Q. 81 - Precalculus Enhanced with Graphing Utilities

Question

Three painters, Beth, Bill, and Edie, working together, can paint the exterior of a home in 10 hours (hr). Bill and Edie together have painted a similar house in 15 hr. One day, all three worked on this same kind of house for 4 hr, after which Edie left. Beth and Bill required 8 more hr to finish. Assuming no gain or loss in efficiency, how long should it take each person to complete such a job

alone?

Step-by-Step Solution

Answer

It will take Beth 30 hours, Bill 24 hours, Edie takes 40 hours to complete the job alone

Step 1: Given information

Three painters, Beth, Bill, and Edie, working together, can paint the exterior of a home in 10 hours (hr). Bill and Edie together have painted a similar house in 15 hr.

Step 2: Find the equations

let x be no of hours Beth completes the job alone

Let y be no of hours Bill completes the job alone 

Let z no of hours Edie completes the job alone 

From the given conditions the equations can be given as

$\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{10} \left(1\right)$

Together bill and Edie painted a similar house in 15 hours

$\frac{1}{y}+\frac{1}{z}=\frac{1}{15}$                       (2)

All three worked for 4 hours and then Edie left Beth and Bill  required 8 hours to paint the house

$\frac{12}{x}+\frac{12}{y}+\frac{4}{z}=1$ 

Step 3: Solve the equations

We substitute $\frac{1}{x}=u,\frac{1}{y}=v,\frac{1}{z}=w$

We get,

$$\begin{gathered} u+v+w=\frac{1}{10} \left(4\right) \\ v+w=\frac{1}{15} \left(5\right) \\ 12u+12v+4w=1 \left(6\right) \end{gathered}$$

Substituting 5 in 4 we get,

$$\begin{gathered} u+\frac{1}{15}=\frac{1}{10} \\ u=\frac{1}{30} \end{gathered}$$

Substitute u in equation 6 we get,

$$\begin{gathered} \frac{2}{5}+12v+4w=1 \\ 12v+4w=\frac{3}{5} \left(7\right) \end{gathered}$$ 

Multiply equation 5 by -4 and add it to equation 7 we get,

$-4v-4w=-\frac{4}{15}+12v+4w=\frac{3}{5}8v=\frac{1}{3}$

Hence $v=\frac{1}{24}$

Find The value of w

$$\begin{gathered} v+w=\frac{1}{15} \\ w=\frac{1}{15}-\frac{1}{24} \\ w=\frac{1}{40} \end{gathered}$$

Step 4: Resubstitute to find the values of x, y, z

We get,

$$\begin{gathered} u=\frac{1}{x} \\ u=30 \\ v=\frac{1}{y} \\ v=24 \\ w=\frac{1}{z} \\ w=40 \end{gathered}$$

Step 5: Conclusion

It will take Beth 30 hours, Bill 24 hours, Edie takes 40 hours to complete the job alone