A and B together complete a work in 4 days, B and C together in 6 days

Question

A and B together complete a work in 4 days, B and C together in 6 days, C and A together in 5 days. Working independently, who will finish the work in the least time and in how many days?

A. A, 120/17 days
B. A, 120/7 days
C. B, 120/17 days
D. B, 120/13 days
E. C, 120/7 days

KarishmaB · Accepted Answer

A and B together take least time - 4 days. So one of them must be the fastest.
Next C and A take 5 days (less than what B and C take together). So A must be faster than B.

Hence A is the fastest. Choose between options (A) and (B).

A and B together take 4 days and A is faster than B. Then, definitely, A takes less than 8 days because if their speeds were the same, both would take 8 days working independently. 
Only option (A) has less than 8 days so answer (A)

manasi05 · Answer

A and B together complete a work in 4 days
So their combined work rate will be 1/4
A+B= 1/4
B and C together complete a work in 6 days
So their combined work rate will be 1/6
B+C = 1/6
C and A together complete a work in 5 days
So their combined work rate will be 1/5
A+C = 1/5
Find their combined work by adding up the equation above
2(A+B+C) = 74/120
A+B+C = 37/120
Now equate for B and C in terms of A in the above equation
A+ 1/4 - A + 1/5 - A = 37/120
A = 17/120
B = 13/120
C = 7/120

So the number of days for A will be 120/17, B will be 120/13, C will be 120/7
The least value is 120/17
So A will finish the work in the least time in 120/17 days.
IMO answer is A

manasi05 · Answer

We can solve this question by the backsolving method also.
In answer choices, we are given 3 different values 120/17, 120/7, and 120/13. The least value among these is 120/17. So we can get rid of the answer choice E
From our answer choices, 120/17 is the value of either A or B
When the value of Work is one then work rate and time are reciprocal of each other.
From question stem, we also know that 
A and B together complete a work in 4 days
So their combined work rate will be 1/4
Now the reciprocal of answer choices is the individual rate.
So we substitute the values for A and B from the answer choices and whichever option gives us a result of 1/4 will be the correct value. Using that we will know 120/17 is the value of A and 120/13 is the value for B
So the answer will be A

This question can be solved quickly using this method. I didn't realize to use it when I started to solve the question. After I got the answer then I realized I could have solved it this way and would have saved time.

IMO answer will be A

imeanup · Answer

First, take the LCM of A+B, B+C, C+A in order to find the total work done.

Combined to find the efficiency of A+B+C working together.

Sub. the value to find the efficiency of the individual.

Sub. in total work to find the days required.

Please refer to the below image for a better understanding.

Smile38 · Answer

ALTERNATIVE METHOD 
Let total Work done be 120 { which is LCM of 4, 6 and 5 days}

(rate= work/time)

adding all, 2(A+B+C) = 74
 A+B+C = 37

Now, rates of
  A= 37-20 =17 
 B= 37-24 =13
 C= 37-30 =7

As we can see A has highest rate, therefore it will take least time

work/rate = time
  120/17= time
 
ANSWER = A

Aboyhasnoname · Answer

If you are short on time...We can narrow down our options to A and B...Since A is the fastest..Here is how...
A and B together complete a work in 4 days, 
B and C together in 6 days, 
This means C is slower than B..as its slowing down.... 
/
C and A together in 5 days.
This Means C is slower than A as its slowing down..
/
C and A together in 5 days. B and C together in 6 days, 
This means B is also slower than A ..as B is slowing down.....So if we are short on time...We can see...between A and B is our right answer... Increases the probability.....

soniasw16 · Answer

But how do you know 12/17 is A and 120/13 is B? It values could be vice versa based on your method.

soniasw16 · Answer

How did you get "both would take 8 days working independently"? Is there a formula for this or did you just trial and error some values?

A and B together complete a work in 4 days, B and C together in 6 days

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A and B together complete a work in 4 days, B and C together in 6 days

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