Mathematics

Let the speed of the boat in still water be b and the speed of the current be c. Let the distance between X and Y be d. Since Y is equidistant from X and Z, distance XZ = 2d. Time taken from X to Y and back is 10 hours: (d/(b-c)) + (d/(b+c)) = 10 Time taken from X to Z is 4 hours: 2d/(b-c) = 4 From the second equation, d = 2(b-c). Substitute d in the first equation: (2(b-c)/(b-c)) + (2(b-c)/(b+c)) = 10 => 2 + 2(b-c)/(b+c) = 10 => 2(b-c)/(b+c) = 8 => (b-c)/(b+c) = 4 This is not possible as the ratio must be less than 1. Re-examining the problem, the time from X to Z is 4 hours downstream, so speed downstream is (b+c), and time = distance/speed = 2d/(b+c) = 4. So, 2d/(b+c) = 4 => d = 2(b+c). Now substitute d in the first equation: (d/(b-c)) + (d/(b+c)) = 10 => (2(b+c)/(b-c)) + (2(b+c)/(b+c)) = 10 => 2(b+c)/(b-c) + 2 = 10 => 2(b+c)/(b-c) = 8 => (b+c)/(b-c) = 4 Let r = b/c. Then (r+1)/(r-1) = 4 => r+1 = 4r - 4 => 3r = 5 => r = 5/3 Therefore, the ratio of the speed of the boatman in still water to the speed of the current is 5:3.

The shares of A, B, C, and D are in the ratio 5:2:4:3. Let the common multiplier be x. Then C's share is 4x and D's share is 3x. Given C gets Rs. 1000 more than D, we have 4x - 3x = 1000, so x = 1000. Therefore, B's share is 2x = 2 * 1000 = Rs. 2000. Among the options, Rs. 2000 corresponds to option C, so the correct answer is C.

The fourth proportion to 3, 7, and 9 is found by solving the proportion 3 : 7 = 9 : x. Cross-multiplying gives 3x = 7 × 9, so x = 63.

Let the number of 25p, 10p, and 5p coins be x, 2x, and 3x respectively. The total value is 25x + 10(2x) + 5(3x) = 25x + 20x + 15x = 60x pence. Since Rs. 30 = 3000 pence, 60x = 3000, so x = 50. Therefore, the number of 5p coins is 3x = 150.

Given the equation XY = kZ, and the initial ratio X:Y:Z = 2:3:4 with X = 4, we find the values of Y and Z. Since X corresponds to 2 parts and equals 4, each part is 2. Therefore, Y = 3 parts = 6 and Z = 4 parts = 8. Using XY = kZ, we get 4 * 6 = k * 8, so k = 3. When Y changes to 12 and Z remains 8, the equation becomes X * 12 = 3 * 8 = 24. Solving for X gives X = 24 / 12 = 2. Thus, the new value of X is 2, which corresponds to option A.

Let the third number be x. The first number is 20% more than x, which is 1.2x. The second number is 50% more than x, which is 1.5x. The ratio of the two numbers is therefore 1.2x : 1.5x = 1.2 : 1.5 = 12 : 15 = 4 : 5. Hence, the correct ratio is 4 : 5, which corresponds to option C.

Given a:b = 3:5, let a=3x and b=5x. After adding 2, (3x+2)/(5x+2) = 2/3. Cross-multiplying gives 9x + 6 = 10x + 4, so x = 2. Thus, b = 5x = 10.

When walking at 80% speed, the man takes longer to reach his office. Let the usual time be T minutes. At 80% speed, time taken is T/0.8 = 1.25T. The delay is 1.25T - T = 0.25T, which equals 10 minutes. Solving 0.25T = 10 gives T = 40 minutes. However, this contradicts the initial correction, so let's re-express carefully: If usual speed is S and usual time is T, distance D = S × T. At 80% speed, speed is 0.8S, time taken is D / (0.8S) = T / 0.8 = 1.25T. The delay is 1.25T - T = 0.25T = 10 minutes, so T = 40 minutes. Therefore, the usual time is 40 minutes, which corresponds to option C. The original correct answer C is correct.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

If a 3/4 full tank allows the car to travel 360 km, then a full tank would allow it to travel (360 km) ÷ (3/4) = 480 km. Therefore, with a 2/3 full tank, the car can travel 480 km × (2/3) = 320 km. However, since 320 km is option B, but the question asks how many km with 2/3 full tank, the correct answer is 320 km (option B). The original correct_answer was B, which matches this calculation, so the original answer is correct. The explanation was missing and has been added. Difficulty is set to easy based on the straightforward proportional calculation.

Let the two numbers be 5x and 9x. After decreasing each by 5, the ratio becomes (5x - 5) : (9x - 5) = 5 : 11. Solving the equation (5x - 5)/(9x - 5) = 5/11 gives x = 7. Therefore, the numbers are 5*7 = 35 and 9*7 = 63. However, none of the options match these numbers, so let's check the options by substituting back into the ratio condition. Option B (21, 37) gives (21 - 5) : (37 - 5) = 16 : 32 = 1 : 2, not 5 : 11. Option D (15, 27) gives (15 - 5) : (27 - 5) = 10 : 22 = 5 : 11, which matches the second ratio. But the original ratio 15 : 27 simplifies to 5 : 9, which matches the first ratio. Therefore, the correct answer is D (15, 27).

Using the allegation method, the difference between the cost of the second type and the mixture is 19 - 14.2 = 4.8, and the difference between the mixture and the first type is 14.2 - 13 = 1.2. The ratio of the quantities mixed is therefore 4.8 : 1.2, which simplifies to 4 : 1. This means the two types of rice are mixed in the ratio 4:1 (first type : second type) to get the mixture costing Rs. 14.2 per kg. This matches option B.

The distance covered is the same, so speed and time are inversely proportional. Original speed is 24 kmph for 50 minutes; to cover the same distance in 40 minutes, speed = (24 kmph × 50) / 40 = 30 kmph.

Let the number be x. 60% of x is 0.6x. According to the problem, 0.6x - 40 = 50. Adding 40 to both sides gives 0.6x = 90. Dividing both sides by 0.6 gives x = 150. However, 150 is option A, so we need to re-check the calculation. Actually, 0.6x - 40 = 50 implies 0.6x = 90, so x = 90 / 0.6 = 150. Therefore, the correct answer is A, not B. The initial correction was incorrect. The correct answer is A (150).

The sides of the triangle are in the ratio 4:3:2. Let the sides be 4x, 3x, and 2x. The perimeter is the sum of the sides: 4x + 3x + 2x = 9x. Given the perimeter is 81 cm, we have 9x = 81, so x = 9 cm. The largest side is 4x = 4 * 9 = 36 cm. Therefore, the correct answer is 36 cm, which corresponds to option C.

Given X:Y = 2:3 and Y:Z = 4:5, we first equalize the Y terms by finding the least common multiple of 3 and 4, which is 12. Scaling X:Y to 8:12 and Y:Z to 12:15, we combine to get X:Y:Z = 8:12:15.