How Long Would it Take Jen to Paint the Fence Alone?

Imagine a scenario where Joshua can paint a fence in 5 hours, working alone, but together with Jen, they can complete the task in just 2 hours. How could we determine how long it would take Jen to paint the fence on her own? Let's break down the problem and solve it step-by-step using the principles of work rate.

Understanding Work Rate

The rate at which work is completed is a crucial concept when dealing with such problems. By understanding the individual and combined work rates, we can derive the required time for one person to complete the task alone.

Calculating Joshua's Work Rate

Joshua's work rate, denoted as ( R_J ), is the proportion of the fence he can paint in one hour. Given that he can paint a complete fence in 5 hours, his work rate is:

( R_J frac{1 text{ fence}}{5 text{ hours}} 0.2 text{ fences per hour} )

Calculating the Combined Work Rate

When Joshua and Jen work together, they complete the fence in 2 hours. Their combined work rate, denoted as ( R_{JJ} ), is:

( R_{JJ} frac{1 text{ fence}}{2 text{ hours}} 0.5 text{ fences per hour} )

Determining Jen's Work Rate

To find Jen's work rate, denoted as ( R_{Je} ), we use the equation:

( R_J R_{Je} R_{JJ} )

Substituting the known values:

( 0.2 R_{Je} 0.5 )

Thus, solving for ( R_{Je} ):

( R_{Je} 0.5 - 0.2 0.3 text{ fences per hour} )

Calculating the Time for Jen to Paint Alone

With Jen's work rate known, we can calculate the time it would take her to paint the fence alone. If ( R_{Je} ) is 0.3 fences per hour, the time ( T_{Je} ) it takes her to paint one fence is:

( T_{Je} frac{1 text{ fence}}{R_{Je}} frac{1}{0.3} approx 3.33 text{ hours} )

Around 3 hours and 20 minutes, the time it takes Jen to paint the fence on her own is approximately 3.33 hours.

Alternative Calculations

Another way to approach the problem is through the combined work rates and the individual contributions. If Joshua alone paints ( frac{1}{5} ) of the fence in one hour, in two hours, he paints ( frac{2}{5} ) of the fence. Consequently, the remaining ( frac{3}{5} ) of the fence must be painted by Jen, which she completes in 2 hours. Therefore, her work rate is:

( frac{3}{5 times 2} frac{3}{10} text{ fences per hour} )

Then, the time it would take her to paint the entire fence alone is:

( T_{Je} frac{1}{0.3} 3.33 text{ hours} )

This confirms that it would take Jen, approximately 3 hours and 20 minutes, to paint the fence alone.

Conclusion

By understanding and applying the concept of work rates, we can easily determine the time it takes for one person to complete a task when given the time taken by multiple individuals together. In this case, it was calculated that it takes Jen approximately 3 hours and 20 minutes to paint the fence alone.