Question 951733
if sally can paint a room in 6 hours while it takes steve 7 hours to paint the
same room. how long would it take them to paint the room if they worked
together?
<pre>
There are two ways to do it, one with algebra and one way without algebra. 
I'll show both ways.

WITHOUT algebra:

The least common multiple of 6 hours and 7 hours is 42 hours.  So in a 42 hour
period working together, Sally could paint 7 rooms while Steve paints 6.  So
together they would have painted 13 rooms in 42 hours, so together they could
paint 1 room in {{{42/13}}} hours or {{{3&3/13}}} hours. 

WITH algebra:

Sally can paint 1 room in 6 hours, so

Sally's painting rate is {{{matrix(1,2,1,room)/matrix(1,2,6,hours)}}}, or

{{{matrix(1,2,1/6,room/hour)}}}

Steve can paint a room in 7 hours, so

Steve's painting rate is {{{matrix(1,2,1,room)/matrix(1,2,7,hours)}}}.

or {{{matrix(1,2,1/7,room/hour)}}}
 
Let x = the number of hours it would take them to paint the room.

Their combined painting rate is {{{matrix(1,2,1,room)/matrix(1,2,x,hours)}}}.

or {{{matrix(1,2,1/x,room/hour)}}}

The equation comes from:

{{{(matrix(3,1,"Sally's",painting, rate))}}}{{{""+""}}}{{{(matrix(3,1,"Steve's",painting, rate))}}}{{{""=""}}}{{{(matrix(4,1,their,combined,painting, rate))}}}

{{{1/7}}}{{{""+""}}}{{{1/6}}}{{{""=""}}}{{{1/x}}}

Multiply through by LCD 42x

{{{6x}}}{{{""+""}}}{{{7x}}}{{{""=""}}}{{{42}}}

{{{13x}}}{{{""=""}}}{{{42}}}

{{{x}}}{{{""=""}}}{{{42/13}}}

{{{x}}}{{{""=""}}}{{{3&3/13}}}{{{hours}}}

Edwin</pre>