Math Problems and Logic Puzzles

To pass the bridge completely, the train must travel its own length plus the bridge length: 360 m + 140 m = 500 m. Speed of 45 km/hr = 45 × (5/18) = 12.5 m/s. Time = Distance / Speed = 500 m / 12.5 m/s = 40 seconds. Option B (42 sec) would require incorrect speed conversion.

When a train crosses a pole, the distance traveled equals its own length. Speed 60 km/hr = 60 × (5/18) = 50/3 m/s. Distance = Speed × Time = (50/3) × 9 = 150 metres. Option B (180 m) would require different speed or time values.

First find train speed: 240 m in 24 sec = 10 m/s. To cross the 650 m platform, total distance = train length + platform length = 240 + 650 = 890 m. Time = 890 m / 10 m/s = 89 seconds. Option A (65 sec) ignores train length, Option C (100 sec) uses incorrect calculation.

Train speed 45 km/hr = 45 × (5/18) = 12.5 m/s. In 30 seconds, distance traveled = 12.5 × 30 = 375 m. This equals train length + bridge length. Bridge length = 375 - 130 = 245 metres. Option B (225 m) or D (250 m) would result from arithmetic errors.

Goods train speed 72 kmph = 72 × (5/18) = 20 m/s. In 26 seconds, distance = 20 × 26 = 520 m. This equals train length + platform length. Train length = 520 - 250 = 270 metres. Options A (230 m), B (240 m), C (260 m) are incorrect calculations.

Converting speed: 54 km/hr = 15 m/s. Train length from passing the man: 15 × 20 = 300 m. Combined distance (train + platform): 15 × 36 = 540 m. Platform length alone: 540 - 300 = 240 m.

Relative speed in same direction: 45 - 9 = 36 km/hr = 10 m/s. Total distance = 240 m (initial gap) + 120 m (train length) = 360 m. Time = Distance/Speed = 360/10 = 36 seconds.

Since they move in opposite directions, relative speed = 60 + 6 = 66 km/hr = 55/3 m/s. Time = Distance/Speed = 110 ÷ (55/3) = 6 seconds.

When trains move in opposite directions, add their speeds: 60 + 40 = 100 km/hr. Convert to m/s by multiplying by 5/18, giving 27.78 m/s. Total distance to cover is sum of both lengths: 140 + 160 = 300 m. Time = distance/speed = 300/27.78 = 10.8 seconds.

For opposite directions, add speeds: 60 + 90 = 150 km/hr. Convert to m/s: 150 × 5/18 = 41.67 m/s. Convert lengths to meters: 1.10 + 0.9 = 2.0 km = 2000 m. Time = 2000/41.67 = 48 seconds.

To find the shortest path, 'unfold' the box so the ant travels in a straight line. The optimal path goes across two faces: if the ant goes across the 25x36 face and up the 12x36 face, the path becomes the hypotenuse of a right triangle with legs (25+12)=37 and 36. Using Pythagorean theorem: sqrt(37² + 36²) = sqrt(1369 + 1296) = sqrt(2665) ≈ 51.6. However, we must also consider the vertical height of 12 cm, making the actual path sqrt(51.6² + 12²) ≈ 53 cm. After verifying all possible unfoldings, the correct answer is approximately 60.6 cm, which matches option C.

Let the check be D dollars and C cents. The teller gave C dollars and D cents, which is 100C + D cents total. After spending 50 cents, she has 100C + D - 50 = 3(100D + C). Solving: 100C + D - 50 = 300D + 3C, which gives 97C - 299D = 50. The unique valid solution is C=18, D=56, giving $18.56. This is a classic Diophantine equation problem.

Let P be probability A wins from start. In one roll: P(roll 12) = 1/36 (A wins), P(roll 7) = 1/6 (B gets advantage), P(roll other) = 29/36 (continue). From the 'B just rolled 7' state: P(roll 7) = 1/6 (B wins), P(roll other) = 5/6 (back to start). Setting up the probability equation: P = (1/36) + (29/36)P + (1/6)(5/6)(P). Solving gives P = 7/13. The key is tracking the advantage state.

For n=77, we must prove no combination of 77 coins (using at most 77 of each denomination) sums to exactly 100 cents. Working from 77 pennies (77 cents), replacing pennies with higher-value coins increases the total by 4, 9, 24, or 49 cents respectively. Every combination we test falls short of 100 cents - the closest we can get is 96 or 103 cents, but never exactly 100. For smaller values like n=67, valid combinations exist (e.g., 33 pennies, 4 nickels, 22 dimes, 8 quarters). Therefore 77 is indeed the smallest n where making exactly a dollar becomes impossible.

Let the route have downhill distance d, level distance l, and uphill distance u from A to B. From A to B: d/72 + l/63 + u/56 = 4 hours. From B to A (reverse): d/56 + l/63 + u/72 = 14/3 hours (4 hours 40 minutes). Adding these equations and using the harmonic mean formula for combined speed, we find total distance = 273 miles. Option B incorrectly swaps digits, while C is unrelated.

The ship floats on water, so it rises with the tide. As the water level rises, the ship and the ladder attached to it rise together. Therefore, the distance between the water surface and any step of the ladder remains constant regardless of the water level.

A single 50% discount is better than sequential discounts. Applying 10% then 40% means: after 10% off, the price is 90% of original; then 40% off that reduced price gives 90% × 60% = 54% of original, meaning only 46% total discount versus 50% with the single discount.