Mathematics

True discount is directly proportional to the time period when the rate of interest and principal remain the same. Since the time is halved, the true discount will also be halved. Half of Rs. 20 is Rs. 10, so the correct answer is Rs. 10.

Initially, the provisions are enough for 12 people for 15 days, which totals 12 × 15 = 180 person-days of food. When 3 guests come for 2 days, the total number of people during those 2 days is 12 + 3 = 15. They consume 15 × 2 = 30 person-days of food in those 2 days. This leaves 180 - 30 = 150 person-days of food remaining. After the guests leave, only the original 12 people remain, so the remaining provisions will last 150 ÷ 12 = 12.5 days. Therefore, the total duration the provisions last is the initial 2 days plus 12.5 days, but since the question asks how long the provisions will last now (after guests came), the answer is 12.5 days, which corresponds to option D.

To express 1.09 as a percentage, multiply by 100. Thus, 1.09 × 100 = 109%, which corresponds to option C.

Initially, the food lasts 13 days for all men. After 1 day, 2/5 of the men desert, so 3/5 remain. The food consumed in the first day reduces the quantity for the remaining men. The remaining food will last longer for fewer men. Calculating accordingly, the food lasts 20 days after the desertion.

The difference between simple interest (SI) and true discount (TD) is equal to the interest on the true discount for the given time. Using the formula SI - TD = (TD × R × T)/100, and given SI = 85 and TD = 80, solving for the sum (principal) yields Rs. 1800.

Given that 15 cattle can graze 1 field in 20 days, the total work to graze one field is 15 × 20 = 300 cattle-days. For 3 such fields, the total work is 300 × 3 = 900 cattle-days. With 20 cattle, the number of days required to graze 3 fields is 900 ÷ 20 = 45 days. Therefore, the correct answer is 45 days, which corresponds to option B.

Since 18 cattle graze 1 field in 12 days, the total cattle-days to graze 1 field is 18 × 12 = 216. For 3 fields, total cattle-days needed is 216 × 3 = 648. With 6 cattle grazing, the days required is 648 ÷ 6 = 108 days (x = 108).

The first discount of 10% on Rs. 720 reduces the price to Rs. 648. The final price after the second discount is Rs. 550.80. The second discount rate is calculated as ((648 - 550.80) / 648) × 100 = 15%.

The difference in interest for 2 years at 4% higher rate is Rs. 60. Using the formula for simple interest difference: Principal × Rate difference × Time / 100 = 60. Substituting Rate difference = 4%, Time = 2 years, we get Principal = (60 × 100) / (4 × 2) = Rs. 750.

The given capitals are in the ratio 6 : 8 : 10. To simplify this ratio, divide each term by their greatest common factor, which is 2. This gives the simplified ratio 3 : 4 : 5. Standard practice in mathematics is to express ratios in their simplest form, so the correct answer is option A (3:4:5).