Question

Two men and 7 children complete a certain piece of work in 4 days while 4 men and 4 children complete the same work in only 3 days. The number of days required by 1 man to complete the work is
A) 60 days
B) 15 days
C) 6 days
D) 51 days

Answer 1

Hint: We will be making 2 linear equations in 2 variables as we have two different quantities (man and child) according to the given 2 cases. By the solution of these equations, we can find the values of variables which will give us the required answer.
We will consider the whole work as 1 unit or 1 WD.

Complete step-by-step answer:
We will first suppose the work done by a man and a child in one day be x and y respectively.
🡪 Work done by one man in one day = x
🡪 Work done by one child in one day = y
The linear equations according to the statements can be given as:
i) Two men and 7 children can complete the work in 4 days, in 1 day the work completed will be
\[
  4{\text{ }}days = 1{\text{ WD}} \\
  {\text{1 }}day = \dfrac{1}{4}WD \;
 \]
$ \Rightarrow 2x + 7y = \dfrac{1}{4}...(1)$
ii) 4 men and 4 children can complete the same work in 3 days in 1 day the work completed will be
\[
  {\text{3 }}days = 1{\text{ WD}} \\
  {\text{1 }}day = \dfrac{1}{3}{\text{ WD}} \\
 \]
$ \Rightarrow 4x + 4y = \dfrac{1}{3}...(2)$
Solving (1) and (2), we get:
Multiplying (1) by 2 and the subtracting it from (2)
$
   \Rightarrow 4x + 14y - 4x - 4y = \dfrac{1}{2} - \dfrac{1}{3} \\
   \Rightarrow 10y = \dfrac{{3 - 2}}{6} \\
   \Rightarrow 10y = \dfrac{1}{6} \\
   \Rightarrow y = \dfrac{1}{{60}} \;
 $
We are required to find the number of days required by 1 man to complete the work, for that we have to find the value of variable x.
Substituting the value of y in (1) to find the value of x:
$
   \Rightarrow 2x + 7\left( {\dfrac{1}{{60}}} \right) = \dfrac{1}{4} \\
   \Rightarrow 2x = \dfrac{1}{4} - \dfrac{7}{{60}} \\
   \Rightarrow 2x = \dfrac{{15 - 7}}{{60}} \\
   \Rightarrow 2x = \dfrac{8}{{60}} \\
   \Rightarrow 2x = \dfrac{2}{{15}} \\
   \Rightarrow x = \dfrac{1}{{15}} \;
 $
Thus, 1 man completes $\dfrac{1}{{15}}\left( {WD} \right)$ of the work in 1 day. The complete work (1 WD) will be done in:
\[
  \dfrac{1}{{15}}{\text{ WD}} = 1{\text{ }}day \\
  {\text{1 WD}} = 1 \times 15{\text{ }}days \\
  {\text{1 WD}} = 15{\text{ }}days \\
 \]
So, the correct answer is “Option B”.

Note: We call the relationship between the number of men and work done as inverse proportionality because when the number of people are more, the work is done in less number of days and vice – versa. In unitary method, remember that the known value is kept on the right hand side while the unknown i.e. value to be found is kept on the right hand side.