Chapter 11.1, Problem 81AYU

To determine

To calculate: The time taken by Beth, Bill and Edie to complete the job alone.

Answer to Problem 81AYU

The time taken by Beth is 30 hours, by Bill is 20.03 hours and by Edie is 40 hours to complete the job alone.

Explanation of Solution

Given information:

In 10 hours Beth, Bill and Edie can paint exterior of a home. Bill and Edie can complete the similar task in 15 hours. One fine day all three worked for 4 hours and then Edie left so it took 8 hours to complete the task by Beth and Bill.

Calculation:

Consider the provided information that in 10 hours Beth, Bill and Edie can paint exterior of a home. Bill and Edie can complete the similar task in 15 hours. One fine day all three worked for 4 hours and then Edie left so it took 8 hours to complete the task by Beth and Bill.

Let in one hour the amount of work done by Beth is x, by Bill is y and by Edie is z.

If all three work together and work done in 1 hour is $x+y+z=1$ .

In 10 hours Beth, Bill and Edie can paint exterior of a home.

So, the equation is $10x+10y+10z=1$

Bill and Edie can complete the similar task in 15 hours.

So, the equation is $15y+15z=1$

One fine day all three worked for 4 hours and then Edie left so it took 8 hours to complete the task by Beth and Bill.

So, the equation is,

  $\begin{array}{c}4\left(x+y+z\right)+8\left(x+y\right)=1\\ 12x+12y+4z=1\end{array}$

So, the system of equation so formed is,

  $\begin{array}{c}10x+10y+10z=1\\ 15y+15z=1\\ 12x+12y+4z=1\end{array}$

Multiply $10x+10y+10z=1$ by 3 and $15y+15z=1$ by 2 and then subtract.

  $x=0.0333$

Therefore, time taken by Beth is $\frac{1}{0.033}=30\text{ hrs}$ .

Multiply $10x+10y+10z=1$ by 6 and $12x+12y+4z=1$ by 5 and subtract.

  $z=0.025$

Therefore, time taken by Edie is $\frac{1}{0.025}=40\text{ hrs}$ .

Now substitute $z=0.025$ in equation $15y+15z=1$ .

Therefore, $y=0.0416$

Therefore, time taken by Bill is $\frac{1}{0.0416}=24.03\text{ hrs}$ .

Thus, the amount placed in treasury bill is $\$8000$ , in treasury bond is $\$7000$ and corporate bond is $\$5000$ .