Question 880396

A can do a piece of work in 2/3 as many days as B, and B can do the same work in 4/5 as many days as C. Together they can do the work in 3 7/11 days. In how many days can A do the work alone?
<pre>
Suppose C can do one complete job in x days.

Then C's work rate in jobs per day is

{{{matrix(1,2,1,job)/matrix(1,2,x,days)}}}{{{""=""}}}{{{matrix(1,2,1/x,jobs/day)}}}
</pre>
 B can do the same work in 4/5 as many days as C
<pre>

So B can do the job in {{{expr(4/5)x}}} days.

Then B's work rate in jobs per day is

{{{matrix(1,2,1,job)/matrix(1,2,expr(4/5)x,days)}}}{{{""=""}}}{{{matrix(1,2,1/(expr(4/5)x),jobs/day)}}}

Simplify that compound fraction by multiplying top and bottom by 5

So B's work rate is

{{{matrix(1,2,5/(4x),jobs/day)}}}

</pre>
A can do a piece of work in 2/3 as many days as B
<pre>

Since B can do the job in {{{expr(4/5)x}}} days,

A can do the job in {{{expr(2/3)(expr(4/5)x)}}} days.

So A can do the job in {{{expr(8/15)x}}} days

---

Then A's work rate in jobs per day is

{{{matrix(1,2,1,job)/matrix(1,2,expr(8/15)x,days)}}}{{{""=""}}}{{{matrix(1,2,1/(expr(8/15)x),jobs/day)}}}

Simplify that compound fraction by multiplying top and bottom by 15

So A's work rate is

{{{matrix(1,2,15/(8x),jobs/day)}}}

---
</pre>
Together they can do the work in 3 7/11 days.
<pre>
Change {{{3&7/11}}} to improper fraction {{{40/11}}}

Then all three's combine work rate in jobs per day is

{{{matrix(1,2,1,job)/matrix(1,2,expr(40/11),days)}}}{{{""=""}}}{{{matrix(1,2,1/expr(40/11),jobs/day)}}}

Simplify that compound fraction by multiplying top and bottom by 11

So their combined work rate is

{{{matrix(1,2,11/40,jobs/day)}}}

---

The equation comes from:

{{{(matrix(6,1,
"A's",work,"rate,",which, is,matrix(1,2,15/(8x),jobs/day)))}}}{{{""+""}}}{{{(matrix(6,1,
"B's",work,"rate,",which, is,matrix(1,2,5/(4x),jobs/day)))}}}{{{""+""}}}{{{(matrix(6,1,
"C's",work,"rate,",which, is,matrix(1,2,1/x,jobs/day)))}}} {{{""=""}}} {{{(matrix(7,1,
Their,combined,work,"rate,",which, is,matrix(1,2,11/40,jobs/day)))}}}

{{{matrix(1,9,

15/(8x),""+"",5/(4x),""+"",1/x,"",""="","",11/40)}}}

Multiply through by the LCD of {{{red(40x)}}}:

{{{matrix(1,9,

red(40x)*expr(15/(8x)),""+"",red(40x)*expr(5/(4x)),""+"",red(40x)*expr(1/x),"",""="","",red(40x)*expr(11/40))}}}

{{{matrix(1,9,

75,""+"",50,""+"",40,"",""="","",11x)}}}

{{{matrix(1,5,
165,"",""="","",11x)}}}

{{{matrix(1,5,
165/11,"",""="","",x)}}}

{{{matrix(1,5,
15,"",""="","",x)}}}

So C can finish the job in 15 days.

The qustion is:
</pre>
In how many days can A do the work alone?
<pre>
A can do the job in {{{expr(8/15)x}}} days, or {{{expr(8/15)15}}}

or 8 days.  That's the answer.

------------------------------------

To check we need how many days it takes B to do the job alone:

B can do the job in {{{expr(4/5)x}}} days.

That's {{{expr(4/5)15)}}} or 12 days.

To check, add their rates in jobs per day to see if we get {{{11/40}}}

{{{matrix(1,9,

1/8,""+"",1/12,""+"",1/15,"",""="","",11/40)}}}

That checks, so we are right.

A can complete the job in 8 days.

Edwin</pre>