Question

A takes twice as much time as B or thrice as much time as C to finish a piece of work. Working together, they can finish the work in 2 days. B can do the work alone in
(a) 3 days
(b) 7 days
(c) 4 days
(d) 6 days

Answer 1

Hint: Find the units of work done by each of them per day then we can find number of units of total work. Do this by using per day fraction of work.
\[Per\text{ }day\text{ }fraction\text{ }=\text{ }\dfrac{1}{Time\text{ }taken}\]

Complete step-by-step answer:

They will work together for 2 days and finish the work.
Case 1: Let the time taken by A to finish a piece of work be a. Then the per day fraction of work done by A will be $\left( \dfrac{1}{a} \right)$
Case 2: Let the time taken by B to finish a piece of work be b. Then the per day fraction of work done by B will be $\left( \dfrac{1}{b} \right)$
Case 3: Let the time taken by C to finish a piece of work be c. Then the per day fraction of work done by C will be $\left( \dfrac{1}{c} \right)$
So,
Total per day fraction of work will be sum of all cases
Total per day fraction of work = (Case 1) + (Case 2) + (Case 3)
Total per day fraction of work = $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}$
We know:
(Total time).(Total per day fraction of work) = 1
Here 1 implies that work is completed.
So, by substituting 2 days into equation, we get:
$2\left( \dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c} \right)=1.........(i)$
Given:
A take twice as much time as B.
A take thrice as much time as C.
By using above conditions, we get:
a = 2b and a = 3c
By substituting these into equation(i), we get
$2\left( \dfrac{1}{a}+\dfrac{1}{\dfrac{a}{2}}+\dfrac{1}{\dfrac{a}{3}} \right)=1$
By solving this, we get:
$2\left( \dfrac{1}{a}+\dfrac{2}{a}+\dfrac{3}{a} \right)=1$
By taking least common multiple and then adding the fractions, we get:
$2\left( \dfrac{6}{a} \right)=1$
By multiplying both sides with a, we get:
$\dfrac{12}{a}.a=a$
a = 12
By using given conditions, we found relation between a and b:
a = 2b
By substituting value of a, we get:
12 = 2b
By dividing 2 on both sides, we get:
$b=\dfrac{12}{2}=6$
Therefore, it takes 6 days for B to complete the work alone.
Option (d) is correct.

Note: Do not confuse time taken and per day fraction of work. The simple relation is stated in the hint.