of 10% per annum, which corresponds to option D.

Using simple interest, if a sum triples itself in 15 years 6 months (15.5 years), it means the interest earned is twice the principal in that time. The simple interest formula is A = P + P*r*t, where A is the amount, P is the principal, r is the rate, and t is time. Since the amount triples, 3P = P + P*r*15.5, which simplifies to r*15.5 = 2, so r = 2/15.5. To find the time to double the principal (amount = 2P), we set 2P = P + P*r*t, which simplifies to r*t = 1, so t = 1/r = 15.5/2 = 7.75 years, or 7 years 9 months. Therefore, the correct answer is B.

The shopkeeper sells the mixture at Rs. 177 per kg with an 18% profit, so the cost price of the mixture is Rs. 177 / 1.18 = Rs. 150 per kg. Let the cost of the other brand be x. The mixture cost per kg = (2*200 + 3*x) / 5 = 150. Solving gives x = Rs. 120 per kg.

True discount (TD) is the difference between the sum due and its present worth (PW). Using the formula TD = (Sum due × Rate × Time) / (100 + Rate × Time), we can find the sum due. Given TD = 168, Rate = 14%, Time = 2 years, solving for sum due gives Rs. 2400.

The difference between the amounts after 4 years and 3 years is Rs. 854 - Rs. 815 = Rs. 39, which represents the simple interest for 1 year. Therefore, the interest for 3 years is 3 × 39 = Rs. 117. Subtracting this interest from the amount after 3 years gives the principal: Rs. 815 - Rs. 117 = Rs. 698. This matches option D, confirming that the principal sum is Rs. 698.

The problem states that the rate of interest (R) and the time (T) in years are equal. Using the simple interest formula: Interest = (Principal × Rate × Time) / 100, and substituting R = T, we get 432 = (1200 × R × R) / 100, which simplifies to 432 = 12R². Solving for R² gives R² = 36, so R = 6. This means the rate of interest is 6%. Substituting back confirms the interest paid is Rs. 432. Therefore, the correct answer is B (6%).

The average is calculated by summing the numbers (1+2+3+4+5 = 15) and dividing by the count (5), resulting in 15/5 = 3. Therefore, option C is correct.

The sum becomes 7/5 of itself, meaning the interest is 7/5 - 1 = 2/5 of the principal. Using the simple interest formula I = PRT, we have (2/5)P = P × R × 4, which simplifies to R = (2/5)/4 = 2/20 = 1/10 = 10%. Hence, the rate per annum is 10%.