Question

A man and a boy can do a piece of work in 24 days. If the man works alone for the last 4 days, it is complete in 5 days. How long would the boy take to do it alone?

Answer 1

Hint: Given that the man and a boy can do a piece of work in 24 days.
First, we take the total piece of work as 1unit.
Then by the given information, the man and the boy can do 1 unit in 24 days, so we can find the unit of work done by them in 1 day. Similarly, we can calculate the unit of work done by them in $24-4=20$ days.
Then we can find the remaining work which is done by the man alone in 5 days.
By subtracting the 1-day work of the man from the work of the man and the boy, we get the boy’s 1-day work.
Then we can find easily how many days the boy needs to do the 1 unit of work.

Complete step by step solution:
It is given that, a man and a boy can do a piece of work in 24 days.
If we take the whole work as 1 unit then,
Total work done by the man and the boy in 1 day =\[\dfrac{1}{{24}}\] unit of work
Total work done by the man and the boy in 20 day =\[\dfrac{1}{{24}} \times 20\] units of work
On solving we get,
Work done by the man and boy in 20 days is \[ = \dfrac{5}{6}unit\]of work
Let us find the remaining work,
Remaining work is = \[(1 - \dfrac{5}{6})unit = \dfrac{{6 - 5}}{6}unit = \dfrac{1}{6}unit\]
Remaining \[\dfrac{1}{6}\] of the work is done by the man alone in 5 days,
That is the work done by man alone in 1 day \[\dfrac{1}{6} \times \dfrac{1}{5} = \dfrac{1}{{30}}unit\] of work
∴ The Boy’s one day's work = Total work done by the man and the boy in 1 day - The work done by the man alone in 1 day
The work done by the boy in one day \[ = \dfrac{1}{{24}} - \dfrac{1}{{30}}\]
On solving we get,
\[ = \dfrac{{5 - 4}}{{120}}\]
Hence the work by the boy in one day \[ = \dfrac{1}{{120}}unit\]

$\therefore$ Boy alone would take 120 days to complete the work.

Note:
We have used, total piece of work as 1unit. Here the denominator term is taken as the number of days required to complete the work. That is $\dfrac{1}{a}$ denotes that 1 unit of work is completed in “$a$” days. We here use the logic that if the number of men working increases then the number of the working days decreases. Hence the boy completes the work later when compared to the man.