100m Race Dynamics

This problem involves calculating relative distances covered by runners in a race based on given margins.

100m Race: A vs B Performance

When runner A completes the 100 m race, runner B is 10 m behind.

• A covers 100 m.
• B covers $100 \text{ m} - 10 \text{ m} = 90 \text{ m}$.

This implies the ratio of distances covered by A and B in the same time is $\frac{\text{Distance}_A}{\text{Distance}_B} = \frac{100}{90}$.

100m Race: B vs C Performance

When runner B completes a race (equivalent to 100 m for comparison), runner C is 5 m behind.

• B covers 100 m.
• C covers $100 \text{ m} - 5 \text{ m} = 95 \text{ m}$.

This implies the ratio of distances covered by B and C in the same time is $\frac{\text{Distance}_B}{\text{Distance}_C} = \frac{100}{95}$.

100m Race: Calculating A vs C Margin

We need to determine how far C runs when A finishes the 100 m race.

From the A vs B analysis, when A runs 100 m, B runs 90 m.

Now, we use the B vs C ratio to find C's distance when B runs 90 m:

The ratio $\frac{\text{Distance}_C}{\text{Distance}_B} = \frac{95}{100}$.

Therefore, when $\text{Distance}_B = 90 \text{ m}$, the distance C covers is:

$\text{Distance}_C = 90 \text{ m} \times \frac{95}{100}$

$\text{Distance}_C = 90 \times 0.95 = 85.5 \text{ m}$

Final Beat Margin: A vs C

When A completes the 100 m race, C has run 85.5 m.

The distance A beats C by is the difference between the race distance and C's covered distance:

$100 \text{ m} - 85.5 \text{ m} = 14.5 \text{ m}$