Question

A is thrice as efficient as B and B is twice as efficient as C. If A, B and C work together, how long will they take to complete a job which B completes in 10 days?
(a) \[\dfrac{20}{9}days\]
(b) \[\dfrac{11}{9}days\]
(c) 3 days
(d) None of these

Answer 1

Hint:First of all, assume the efficiency of ‘C’ as ‘w’ work/day and write the efficiency of A and B in terms of that. Now assume that ‘A’, ‘B’ and ‘C’ work together for n days and equate their total work in n days for the work done by ‘B’ in 10 days. Use work done = (efficiency) \[\times \] (time) to calculate work done by each one of them.

Complete step-by-step answer:
In this question, we are given that A is thrice as efficient as B and B is twice as efficient as C. If A, B, and C work together, we have to find the time taken by them to complete a job which B completes in 10 days.
First of all, we must know what efficiency is. Efficiency is the amount of work done by a person/thing at a particular time. In other words, we can say that efficiency is the ability to do work, successfully, and without waste.
We are given that A, B, and C are doing some work together. So, let us assume that the efficiency of C is ‘w’ work/day. As we are given that B is twice as efficient as C. So, we get,
The efficiency of B = 2 (efficiency of C)
= 2 (W) work / day
= 2W work / day
As we are given that A is thrice as efficient as B. So, we get,
The efficiency of A = 3 (efficiency of B)
= 3 (2W) work/day
= 6W work/day
Now, let us assume that A, B, and C work together for ‘n’ days. We have found that the efficiency of A, B, and C are 6W work/ day, 2W work/day, and W work/day. So, we get,
Work done by A in n days = (4W)n = 6Wn work
Work done by B in n days = (2W)n = 2Wn work
Work done by C in n days = (W)n = Wn work
So, we get the work done by A, B, and C together in n days,
\[{{W}_{o}}=6Wn+2Wn+Wn\]
So, we get,
\[{{W}_{o}}=9Wn....\left( i \right)\]
Now, we have found that the efficiency of B is 2W work/day. So we get,
Work done by B in 10 days = (2W) 10 = 20W …..(ii)
Now, we are given that work done by A, B, and C together is equal to the work done by B in 10 days. So, now we will equate equation (i) and (ii), we get,
9Wn = 20W
By dividing 9W on both the sides of the above equation, we get,
\[n=\dfrac{20W}{9W}\]
\[n=\dfrac{20}{9}days\]
Hence, A, B, and C will take \[\dfrac{20}{9}days\] to complete a job which B completes in 10 days.
Hence, option (a) is the right answer.

Note: In this question, it is better to assume C’s efficiency and calculate efficiency of A and B in terms of C’s efficiency because if we assume A’s and B’s efficiency then another efficiency would come in a fraction which could create confusion while solving the problem. Also, take great care while writing the equation according to the information given in the question. Also remember the formula (efficiency) (time) = (Work done).