Question

A shoe repairman is working with his assistant, who takes 1.5 times as long to repair a pair of shoes. Together they can fix 20 pairs of shoes in an eight-hour day. How long does it take the repairman to fix one pair of shoes by himself?

Answer 1

Answer: $\dfrac{2}{3}\text{ hour}$

Step-by-step explanation:

Let x be the time taken by the shoe repairman to repair 20 pairs of shoes and 1.5 x be the the time taken by his assistant to repair the shoes.

Since they took 8 hours to complete the whole job , so we have the following equation:-

$\dfrac{1}{\text{Time taken by shoes repairman}}+\dfrac{1}{\text{Time taken by assistant}}=\dfrac{1}{8}\\\\ \dfrac{1}{x}+\dfrac{1}{1.5x}=\dfrac{1}{8}\\\\ \dfrac{1+1.5}{1.5x}=\dfrac{1}{8}\\\\ x=\dfrac{8\times2.5}{1.5}=\dfrac{40}{3}$

i.e. Time taken by shoe repairman to repair 20 pairs of shoes  = $\dfrac{40}{3}\text{ hours}$

By unitary method ,

Then, Time taken by shoe repairman to repair 1 pair of shoes  = $\dfrac{40}{3}\div 20\text{ hours}$

$=\dfrac{40}{3}\times\dfrac{1}{20}=\dfrac{2}{3}\text{ hour}$

Hence, Time taken by shoe repairman to repair 1 pair of shoes =$\dfrac{2}{3}\text{ hour}$

Answer 2

Suppose the time taken for the repairman to fix one pair of shoes is x, time taken by the assistant will be 1.5x. 
The number of shoes repaired by shoe man in an hour will be:
1/x 
the number of shoes prepared by assistant in 1 hour will be:
1/1.5x
the number of shoes repaired by both of them in 1 hour will be:
8/20
hence;
1/1.5x+1/x=8/20
2.5/(1.5x)=2/5
cross-multiplying our equation we get:
2.5*5=1.5x*2
7.5=3x
x=7.5/3
x=2.5 hours
therefore we conclude that it takes the repairman 2.5 hours to fix one pair of shoes.