Question

A man, a woman and a boy can complete a job in 3, 4 and 12 days respectively. How many boys must assist one man and one woman to complete the job $\dfrac{1}{4}\text{th}$ of a day?
a) 1
b) 4
c) 19
d) 41

Answer 1

Hint: In this question, we are given the number of days taken by a man, a woman and a boy to complete the work individually. Therefore, we should try to find out the fraction of the total work that can be completed by an individual man, woman and boy in $\dfrac{1}{4}\text{th}$ of a day. Then, as we are given that only one man and one woman should work, we can take the number of boys required to be x and then find x such that the total fraction of the work completed by all of them is 1.

Complete step by step solution:
We are given that
No. of days taken by a man to complete the work $=3$
Therefore, using unitary method approach, we can write that
$\Rightarrow $ Fraction of the work completed by a man in one day $=\dfrac{1}{3}$
$\Rightarrow $ Fraction of the work completed by a man in $\dfrac{1}{4}\text{th}$ of a day$=\dfrac{1}{3}\times \dfrac{1}{4}=\dfrac{1}{12}.................................(1.1)$
Also, we have been given that
No. of days taken by a woman to complete the work $=4$
Therefore, using unitary method approach, we can write that
$\Rightarrow $ Fraction of the work completed by a woman in one day $=\dfrac{1}{4}$
$\Rightarrow $ Fraction of the work completed by a woman in $\dfrac{1}{4}\text{th}$of a day$=\dfrac{1}{4}\times \dfrac{1}{4}=\dfrac{1}{16}.................................(1.2)$
No. of days taken by a boy to complete the work $=12$
Therefore, using unitary method approach, we can write that
$\Rightarrow $ Fraction of the work completed by a boy in one day$=\dfrac{1}{12}$
$\Rightarrow $ Fraction of the work completed by a boy in $\dfrac{1}{4}\text{th}$of a day$=\dfrac{1}{12}\times \dfrac{1}{4}=\dfrac{1}{48}.................................(1.3)$
Let there be x boys required with one man and one woman to complete the task. As the work should be completed, the total fraction of the work completed should be equal to one.
Therefore, using (1.1), (1.2) and (1.3), in $\dfrac{1}{4}\text{th}$ of a day, we get
Fraction of work completed by one man+ Fraction of work completed by one woman + Fraction of work completed by x boys $=1\times \dfrac{1}{12}+1\times \dfrac{1}{16}+x\times \dfrac{1}{48}=1$
$\Rightarrow x\times \dfrac{1}{48}=1-\dfrac{1}{12}-\dfrac{1}{16}=\dfrac{12\times 16-16-12}{12\times 16}=\dfrac{164}{192}$ …………………….(1.4)
Where in the last step we have multiplied the denominators to make the denominators of all the fractions the same and have multiplied appropriate factors in the numerator to keep the values of the individual fractions constant.
Therefore, from (1.4), we should have
$x=48\times \dfrac{164}{192}=41$
Thus the answer should be 41 which matches option (d) in the question.

Note: We should note that we could also have found the fraction of the total work done by a man, woman and a boy in an hour and then found out the fraction of the work done by them in 6 hours which is $\dfrac{1}{4}\text{th}$ of a day and then followed the other steps to obtain the answer. However, the answer would still have remained the same.