SOLUTION: abbie paints twice as fast as beth and three times as fast as cathie. if it takes them 60 minutes to paint a living room with all three working together,how long would it take each

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Question 1197111: abbie paints twice as fast as beth and three times as fast as cathie. if it takes them 60 minutes to paint a living room with all three working together,how long would it take each woman working alone?
Found 3 solutions by ikleyn, josgarithmetic, math_tutor2020:
Answer by ikleyn(52781)   (Show Source):
You can put this solution on YOUR website!
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abbie paints twice as fast as beth and three times as fast as cathie.
if it takes them 60 minutes to paint a living room with all three working together,
how long would it take each woman working alone?
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Their rates of work are in proportion A : B : C = 6 : 3 : 2. So, let 2x be the Cathie's rate of work; 3x be Beth's rate of work and 6x be Abbie's rate of work. According to the condition, 6x + 3x + 2x = of the job 11x = x = . So, Abbie's rate of work is = of the job per minute; Beth's rate of work is = of the job per minute; Cathie rate of work is = of the job per minute. It means that Abbie needs 110 minutes = 1 hour and 50 minutes to complete the job alone; Beth needs 220 minutes = 3 hour and 40 minutes to complete the job alone; Cathie needs 330 minutes = 5 hour and 30 minutes to complete the job alone.

Solved.

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As this problem is twisted, it shows me that it is slightly higher that the average school level (with a claim).

Therefore, my response is written in an adequate style, without chewing.

Answer by josgarithmetic(39617)   (Show Source):
You can put this solution on YOUR website!
twice as fast and three times as fast,...

Abbie 2B Beth B Cathie 2B/3

60 minutes for the three working together to do 1 job.

-

---------- Beth need 220 minutes to do 1 job alone.
The other two follow from using this value.

Answer by math_tutor2020(3817)   (Show Source):
You can put this solution on YOUR website!

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.

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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
Abbie does 6 parts
Beth does 3 parts
Cathie does 2 parts
When the girls work together,
Abbie does 6/11 of the job
Beth does 3/11 of the job
Cathie does 2/11 of the job

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
Abbie's portion is (6/11)*3300 = 1800 sq ft
Beth's portion is (3/11)*3300 = 900 sq ft
Cathie's portion is (2/11)*3300 = 600 sq ft
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
Abbie's unit rate is 1800/60 = 30 sq ft per min
Beth's unit rate is 900/60 = 15 sq ft per min
Cathie's unit rate is 600/60 = 10 sq ft per min
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

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Answers:
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