Chapter 11.5, Problem 45PPE

To determine

How long would it take her apprentice to wash all the windows alone?

Answer to Problem 45PPE

11h 20 min

Explanation of Solution

Given:

Sumi can wash the windows of an office building in $\frac{3}{4}$ the time it takes her apprentice. One day they worked together for 2h 16 minutes and then Sumi continued alone and it took her 4 h 32 min to complete the job.

Concept Used:

• Least Common Denominator (LCD) is the smallest number that is divisible by the denominator of all the fractions that are being added or subtracted.
• Solving an equation containing rational expressions
• Multiply both sides of equations by LCD(least common denominator) of all denominators.
• Remove any grouping symbols and solve the resulting equation for the variable asked in equation

Calculation:

Let it takes x minutes to her apprentice to wash all the windows alone. So, in 1 minute she will do $\frac{1}{x}$ part of washing.

Then the time taken by Sumi to wash all window would be $\frac{3}{4}x$ . And her 1 minute part of job would be $\frac{4}{3x}$ .

On a particular day they worked together for 2 h 16 min = 136 min. So, they together completed $136\left(\frac{1}{x}+\frac{4}{3x}\right)=136\left(\frac{3+4}{3x}\right)=\frac{952}{3x}$ part of the job.

So remaining job is $1-\frac{952}{3x}=\frac{3x-952}{3x}$ part.

This remaining part Sumi completed in 4 h 32 min = 272 minutes.

Now Sumi do in 272 min $=\frac{3x-952}{3x}$ part

So, in 1 min she will do $=\frac{\frac{3x-952}{3x}}{272}=\frac{3x-952}{816x}$ part

Now as per our initial assumption, Sumi’s 1 min work is $\frac{4}{3x}$ part. So on equating, it gives

  $\begin{array}{c}\frac{3x-952}{816x}=\frac{4}{3x}\\ \frac{3x-952}{816}=\frac{4}{3}\\ 3\left(3x-952\right)=4\left(816\right)\\ 9x-2856=3264\\ 9x=6120\\ x=\frac{6120}{9}\\ x=680\end{array}$

So, it will take 680 minutes or 11h 20 min to her apprentice to wash all the windows alone.