A and B together can do a piece of work in 12 days, which B and C tog

Question

A and B together can do a piece of work in 12 days, which B and C together can do in 16 days. After A has been working at it for 5 days and B for 7 days, C finishes in 13 days. In how many days C alone will do the work?

(A) 16
(B) 24
(C) 36
(D) 48
(E) 50

(A) 16 
(B) 24
(C) 36 
(D) 48
(E) 50

sujoykrdatta · Accepted Answer

Let us look at a method of solving this question without making equations using variables

A and B together can do a work in 12 days 
B and C together can do it in 16 days
Let us assume the work to be a multiple of 12 and 16 = 48 units (to make our calculations easier)

Thus,  work done by A and B together in 1 day = 48/12 = 4 units  ... (i)
 Work done by B and C together in 1 day = 48/16 = 3 units  ... (ii)

We know that the work was completed in the following fashion: 
A for 5 days, B for 7 days and C for 13 days
We can rearrange and regroup the work without changing the work done. We rearrange in such a way that we can group A and B together and then B and C together:

Total work 
= Work done by A (5 days) + B (7 days) + C (13 days)
= Work done by A (5 days) + B (5 days) + B (2 days) + C (13 days)
= Work done by   +   + C (11 days)
= Work done by A and B together (5 days) + B and C together (2 days) + C (11 days)

Using (i) and (ii) and equating the total work, we have: 
48 = (4 x 5) + (3 x 2) + C (11 days)
=> C (11 days) = 48 - 20 - 6 = 22
=> Work done by C in 1 day = 22/11 = 2

=> Time taken by C to complete the work alone = 48/2 = 24 days

Answer B

Nidzo · Answer

Make all of the joint rates be under 48 (LCM of 12 and 16)

1/a + 1/b = 1/12 -> 1/a + 1/b = 4/48  (1)

1/b + 1/c = 1/16 -> 1/b + 1/c = 3/48  (2)

5/a + 7/b + 13/c = 48/48  (3)

Multiply  (1)  by 5 to get 5/a + 5/b = 20/48  (4)

Multiply  (2)  by 2 to get 2/b + 2/c = 6/48  (5)

Add  (4)  and  (5)  to get 5/a + 7/b + 2/c = 26/48  (5)

Subtract  (5)  from  (3)  to get: 
11/c = 22/48
11/c = 11/24
c = 24

Answer B

Jihyo · Answer

I don't understand the question... I get that first 5 days A and B work together,then B works for two days (So total 7 days)
But when do C work? After 7 days have finished or their work times overlap ?

Dheeraj_079 · Answer

Let the time taken for a alone be a days, b alone be b days and c alone be c days.
From the question, 
1/a + 1/b = 1/12
1/b + 1/c = 1/16 and
5/a +7/b+13/c = 1
Solving the equations we get the value c = 24
So the option B is the answer

BrushMyQuant · Answer

Let's assume following variables/values 
 
 Rate of A = A 
 Rate of B = B 
 Rate of C = C 
 Work Done = 1 (As Work done by everyone is same)  
Formula to be used  Rate * Time = Work

A and B together can do a piece of work in 12 days

=> Combined rate of A and B = A + B
=> Rate * Time = Work
=> (A + B) * 12 = 1
=> A + B = 1/12 ...(1)

B and C together can do in 16 days

=> Combined rate of B and C = B + C
=> Rate * Time = Work
=> (B + C) * 16 = 1
=> B + C = 1/16 ...(2)

After A has been working at it for 5 days and B for 7 days, C finishes in 13 days

Work done by A in 5 days = A * 5 = 5A
Work done by B in 7 days = B * 7 = 7B
Work done by C in 13 days = C * 13 = 13 C

Total Work done = 5A + 7B + 13C = Total Work = 1 
=> 5A + 5B + 2B + 2C + 11C = 1
=> 5*(A + B) + 2*(B + C) + 11C = 1
=> 5*(1/12) + 2*(1/16) + 11C = 1
=> 5/12 + 1/8 + 11C = 1
=> 11C = 1 - 5/12 - 1/8
=> 11C = 24/24 - 10/24 - 3/24 = 11/24
=> C = 1/24

In how many days C alone will do the work?

=> Rate * Time = Work
=> 1/24 * Time = 1
=> Time = 24 days

So,  Answer will be B 
Hope it helps!

Watch following video to MASTER Work Rate Problems

https://www.youtube.com/watch?v=dRxzi_x3ZwY

Gambrosio · Answer

You can use the combined times directly. A works 5 and B 7 days - take it as A+B for 5 days: 5/12 done. Rest is 2 days for B and 13 days for C - this equals 2 days for B and C: additional 2/16 done. Rest is C alone for 11 days.

Now take 5/12 + 2/16 = 26/48. So: C alone does 22/48 in 11 days, or 2/48 = 1/24 a day.

Therefore, C alone takes 24 days. Answer B

A and B together can do a piece of work in 12 days, which B and C tog

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

A and B together can do a piece of work in 12 days, which B and C tog

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards