Question

X and Y can finish a piece of work in \[30\] days. Six days they work together and then X quits the job. 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

Hint: Let's calculate the number of days $Y$ take to complete the piece of work. By using the unitary method we can calculate the work done by $X$ and $Y$ in a day.The unitary method is to find the value of a single unit and then multiply the value of a single unit to the number of units to get the necessary value. We need to calculate the total work done, we will be taking the LCM of the individual time taken. If one person does a piece of work in $X$ days and another person does it in $Y$ days. Then together they can finish that work in \[\dfrac{{XY}}{{(X + Y)}}\] days.

Complete step by step answer:
First we will find the LCM of\[30{\text{ }}and{\text{ }}32\].
So, the prime fraction of \[30 = 2 \times 3 \times 5\]
The prime fraction of \[32 = 2 \times 2 \times 2 \times 2\]
Thus, the LCM\[(30,32)\] is
\[2 \times 2 \times 2 \times 2 \times 2 \times 3 \times 5=480 \]
Let, the total unit of work be \[480\] units. Given that, X and Y complete the work in\[30\] days. Units of work done per day by X and Y together is (X + Y)
\[\dfrac{{480}}{{30}}\]
\[\Rightarrow 16\] units/day

According to the question, efficiency of (X + Y) working for \[6\] days + efficiency of Y working for \[32\] days completes\[480\]units of work
\[\Rightarrow 16 \times 6 + Y \times 32 = 480 \\
\Rightarrow 96 + 32Y = 480 \\ \]
By using transposition, we get,
\[\Rightarrow 32Y = 480 - 96 \\
\Rightarrow 32Y = 384 \\ \]
\[\Rightarrow Y = \dfrac{{384}}{{32}} \\
\Rightarrow Y = 12 \\ \]
Thus, Y can complete\[12\] units/day. Next, we will calculate the days taken by the Y to finish the work. Time taken by Y to complete the whole work,
\[\dfrac{\text{Total work}}{{12}}\]
\[\Rightarrow \dfrac{{480}}{{12}}\]
\[\Rightarrow 40\] days

Hence, the total number of days taken by Y to complete the piece of work alone is \[40\] days.

Note:To calculate time to complete a work in general if a person A completes 1/n th part of work in one day, then the time taken by $A$ to complete the work = n days. If a man takes 10 days to complete a piece of work; then according to the unitary method work is done in a \[1\] day \[ = 1/10\]. On the other hand, if a man completes\[ = 1/10\]th of the work in one day, to complete the work he will take \[10\] days. In case of three persons taking x, y and z days respectively, They can finish the work together in xyz/(xy + yz + xz) days.