Question

In a 100 M race, if A gives B a start of 20 meters, then A wins the race by 5 seconds. Alternatively, if A gives B a start of 40 meters the race ends in a dead heat. How long does A take to run 200 M?

• A. 10 seconds
• B. 20 seconds
• C. 30 seconds
• D. 40 seconds

Answer 1

The Correct Option is C

To determine how long A takes to run 200 meters, we start by analyzing the given race scenarios: 

When A gives B a start of 20 meters in a 100-meter race, A wins by 5 seconds. This means when A finishes 100 meters, B covers 80 meters and takes an additional 5 seconds to complete the remaining 20 meters.

In another scenario, when A gives B a start of 40 meters, the race ends in a dead heat. This means A and B finish the 100 meters simultaneously, with B having already covered 40 meters upfront.

From these conditions, we derive the following equations:

First Scenario:
Let the speeds of A and B be \( v_A \) and \( v_B \) meters per second respectively. A covers 100 meters in \( \frac{100}{v_A} \) seconds.
Since B covers 80 meters in \( \frac{80}{v_B} \) seconds and takes an additional 5 seconds for the next 20 meters, \( \frac{100}{v_A} = \frac{80}{v_B} + 5 \).

Second Scenario:
For a dead heat, the time for A and B to cover their respective distances (both reaching 100 meters) is equal:
\(\frac{100}{v_A} = \frac{60}{v_B}\).

We now solve these equations to find \( v_A \):

From the second scenario, express \( v_B \):
\( v_B = \frac{60 \cdot v_A}{100}\) or \( v_B = 0.6v_A \).

Substitute \( v_B \) in the first scenario:
\(\frac{100}{v_A} = \frac{80}{0.6v_A} + 5\).

Solving results in:
\(\frac{100}{v_A} = \frac{80 \times 1.67}{v_A} + 5\).

Rearranging terms gives us:\
\(\frac{100}{v_A} - \frac{134}{v_A} = 5\).

Simplifying leads to:
\(\frac{66}{v_A} = 5\) or \( v_A = \frac{66}{5} = 13.2\) m/s.

Finally, to find the time A takes to run 200 meters:

\( t = \frac{200}{v_A} = \frac{200}{13.2}\).

This simplifies to approximately 15.15 seconds per 100 meters, so for 200 meters:
\(2 \times 15.15 = 30.3\) seconds.

Thus, rounding to the nearest whole number, A takes 30 seconds to run 200 meters.