Mathematics

To find 25% of 30% of 45%, convert percentages to decimals and multiply: 0.25 × 0.30 × 0.45 = 0.03375.

Since speed is inversely proportional to time for the same distance, the ratio of times taken is the inverse of the ratio of speeds. Given speeds ratio is 3:4, so time ratio is 4:3.

60% of 50 is 0.6 × 50 = 30, and 40% of 30 is 0.4 × 30 = 12. The difference is 30 - 12 = 18, so 60% of 50 is 18 greater than 40% of 30.

The original ratio of boys to girls is 7:8. After a 20% increase in boys, the number becomes 7 × 1.20 = 8.4. After a 10% increase in girls, the number becomes 8 × 1.10 = 8.8. The new ratio is 8.4 : 8.8, which simplifies to 21 : 22 when multiplied by 5, but since the options do not include 21:22 as the correct ratio, let's check the options carefully. Actually, 8.4 : 8.8 simplifies to 21 : 22, which matches option C. Therefore, option C is correct, not B. The initial correction was incorrect. The new ratio is 21:22, so option C is correct.

Given that 96% of the population is 23040, the total population can be found by dividing 23040 by 0.96. This calculation is 23040 ÷ 0.96 = 24000. Therefore, the total population of the village is 24000, which corresponds to option B.

Let the amount received by a singer be x. Then, 3 dancers get as much as 2 singers, so 3 dancers = 2 singers = 2x, thus 1 dancer = 2x/3. Similarly, 3 comperes get as much as 2 dancers, so 3 comperes = 2 dancers = 2 * (2x/3) = 4x/3, thus 1 compere = (4x/3)/3 = 4x/9. Total amount = 4 singers * x + 5 dancers * (2x/3) + 3 comperes * (4x/9) = 4x + (10x/3) + (12x/9) = 4x + 3.333x + 1.333x = 8.666x = Rs.130000. So, x = 130000 / 8.666 = Rs.15000.

The original ratio of seats is 5:8. Increasing 5 by 30% gives 5 × 1.3 = 6.5, and increasing 8 by 25% gives 8 × 1.25 = 10. The new ratio is 6.5:10, which simplifies to 13:20.

Taimoor walked from 6 am to 9 am, so total time is 3 hours. Let the distance to school be x km. Time to school = x/2 hours, time back = x/3 hours. Total time = x/2 + x/3 = 3 hours. Solving gives x = 7.2 km.

To find M:P, first express all ratios with a common term. Given M:N = 3:4, N:O = 5:7, and O:P = 7:9, we align N and O terms: N is 4 in first ratio and 5 in second, so multiply first ratio by 5 and second by 4 to get M:N = 15:20 and N:O = 20:28. Similarly, O:P = 7:9, multiply by 4 to get O:P = 28:36. Now, M:P = 15:36, which simplifies to 5:12. Hence, M:P = 5/12.

Let the three numbers be a, b, and c. We know that a² + b² + c² = 138 and ab + bc + ca = 131. Using the identity (a + b + c)² = a² + b² + c² + 2(ab + bc + ca), we get (a + b + c)² = 138 + 2*131 = 138 + 262 = 400. Therefore, a + b + c = √400 = 20.