A man rides a bike whose front wheel has a circumference of 40 inches and back wheel has a circumference of 70 inches. The bike is ridden along a straight road without any slippage. How many inches will the bike travel by the time the front wheel has made 15 more revolutions than the back wheel?

Introduction / Context:
This is a problem involving rotational motion and distances travelled without slippage. A bike has two wheels of different circumferences, and because there is no slipping, both wheels cover the same linear distance along the road. The front wheel, however, makes more revolutions than the back wheel. We must determine the actual linear distance travelled when the difference in revolutions is 15.

Given Data / Assumptions:

- Circumference of front wheel = 40 inches per revolution.
- Circumference of back wheel = 70 inches per revolution.
- The front wheel has made 15 more revolutions than the back wheel.
- Let F be the number of front wheel revolutions and B be the number of back wheel revolutions.
- The bike moves in a straight line without any slippage, so both wheels cover the same linear distance.

Concept / Approach:
The linear distance travelled is equal to circumference multiplied by the number of revolutions for any wheel. Since both wheels travel the same linear distance, 40 * F must equal 70 * B. We are also given that F = B + 15. These two equations allow us to solve for F and B. Once we know F, we can compute the total distance as 40 * F inches.

Step-by-Step Solution:
Let B be the number of revolutions of the back wheel. Then F, the number of revolutions of the front wheel, is B + 15. Distance travelled according to the front wheel = 40 * F inches. Distance travelled according to the back wheel = 70 * B inches. Because there is no slippage: 40 * F = 70 * B. Substitute F = B + 15: 40 (B + 15) = 70 B. Expand: 40B + 600 = 70B. Rearrange: 600 = 30B, so B = 600 / 30 = 20. Therefore F = B + 15 = 20 + 15 = 35 revolutions. Total distance travelled = circumference of front wheel * F. Distance = 40 * 35 = 1400 inches.

Verification / Alternative check:
We can also verify using the back wheel. If B = 20, distance according to back wheel = 70 * 20 = 1400 inches, which matches the distance computed using the front wheel. The difference in revolutions is F - B = 35 - 20 = 15, exactly as required in the problem statement. This confirms that 1400 inches is consistent with all given data.

Why Other Options Are Wrong:
1100, 1200, and 1300 inches: For these values, when divided by 40 and 70, the resulting numbers of revolutions would not differ by 15. None of them satisfy both the equal distance condition and the 15 revolution difference at the same time.

Common Pitfalls:
Some learners mistakenly assume the wheel with the larger circumference must make more revolutions, which is the opposite of what happens. Others forget to use the fact that both wheels travel the same linear distance and instead treat the revolutions separately. Always set up both equations together: one expressing equal distances and the other expressing the difference in revolutions.

Final Answer:
The bike will have travelled 1400 inches when the front wheel has made 15 more revolutions than the back wheel.