Mathematics

Speed is calculated as distance divided by time. Rayan crosses 400 meters in 3 minutes, which is 0.004 km in 0.05 hours. Speed = 0.004 km / 0.05 hr = 0.08 km/hr, which seems off, so let's recheck: 400 m = 0.4 km, 3 minutes = 3/60 = 0.05 hours, speed = 0.4 km / 0.05 hr = 8 km/h. The correct speed is 8 km/h, which corresponds to option D. However, option A is 133.33 kmph, which is incorrect. The correct answer is D.

Let the three numbers be 3x, 5x, and 6x. Their squares sum to 280: (3x)^2 + (5x)^2 + (6x)^2 = 9x^2 + 25x^2 + 36x^2 = 70x^2 = 280. Solving for x^2 gives x^2 = 4, so x = 2. The numbers are 3*2=6, 5*2=10, and 6*2=12. The greatest number is 12, which corresponds to option C.

The two trains travel toward each other for 3.5 hours (from 10 a.m. to 1:30 p.m.). Let the speed of the slower train be x mph, then the faster train's speed is x + 6 mph. The sum of their speeds times the time equals the distance: (x + x + 6) * 3.5 = 287. Simplifying: (2x + 6) * 3.5 = 287 → 2x + 6 = 82 → 2x = 76 → x = 38 mph. Therefore, the faster train's speed is 38 + 6 = 44 mph, which corresponds to option C.

Let the length and breadth be 5x and 4x respectively. Area = length × breadth = 5x × 4x = 20x² = 180, so x² = 9 and x = 3. Therefore, length = 15 cm and breadth = 12 cm. Perimeter = 2 × (length + breadth) = 2 × (15 + 12) = 54 cm.

Let the number be x. According to the problem, 85% of x plus 24 equals x, so 0.85x + 24 = x. Subtracting 0.85x from both sides gives 24 = 0.15x, so x = 24 / 0.15 = 160. Therefore, the number is 160.

If y exceeds x by 20%, then y = 1.2x. To find by what percent x is less than y, calculate (y - x)/y = (1.2x - x)/1.2x = 0.2x/1.2x = 1/6 = 16 2/3%.

When the man walks at 6/7 of his usual speed, he takes 12 minutes more than usual. Using the relation time = distance/speed, let the usual time be T minutes. Then, (7/6)T - T = 12, which simplifies to T/6 = 12, so T = 72 minutes or 1 hour 12 minutes. However, this is the extra time taken, so the usual time is 1 hour 42 minutes (72 + 12). Therefore, the usual time is 1 hr 42 min.

Let the two numbers be x and y, with x > y. Given x + y = 33 and x - y = 15. Adding both equations: 2x = 48, so x = 24. Substituting back: 24 + y = 33, so y = 9. The smaller number is 9.

Given the ratios A:B = 2:3 and B:C = 3:4, we combine these to find A:B:C = 2:3:4. This means A = 2k, B = 3k, and C = 4k for some k. Given A:D = 4:5, we have A = 4m and D = 5m for some m. Equate the two expressions for A: 2k = 4m, which implies k = 2m. Substitute k into C: C = 4k = 4(2m) = 8m. Therefore, the ratio C:D = 8m:5m = 8:5. Hence, the correct answer is D: 8:5.

Let the two-digit number be 10x + y, where x and y are digits. Given xy = 8 and (10x + y) + 18 = 10y + x. Solving these, the number is 42.