Question 974568
Sumi can wash windows of an office building in 3/4 the time it takes her
apprentice. One day they worked on the building together for 2h 16 min then Sumi
continued alone. It took 4 h 32 min more to complete the job. How long would it
take her apprentice to wash all the windows? 
How do you figure it out and what is the answer?
<pre>
Let x hours represent the solution, how long it would take the apprentice to
wash all the windows

So the apprentice's window-washing rate is 1 set of windows per x hours.

That is to say that the apprentice's window-washing rate is {{{matrix(1,4,1,set,of,windows)/matrix(1,2,x,hours)}}} or {{{matrix(1,2, 1/x, matrix(1,3,set,of,windows)/hr))}}}
</pre>
 Sumi can wash windows of an office building in 3/4 the time it takes her apprentice.
<pre>

Then Sumi's window-washing rate is 1 set of windows per {{{expr(3/4)x}}} hours.

Therefore Sumi's window-washing rate is {{{matrix(1,4,1,set,of,windows)/matrix(1,2,expr(3/4)x,hours)}}} or {{{matrix(1,2, 1/expr(3/4)x, matrix(1,3,set,of,windows)/hr))}}}

We simplify {{{1/expr(3/4)x}}}by multiplying top and bottom by 4 getting
{{{4/(3x)}}}. So Sumi's window-washing rate is

{{{matrix(1,2, 4/(3x), matrix(1,3,set,of,windows)/hr))}}}
</pre>
One day they worked on the building together...
<pre>
So their combined rate was the sum of those rates

{{{matrix(1,2, 1/x+4/(3x), matrix(1,3,set,of,windows)/hr))}}}

We simplify {{{1/x+4/(3x)=3/(3x)+4/(3x)=7/(3x)}}}

So their combined rate was:

{{{matrix(1,2, 7/(3x), matrix(1,3,set,of,windows)/hr))}}}

</pre>
One day they worked on the building together for 2h 16 min 
<pre>
We convert that to an improper fraction of hours.
{{{matrix(1,10,matrix(1,3,2h,16,min), ""="",2&16/60,hrs,""="",2&4/15,hr,""="",34/15,hr)}}} 

Since rate×time = production, during that time the fraction of 
the set of windows they washed working together was

{{{(7/(3x))*(34/15)}}}{{{""=""}}}{{{matrix(1,6,238/(45x),of,the,set,of,windows)}}}

then Sumi continued alone. It took 4 h 32 min more to complete the job.

We convert that to an improper fraction of hours.
{{{matrix(1,10,matrix(1,3,4h,32,min), ""="",4&32/60,hrs,""="",4&8/15,hr,""="",68/15,hr)}}}

Again, since rate×time = production, during that time the fraction of 
the set of windows that Sumi washed working alone was

{{{(4/(3x))*(68/15)}}}{{{""=""}}}{{{matrix(1,6,272/(45x),of,the,set,of,windows)}}}

The equation comes from

{{{(matrix(11,1,

The, fraction,of,the,set,of, windows,they,washed,working,together))}}}{{{""+""}}}{{{(matrix(11,1,

The, fraction,of,the,set,of, windows,Sumi,washed,working,alone))}}}{{{""=""}}}{{{(matrix(5,1,

1,complete,set,of, windows))}}}

{{{matrix(1,7,238/(45x),""+"",272/(45x),"",""="","",1)}}}

Multiply through by 45x

{{{238+272}}}{{{""=""}}}{{{45x}}}

{{{510}}}{{{""=""}}}{{{45x}}}

{{{510/45}}}{{{""=""}}}{{{x}}}

{{{34/3}}}{{{""=""}}}{{{x}}}

{{{11&1/3}}}{{{""=""}}}{{{x}}}

That's 11 hours 20 minutes, so that's how long it would take
the apprentice working alone to wash all the windows.

Edwin</pre>