Question 1197111
<font color=black size=3>
Abbie paints twice as fast as Beth
This means Beth takes twice as long as Abbie


Abbie paints three times as fast as Cathie
So Cathie takes three times as long as Abbie


x = amount of time, in minutes, Abbie needs to do the job by herself
2x = amount of time, in minutes, Beth needs to do the job by herself
3x = amount of time, in minutes, Cathie needs to do the job by herself


The reciprocal of each item represents the unit rate
1/x = unit rate for Abbie
1/(2x) = unit rate for Beth
1/(3x) = unit rate for Cathie
Each unit rate is in "jobs per minute". 


Those fractional unit rates add to this
(1/x) + (1/(2x)) + (1/(3x))
(6/(6x)) + (3/(6x)) + (2/(6x))
(6+3+2)/(6x)
11/(6x)


If the girls work together, without getting in each others' way, then their combined unit rate is 11/(6x) jobs per minute.


Multiply this unit rate with the total time they use (60 min) and set this equal to 1 to represent getting 1 full job done.


So,
(unit rate)*(time) = amount done
(11/(6x))*(60 min) = 1 full job
(11/(6x))*(60) = 1
110/x = 1
110 = x*1
x = 110


Abbie needs 110 minutes to get the job done on her own.
Beth needs 2x = 2*110 = 220 minutes to get the job done on her own.
Cathie needs 3x = 3*110 = 330 minutes to get the job done on her own.


===========================================================================================


Another approach:


a = Abbie's rate 
b = Beth's rate
c = Cathie's rate


Abbie paints twice as fast as Beth, so,
a = 2b
Also, Abbie paints three times as fast as Cathie
a = 3c


Both equations mentioned are equal to 'a', which means we can equate the right hand sides
2b = 3c
Divide both sides by 2 allowing us to solve for b
b = 3c/2 = 1.5c


So we can say
a = 3c
b = 1.5c


The ratio a:b:c can be expressed as 3c:1.5c:c
Multiply all parts by 2 to get 6c:3c:2c
Every part of this ratio involves the variable c in some fashion.


The coefficients add to 6+3+2 = 11
We can think of it like saying<ul><li>Abbie does 6 parts</li><li>Beth does 3 parts</li><li>Cathie does 2 parts</li></ul>When the girls work together,<ul><li>Abbie does 6/11 of the job</li><li>Beth does 3/11 of the job</li><li>Cathie does 2/11 of the job</li></ul>
Let's say the job is to paint 3300 square feet of wall
I picked some large multiple of 11. Feel free to pick something else. 
I picked a multiple of 11 so that multiplying with the fractions leads to a whole number.


Let's now subdivide the work like so<ul><li>Abbie's portion is (6/11)*3300 = 1800 sq ft</li><li>Beth's portion is (3/11)*3300 = 900 sq ft</li><li>Cathie's portion is (2/11)*3300 = 600 sq ft</li></ul>Note that 1800:900:600 reduces to 6:3:2 after dividing all three parts by 300.


Since it took 60 minutes for each woman to do their portions of the wall, this means<ul><li>Abbie's unit rate is 1800/60 = 30 sq ft per min</li><li>Beth's unit rate is 900/60 = 15 sq ft per min</li><li>Cathie's unit rate is 600/60 = 10 sq ft per min</li></ul>The ratio 30:15:10 is equivalent to 6:3:2 after dividing all three parts by 5.


Now let's have them work alone.
If Abbie works alone, then she needs to paint the entire 3300 sq ft. Her unit rate is 30 sq ft per min. 
Therefore, she needs 3300/30 = 110 minutes.
I'm using the idea that 
time = (amount done)/(rate of work)


If Beth works alone, then she needs 3300/15 = 220 minutes


If Cathie works alone, then she needs 3300/10 = 330 minutes


===========================================================================================


<font color=red>Answers:</font>
Abbie needs 110 minutes if she works alone
Beth needs 220 minutes if she works alone
Cathie needs 330 minutes if she works alone



Footnotes:
110 min = 60 min + 50 min = 1 hr + 50 min
220 min = 180 min + 40 min = 3 hr + 40 min
330 min = 300 min + 30 min = 5 hr + 30 min
180 min = 180/60 = 3 hrs
300 min = 300/60 = 5 hrs
</font>