Question

4 men and 6 boys can finish a piece of work in 5 days while 3 men and 4 boys can finish it in 7 days. Find the time taken by 1 man alone or then by 1 boy alone.

Answer 1

Hint- In order to solve this question we will assume the number of days taken by 1 man and 1 boy as a variable then by given condition we will make 2 different equations with 2 variables, by solving this equation we will get the answer.

Complete step-by-step answer:

Number of day taken by 1 men = x days
And of boy = y days
Work done by 1 men in 1 day = $\dfrac{1}{x}$
Work done 1 boy in 1 day = \[\dfrac{1}{y}\]
According to the first condition
$\dfrac{4}{x} + \dfrac{6}{y} = \dfrac{1}{5}.....................(1)$
According to the second condition
$\dfrac{3}{x} + \dfrac{4}{y} = \dfrac{1}{7}...............(2)$
Let $\dfrac{1}{x} = v{\text{ and }}\dfrac{1}{y} = u$
On multiplying equation (1) by 3 and equation (2) by 4, we get
$
  12v + 18u = \dfrac{3}{5}..............(3) \\
  12v + 16u = \dfrac{4}{7}..............(4) \\
$
Subtracting equation (3) from (4)
$
   \Rightarrow 12v - 12v + 18u - 16u = \dfrac{3}{5} - \dfrac{4}{7} \\
   \Rightarrow 2u = \dfrac{1}{{35}} \\
   \Rightarrow u = \dfrac{1}{{70}} \\
  \therefore y = 70 \\
$
Putting value of y in equation (1)
$
   \Rightarrow \dfrac{4}{x} + \dfrac{6}{{70}} = \dfrac{1}{5} \\
   \Rightarrow \dfrac{1}{x} = \dfrac{1}{4} \times \left( {\dfrac{1}{5} - \dfrac{6}{{70}}} \right) \\
   \Rightarrow x = 35 \\
 $
Hence, one man alone can finish work in 35 days and one boy in 70 days.

Note- In order to solve problems related to work and time; learn the concept of work and time i.e. more men can do more work in less time. If M man can do a piece or work in T hours then total effort or work = MT man hours. Time is directly proportional to total work meaning more time more work while time is inversely proportional to number of men. If men increase then time decreases and vice versa.