Discussion Forum

In how many days can the work be completed by $A$ and $B$ together ?

I. $A$ alone can complete the work in $8$ days.

II. If $A$ alone works for $5$ days and $B$ alone works for $6$ days, the work get completed.

III. $B$ alone can complete the work in $16$ days.

Answer:

Explanation:

I. $A$ can complete the job in $8$ days. So, $A$'s $1$ day's work $=\frac{1}{8}$.

II. $A$ works for $5$ days. $B$ works for $6$ days and the work is completed.

III. $B$ can complete the job in $16$ days. So, $B$'s $1$ day work $=\frac{1}{16}$.

I and III : $(A+B)$'s $1$ day's work $=\left(\frac{1}{8}+\frac{1}{16}\right)$ $=\frac{3}{16}$.

$\therefore$ Both can finish the work in $\frac{16}{3}$ days.

II and III : Suppose $A$ takes $x$ days to finish the work.

Then, $\frac{5}{x}+\frac{6}{x}=1$ $\Rightarrow$ $\frac{5}{x}$ $=\left(1-\frac{3}{8}\right)$ $=\frac{5}{8}$ $\Rightarrow x=8$.

$\therefore$ $(A+B)$'s $1$ day's work $=\left(\frac{1}{8}+\frac{1}{16}\right)$ $=\frac{3}{16}$.

$\therefore$ Both can finish it in $\frac{16}{3}$ days.

I and II : $A$'s $1$ day's work $=\frac{1}{8}$. Suppose $B$ takes $x$ days to finish the work.

Then, from II, $\left(5\times \frac{1}{8}+6\times\frac{1}{x}-1\right)$ $\Rightarrow\frac{6}{x}$ $=\left(1-\frac{5}{8}\right)$ $=\frac{3}{8}$

$\Rightarrow x=\left(\frac{8\times 6}{3}\right)$ $=16$.

$\therefore$ $(A+B)$'s $1$ day work $=\left(\frac{1}{8}+\frac{1}{18}\right)$ $=\frac{3}{16}$.

$\therefore$ Both can finish it in $\frac{16}{3}$ days.

Hence, the correct answer is (C).