Question

A 10 hectare field is reaped by 2 men, 3 women and 4 children in 10 days. If a man, a woman and a child work in the ratio 5:4:2 a 16 hectare field will be reaped by 6 men, 4 women and 7 children is
A. 5 days
B. $6\dfrac{1}{2}$ days
C. 7 days
D. 8 days

Answer 1

Hint: In this question, we will consider a variable $x$ such that $5x,4x$ and $2x$ will be the working efficiency of a man, a woman and a child. Then, we will calculate the value of $x$ by the given conditions. And by using that we will be able to calculate the number of days required to reap 16 hectares.

Complete step-by-step answer:

In this question, we are asked to find the number of days which are required to reap a field of 16 hectares. So, let us consider a variable $x$, such that $5x,4x$ and $2x$ will be the working efficiency of a man, a woman and a child. Now, we know that it took 10 days to reap a field of 10 hectares by 2 men, 3 women and 4 children. So, we can write it as,
Area of the field reaped = (working efficiency) $\times$ (Number of days)
So, we get,
$\begin{aligned} & 10=\left( 2\times 5\times x+3\times 4\times +4\times 2\times x \right)\times 10 \\\ & 10=\left( 10x+12x+8x \right)\times 10 \\\ & 10=\left( 30x \right)\times 10 \\\ & 10=300x \\\ & x=\dfrac{10}{300} \\\ & \Rightarrow x=\dfrac{1}{30}.........(i) \\\ \end{aligned}$
Now, we will use the same formula to reap the field of 16 hectares by 6 men, 4 women and 7 children. Let it take D days to reap the field. So, we get,
$\begin{aligned} & 16=\left( 6\times 5\times x+4\times 4\times x+7\times 2\times x \right)\times D \\\ & 16=\left( 30x+16x+14x \right)\times D \\\ & 16=\left( 60x \right)\times D \\\ & D=\dfrac{16}{60x} \\\ \end{aligned}$
Now, we will put $x=\dfrac{1}{30}$ from equation (i) in the above equation. So, we will get,
$\begin{aligned} & D=\dfrac{16}{60\left( \dfrac{1}{30} \right)} \\\ & D=\dfrac{16\times 30}{60} \\\ & D=\dfrac{16}{2} \\\ & D=8 \\\ \end{aligned}$
Hence, it will take 8 days to reap a field of 16 hectares by 6 men, 4 women and 7 children.
Therefore option (D) is the correct answer.

Note: We can also solve this question by directly equating the value of $x$ in both the cases, that is,
$\dfrac{\left( \text{working capability} \right)\text{ }\\!\\!\times\\!\\!\text{ number of days}}{\text{area of the field}}$. So we can write it as $\dfrac{\left( 2\times 5+3\times 4+4\times 2 \right)\times 10}{10}=\dfrac{\left( 6\times 5+4\times 4+7\times 2 \right)\times D}{16}$
And after solving this we will get the value of D, that is, the number of days required to reap a field of 16 hectares with the given conditions.