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work problem

by hemant_rajput » Wed Jan 16, 2013 9:05 am
Q8 . Three members of a family A,B and C work together to all household chores done. The time it takes them to do the work together is 6 hour less than A would have taken working alone, one hour less than B would have taken alone and half the time C would have taken working alone. How long did it take them to do these chores working together?
a. 20 min
b. 30 min
c. 40 min
d. 50 min

[spoiler]oa: c[/spoiler]
Last edited by hemant_rajput on Thu Jan 17, 2013 7:56 am, edited 1 time in total.
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by Abhishek009 » Wed Jan 16, 2013 9:58 am
hemant_rajput wrote:Q8 . Three members of a family A,B and C work together to all household chores done. The time it takes them to do the work together is 6 hour less than A would have taken working alone, one hour less than B would have taken alone. How long did it take them to do these chores working together?
a. 20 min
b. 30 min
c. 40 min
d. 50 min
[spoiler]oa: c[/spoiler]
Is the question complete ?

We need to know the time taken by C as well in any form to frame an equation ..
Abhishek
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by hemant_rajput » Thu Jan 17, 2013 5:51 am
Abhishek009 wrote:
hemant_rajput wrote:Q8 . Three members of a family A,B and C work together to all household chores done. The time it takes them to do the work together is 6 hour less than A would have taken working alone, one hour less than B would have taken alone. How long did it take them to do these chores working together?
a. 20 min
b. 30 min
c. 40 min
d. 50 min
[spoiler]oa: c[/spoiler]
Is the question complete ?

We need to know the time taken by C as well in any form to frame an equation ..
sorry I missed one part of it. I added the missing part.
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by GMATGuruNY » Thu Jan 17, 2013 1:37 pm
hemant_rajput wrote:Q8 . Three members of a family A,B and C work together to all household chores done. The time it takes them to do the work together is 6 hour less than A would have taken working alone, one hour less than B would have taken alone and half the time C would have taken working alone. How long did it take them to do these chores working together?
a. 20 min
b. 30 min
c. 40 min
d. 50 min

[spoiler]oa: c[/spoiler]
We can plug in the answers, which represent the time when all 3 work together.

Answer choice B: 30 minutes
Since A alone takes 6 hours longer, the time for A alone = 1/2 + 6 = 13/2 hours.
Since B alone takes 1 hour longer, the time for B alone = 1/2 + 1 = 3/2 hours.
Since C alone takes twice as long, the time for C alone = 2(1/2) = 1 hour.

Let the job = 39 units.

Rate for A = w/t = 39/(13/2) = 6 units per hour.
Rate for B = w/t = 39/(3/2) = 26 units per hour.
Rate for C = w/t = 39/1 = 39 units per hour.
Combined rate for A+B+C = 6+26+39 = 71 units per hour.

Work produced by A+B+C in 1/2 hour = r*t = 71(1/2) = 35.5 units.
Doesn't work:
According to this answer choice, A+B+C should complete the entire job (39 units) in 1/2 hour.
Eliminate B.

Answer choice B: 40 minutes
Since A alone takes 6 hours longer, the time for A alone = 2/3 + 6 = 20/3 hours.
Since B alone takes 1 hour longer, the time for B alone = 2/3 + 1 = 5/3 hours.
Since C alone takes twice as long, the time for C alone = 2(2/3) = 4/3 hours.

Let the job = 20 units.

Rate for A = 20/(20/3) = 3 units per hour.
Rate for B = 20/(5/3) = 12 units per hour.
Rate for C = 20/(4/3) = 15 units per hour.
Combined rate for A+B+C = 3+12+15 = 30 units per hour.

Work produced by A+B+C in 2/3 hours = r*t = 30(2/3) = 20 units.
Success!
As this answer choice indicates, A+B+C complete the entire job (20 units) in 2/3 hours.

The correct answer is C.
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by Tommy Wallach » Fri Jan 18, 2013 12:15 am
Hey All,

We can also solve this with RTD algebra, though it's a bear! The actual work value doesn't matter, so we can plug in anything:

Rate * Time = Work
A a 60
B b 60
C c 60
All Three d 60

Based on the given info: a = d + 6, b = d + 1, c = 2d

Rate * Time = Work
A d + 6 60
B d + 1 60
C 2d 60
All Three d 60

Finally, we can solve for the rates, because rates are additive, which means we'll be left with one equation/one variable:

Rate * Time = Work
A 60/(d + 6) d + 6 60
B 60/(d + 1) d + 1 60
C 60/2d 2d 60
All Three 60/d d 60

60/(d+6) + 60/(d+1) + 60/2d = 60/d

Let's multiply through by d:

60d/d+6 + 60d/d+1 + 30 = 60

Let's subtract 30:

60d/d+6 + 60d/d+1 = 30

Now let's multiply both sides by (d+1)(d+6) to remove denominators:

60d(d+1) + 60d(d+6) = 30(d+1)(d+6)

Let's finally divide both sides by 30:

2d(d+1) + 2d(d+6) = (d+1)(d+6)

Finally, we can multiply out and put this into quadratic form:

2d^2 + 2d + 2d^2 + 12d = d^2 + 7d + 6
3d^2 + 7d - 6 = 0
(3d + 2)(d - 3) = 0

d = -2/3 or 3

2/3 of an hour = 40 minutes

Woo! Guru's plugging in answer choices method may be a bit cleaner, but for those algebra-lovers out there, you've always got this option!

-t
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by GMATGuruNY » Fri Jan 18, 2013 2:24 am
Three members of a family A,B and C work together to all household chores done. The time it takes them to do the work together is 6 hour less than A would have taken working alone, one hour less than B would have taken alone and half the time C would have taken working alone. How long did it take them to do these chores working together?
a. 20 min
b. 30 min
c. 40 min
d. 50 min
Some test-takers might find the algebra a bit easier with a numerator of 1.

Let the job = 1.
Let t = the time when A, B, and C work together.
Combined rate for A, B and C working together = w/t = 1/t.

Since A alone takes 6 hours longer, the time for A alone = t+6.
Thus:
A's rate alone = 1/(t+6).

Since B alone takes 1 hour longer, the time for B alone = t+1.
Thus:
B's rate alone = 1/(t+1).

Since C alone takes twice as long, the time for C alone = 2t.
Thus:
C's rate alone = 1/(2t).

Since (A's rate alone) + (B's rate alone) + (C's rate alone) = Combined rate for A, B and C, we get:
1/(t+6) + 1/(t+1) + 1/(2t) = 1/t

(t+1) + (t+6) / (t+6)(t+1) = 1/t - 1/(2t)

(2t + 7) / (t² + 7t + 6) = 2/(2t) - 1/(2t)

(2t + 7) / (t² + 7t + 6) = 1/(2t).

Cross-multiplying, we get:

4t² + 14t = t² + 7t + 6

3t² + 7t - 6 = 0

(3t-2)(t+3) = 0

t = 2/3 or t = -3.

Since the value of t must be positive, t = 2/3 of an hour = 40 minutes.

The correct answer is C.
Last edited by GMATGuruNY on Fri Jan 18, 2013 8:01 am, edited 3 times in total.
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by Tommy Wallach » Fri Jan 18, 2013 2:55 am
That's the explanation I already gave. It's certainly cleaner plugging in 1 for the work than 60, but I always like my students to realize that it doesn't actually matter what you plug in, because it simplifies away anyway.

T
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