Question

$X$ and $Y$ can finish a piece of work in $30$ days. Six days they work together and then $X$ quits the work. $Y$ takes $32$ days to finish the work. Calculate how many days $Y$ take to complete the piece of work?

• A. $32$ days.
• B. $10$ days.
• C. $30$ days.
• D. $40$ days.

Answer 1

The correct option is D

$40$ days.

Explanation for the correct option.

Step-1 : Total work and efficiency:

Find the LCM of $30$ and $32$.

Prime factorization of $30$ and $32$ is

$$\begin{gathered} 30=2\times 3\times 5 \\ 32=2\times 2\times 2\times 2\times 2 \\ \mathrm{LCM}\left(30,32\right)=2\times 2\times 2\times 2\times 2\times 3\times 5 \\ \therefore \mathrm{LCM}\left(30,32\right)=480 \end{gathered}$$

Thus, $480$ is the total work unit.

$X$ and $Y$ together finish the work in $30$ days.

Number of units per day.

$(X+Y)=\frac{480}{30}$

$(X+Y)=16\mathrm{units}/\mathrm{day}$

Step-2 Efficiency of $Y$:

Efficiency of $(X+Y)$working for six days $+$ efficiency of $Y$ working for $32$ days completes $480$ units of work

$$\begin{gathered} (16\times 6)+(Y\times 32)=480 \\ \Rightarrow 96+32Y=480 \\ \Rightarrow 32Y=480-96 \\ \Rightarrow Y=\frac{384}{32} \\ \therefore Y=12 \end{gathered}$$

Time taken by $Y$ to complete the whole work

$$\begin{gathered} =\frac{\mathrm{Total}\mathrm{work}}{12} \\ =\frac{480}{12} \\ =40 \end{gathered}$$

Thus, $40$ days, is the total number of days taken by $Y$ to complete the piece of work alone

Hence, option $\mathrm{D}$ is correct.