Question

A and B can do a job together in 7 days. A is $1\dfrac{3}{4}$ times as efficient as B. The same job can be done by A alone in:
A.$9\dfrac{1}{3}$ days
B.11 days
C.$12\dfrac{1}{4}$ days
D.$16\dfrac{1}{3}$ days

Answer 1

Hint: In this question, we need to determine the number of days in which A can complete the work alone. For this, we will use the unitary method for calculating the one-day work of A and B and then, substituting those values in work done in a day by A and B together is the summation of the work done by A alone and B alone.

Complete step-by-step answer:
Let A alone can finish the job in A days and B alone can do the same job in B days.
According to the question, A is $\left( {1\dfrac{3}{4} = \dfrac{7}{4}} \right)$ times as efficient as B, which implies that A can complete the job in less time as compared with B or B takes more time than A to complete the wok.
So, the time taken by B to complete the work alone is given by $B = \dfrac{{7A}}{4}$
Also,
A’s one day work is given as ${A_{one{\text{ day}}}} = \dfrac{1}{A}$
B’s one day work is given as ${B_{one{\text{ day}}}} = \dfrac{1}{B} = \dfrac{4}{{7A}}$
Now, A and B can complete the work working together for 7 days.
So, one day work of A and B working together is given as ${\left( {A + B} \right)_{one{\text{ day}}}} = \dfrac{1}{7}$
As the work is done in a day by A and B is the summation of the work done by A alone and B alone. So, ${\left( {A + B} \right)_{one{\text{ day}}}} = {A_{one{\text{ day}}}} + {B_{one{\text{ day}}}}$.
Now, substituting ${A_{one{\text{ day}}}} = \dfrac{1}{A}$, ${B_{one{\text{ day}}}} = \dfrac{1}{B} = \dfrac{4}{{7A}}$ and ${\left( {A + B} \right)_{one{\text{ day}}}} = \dfrac{1}{7}$ in the equation ${\left( {A + B} \right)_{one{\text{ day}}}} = {A_{one{\text{ day}}}} + {B_{one{\text{ day}}}}$ to determine the total number of days required by A to complete the work alone.
$
  {\left( {A + B} \right)_{one{\text{ day}}}} = {A_{one{\text{ day}}}} + {B_{one{\text{ day}}}} \\
  \dfrac{1}{7} = \dfrac{1}{A} + \dfrac{4}{{7A}} \\
  \dfrac{1}{7} = \dfrac{{7 + 4}}{{7A}} \\
  \dfrac{1}{7} = \dfrac{{11}}{{7A}} \\
  A = 11{\text{ days}} \\
 $
Hence, A can complete the work while working alone in 11 days and is $1\dfrac{3}{4}$ times as efficient as B.
So, the correct answer is “Option B”.

Note: Students must be careful while dealing with the efficiencies of the persons. If the person is more efficient then, he/she can do the work in less time than the person with lesser efficiency. Moreover, here we have converted the mixed fraction in a proper fraction by following the rule $a\dfrac{b}{c} = \dfrac{{ac + b}}{c}$.