2 men and 7 boys can do a piece of work in 4 days. The same work is done in 3 days by 4 men and 4 boys. How long respectively would it take for one man and one boy to do it?

# 2 men and 7 boys can do a piece of work in 4 days. The same work is done in 3 days by 4 men and 4 boys. How long respectively would it take for one man and one boy to do it?

1. A
15 days, 60 days
2. B
20 days, 40 days
3. C
10 days, 30 days
4. D
90 days, 40 days

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### Solution:

Given that 2 men and 7 boys can do a piece of work in 4 days and the same work is done by 4 men and 4 boys in 3 days.
Let's suppose that one man alone can finish the work in x days and one boy alone finish it in y days.
Thus, one man's one day's work is $\frac{1}{x}.$
And one boy's one day's work is $\frac{1}{y}.$
Also, 2 men’s one day's work is $\frac{2}{x}$ and 7 boy’s one day's work is $\frac{7}{y}$.
According to the question,
$\frac{2}{x}+\frac{7}{y}=\frac{1}{4}$

Again, from the question, writing the algebraic representation of the second condition,
$\frac{4}{x}+\frac{4}{y}=\frac{1}{3}$

Substituting $\frac{1}{x}$ for u and for v in equation (1) and (2),
$8u+28v-1=0$
$12u+12v-1=0$
The solution of the equations by cross multiplication method is given by the formula,
$\frac{u}{{b}_{1}{c}_{2}-{b}_{2}{c}_{1}}=\frac{v}{{a}_{2}{c}_{1}-{a}_{1}{c}_{2}}=\frac{1}{{a}_{1}{b}_{2}-{a}_{2}{b}_{1}}$
Here,
Using the cross-multiplication method to find the value of u and v,
$⇒\frac{u}{28\left(-1\right)-12\left(-1\right)}=\frac{v}{12\left(-1\right)-8\left(-1\right)}=\frac{1}{8\left(12\right)-28\left(12\right)}$
$⇒\frac{u}{-28+12}=\frac{v}{-12+8}=\frac{1}{96-326}$
$⇒\frac{u}{-16}=\frac{v}{-4}=\frac{1}{-240}$
$⇒u=\frac{-16}{-240}=\frac{1}{15}$
$⇒v=\frac{-4}{-240}=\frac{1}{60}$
Again, substituting $\frac{1}{x}$ for u and $\frac{1}{y}$ for v, to find the value of x and y.
$⇒\frac{1}{15}=\frac{1}{x}$

$⇒\frac{1}{60}=\frac{1}{y}$
$⇒y=60$
Therefore, we can say that, 1 man takes 15 days to complete the work and 1 boy takes 60 days to complete the same work.
Hence, option (1) is correct.

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