Distance Calculation: Runners' Problem

This problem requires finding the distance covered by two runners, A and B. We are given their respective speeds and the time difference between them covering the same distance.

• Runner A's Speed (\(S_A\)): 15 km/h
• Runner B's Speed (\(S_B\)): 16 km/h
• Time difference (\(T_A - T_B\)): 16 minutes
• The distance (\(D\)) is the same for both runners.

Time Difference: Minutes to Hours Conversion

To maintain consistency with the speeds given in kilometers per hour (km/h), the time difference must be converted from minutes to hours.

Time difference = 16 minutes = \( \frac{16}{60} \) hours = \( \frac{4}{15} \) hours.

Distance Equation Formulation

The fundamental relationship between distance, speed, and time is Time = Distance / Speed (\( T = \frac{D}{S} \)). Let \(T_A\) be the time taken by runner A and \(T_B\) be the time taken by runner B.

• Time for Runner A: \( T_A = \frac{D}{15} \)
• Time for Runner B: \( T_B = \frac{D}{16} \)

Since Runner A takes 16 minutes (i.e., \( \frac{4}{15} \) hours) longer than Runner B, we can write the equation:

\( T_A - T_B = \frac{4}{15} \)

Substitute the expressions for \( T_A \) and \( T_B \):

\( \frac{D}{15} - \frac{D}{16} = \frac{4}{15} \)

Solving the Distance Equation

To find the distance \(D\), we need to solve the equation. First, find a common denominator for the terms involving \(D\) on the left side.

The least common multiple of 15 and 16 is \( 15 \times 16 = 240 \). Adjust the fractions:

\( \frac{16D}{240} - \frac{15D}{240} = \frac{4}{15} \)

Combine the terms:

\( \frac{16D - 15D}{240} = \frac{4}{15} \)

\( \frac{D}{240} = \frac{4}{15} \)

Isolate \(D\) by multiplying both sides by 240:

\( D = \frac{4}{15} \times 240 \)

Simplify the calculation:

\( D = 4 \times \frac{240}{15} \)

\( D = 4 \times 16 \)

\( D = 64 \)

Final Calculated Distance

The calculated distance covered by both runners is 64 km.