Question

A is twice as good a workman as B and together they finish a price of work in $14$ days. In how many days can A alone finish the work?
(A) $13$
(B) $15$
(C) $17$
(D) $21$

Answer 1

Hint: Here we have to assume the work done by B in $1$ day be any variable. Then find the work done by A in $1$ day using the given condition in terms of variable.
Find the work done by A and B together in $1$ day. Then equate the sum of work done by A in $1$ day and work done by B in $1$ day to the work done by A and B together in $1$ day.
 Find the assumed variable. Finally we get the required answer.

Complete step-by-step solution:
It is given that A is twice as good a workman as B and together they finish a price of work in $14$ days.
We have to find out the time taken by A to finish the work.
Assume the work done by B in $1$ day be any variable.
Let the work done by B in $1$ day be $x$.
Now we have to find the work done by A in $1$ day using the given condition in terms of variable.
As A is twice as good a workman as B.
So, we can write it as mathematically,
Work done by A in $1$ day is$2x$.
Also, we have to find the work done by A and B together in $1$ day.
Since, it is given that A and B together finishes a price of work in $14$ days
So, work done by A and B together in $14$ days =$1$.
Work done by A and B together in $1$ day$ = \dfrac{1}{{14}}$
Now we can find the sum of work done by A in $1$ day and work done by B in $1$ day.
Since, work done by B in $1$ day = $x$ and work done by A in $1$ day = $2x$.
So, work done by A in $1$ day + work done by B in $1$ day
\[ \Rightarrow 2x + x\]
Let us the add the terms and we get,
$ \Rightarrow 3x$
As Work done by A and B together in $1$ day $ = \dfrac{1}{{14}}$.
So we can write it as,
$3x = \dfrac{1}{{14}}$
Divide both sides of the equation by $3$.
$x = \dfrac{1}{{42}}$
So, work done by B in $1$ day$ = \dfrac{1}{{42}}$
Find the work done by A in $1$ day.
Since, work done by A in $1$ day = $2x$ and $x = \dfrac{1}{{42}}$.
So, work done by A in $1$ day$ = 2 \times \dfrac{1}{{42}} = \dfrac{1}{{21}}$.
Thus, A alone can finish the work in \[21\] days.
Hence, (D) is the correct option.

Note: If work remains constant, Men and Days are inversely proportional, i.e., if the number of men decreases, the number of days required to complete the same work increases in inverse proportion and vice-versa.