Questions: Janelle has 3 hours to spend training for an upcoming race. She completes her training by running full speed the distance of the race and walking back the same distance to cool down. If she runs at a speed of 9 mph and walks back at a speed of 3 mph, how long should she plan to spend walking back?

Solution Steps

Janelle has a total of 3 hours to complete her training, which consists of running a certain distance at 9 mph and walking back the same distance at 3 mph. We need to determine how long she should plan to spend walking back.

Let \( d \) be the distance of the race in miles. The time taken to run the distance is given by:

\[ t_{\text{run}} = \frac{d}{9} \]

The time taken to walk back the same distance is:

\[ t_{\text{walk}} = \frac{d}{3} \]

The total time for the training is 3 hours, so:

\[ t_{\text{run}} + t_{\text{walk}} = 3 \]

Substituting the expressions for \( t_{\text{run}} \) and \( t_{\text{walk}} \):

\[ \frac{d}{9} + \frac{d}{3} = 3 \]

To solve for \( d \), first find a common denominator for the fractions:

\[ \frac{d}{9} + \frac{3d}{9} = 3 \]

Combine the fractions:

\[ \frac{4d}{9} = 3 \]

Multiply both sides by 9 to clear the fraction:

\[ 4d = 27 \]

Divide both sides by 4 to solve for \( d \):

\[ d = \frac{27}{4} = 6.75 \]

Now that we have \( d = 6.75 \) miles, calculate the time spent walking back:

\[ t_{\text{walk}} = \frac{d}{3} = \frac{6.75}{3} = 2.25 \text{ hours} \]

Final Answer