Expert: Josh - 12/10/2007

Question
Rick and Juanita complete a job together in 6 hours. It takes rick 9 hours longer than Juanitia to do it alone. How long does it take Juanita.

Answer
Hi Doug,

Let us assume that Rick and Juanita work at a rate of R1 and R2 respectively. Working independently, Rick and Juanita finish the job by themselves in T1 and T2 hours respectively.

The underlying principle is that the amount of work done equals the work rate, multiplied by the time taken to do the job.

Since the amount of work they end up doing is the same,
R1*T1 = R2*T2.

In fact, when they work together, they finish the job in T=6 hours at a combined rate of (R1+R2). So, we know the following:

R1*T1 = R2*T2 = (R1+R2)*T, where T=6.

Finally, Rick takes 9 hours longer than Juanita to complete the job on his own. This means T1 = T2+9. Putting this into the equations above,

R1*(T2+9) = R2*(T2) = 6(R1+R2), where R1 and R2 are unknown and T2 (time taken for Juanita to complete the job) is what we need to determine.

Dividing throughout by R2, we can either write the answer as
T2 = 6(R1+R2)/R2 ...using the second and third terms OR
T2=[6(R1+R2)/R1]-9 ...using the first and third terms

with R1 and R2 both positive (work rates) and R1 < R2.

We do not have sufficient information to determine the value of T2 unless we know the ratio of R1/R2, i.e., how much faster Juanita works with respect to Rick.

e.g., suppose we were told that Rick only gets through half as much as Juanita in a fixed period of time. Then, R1=0.5*R2. In this case,

Time taken Juanita to finish the job on her own would be
T2 = 6(R1+R2)/R2 = 6(1.5*R2)/R2 = 9

Time taken Rick to finish the job on his own would be
T1 = T2+9 = 18.

If the amount of work to be done is denoted by W, then Rick's work rate is W/18; Juanita's work rate is W/9. i.e., he finishes 1/9 of the total work in an hour.

Now, you can see that their combined work rate is R1+R2 = W/9 + w/18 = (3/18)W = W/6. So, it would take them 6 hours to get W done working together.

Let me know if you have any questions