Question

Aaron will jog from home at \( x \) miles per hour and then walk back home by the same route at \( y \) miles per hour. How many miles from home can Aaron jog so that he spends a total of \( t \) hours jogging and walking?

• A. \( \frac{xt}{y} \)
• B. \( \frac{x + t}{xy} \)
• C. \( \frac{xyt}{x + y} \)
• D. \( \frac{x + y + t}{xy} \)
• E. \( \frac{y + t}{x + y} \)

Hint:

When dealing with time, distance, and speed, use the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \) and solve for the unknown.

Answer 1

The Correct Option is C

Step 1: Understand the problem.
Let the distance Aaron jogs be \( d \). The time taken to jog this distance is: \[ \text{Time jogging} = \frac{d}{x} \] The time taken to walk back the same distance is: \[ \text{Time walking} = \frac{d}{y} \] The total time spent jogging and walking is given as \( t \), so: \[ \frac{d}{x} + \frac{d}{y} = t \] Step 2: Solve for \( d \).
Factor out \( d \) from the left-hand side: \[ d \left( \frac{1}{x} + \frac{1}{y} \right) = t \] Simplify the expression inside the parentheses: \[ d \left( \frac{x + y}{xy} \right) = t \] Now, solve for \( d \): \[ d = \frac{t \cdot xy}{x + y} \] \[ \boxed{\frac{xyt}{x + y}} \]