Question

Given,40 men take 8 days to earn Rs.2000. How many men will earn Rs.200 in 2 days?

Answer 1

Hint: This type of problem can be solved using the concept of proportionality. First, consider x to be the total number of men to earn Rs.200 in 2 days. When the earnings are fixed, the total number of men is directly proportional to the number of days they work. Therefore, \[40\propto 8\] and \[x\propto 2\]. And when the number of days is fixed, then the total number of men is directly proportional to the amount they earn. That is \[40\propto 2000\] and \[x\propto 200\]. We have to divide both the cases to remove the proportionality, that is \[\dfrac{x}{40}=\dfrac{2}{8}\] and \[\dfrac{x}{40}=\dfrac{200}{2000}\]. Thus, we get \[\dfrac{x}{40}=\dfrac{2}{8}=\dfrac{200}{2000}\]. Using the rule ‘when \[\dfrac{a}{b}=\dfrac{c}{d}=\dfrac{e}{f}\], then \[\dfrac{a}{b}=\dfrac{c}{d}\times \dfrac{e}{f}\]’, we get \[\dfrac{x}{40}=\dfrac{2}{8}\times \dfrac{200}{2000}\]. Simplify the RHS of the expression and multiply the whole expression by 40. We get the value of x which is the required answer.

Complete step by step answer:
According to the question, we are asked to find the total number of men who earned Rs.200 in 2 days.
 We have been given 40 men 8 days to earn Rs.2000.
Let us first assume x to be the number of men who earned Rs.200 in 2 days.
We have to find the value of x.
We can solve this question by using the proportionality.
Here, when the earing is fixed, the total number of men is directly proportional to the number of days they work.
When the number of men is x, then the total number of days is 2.
And when the number of men is 40, then the total number of days is 8.
Since earrings are fixed, we have to use proportionality.
That is \[x\propto 2\] and \[40\propto 8\].
To remove the proportionality, we can divide both the expressions.
\[\Rightarrow \dfrac{x}{40}=\dfrac{2}{8}\] -------------(1)
Now, let us consider the total number of days the men worked to be constant.
Then the total number of men will be directly proportional to their earnings.
When the number of men is x, then their earning is Rs.200.
And when the number of men is 40, then their earnings are Rs.2000.
Since the total number of days is assumed to be constant, we have to use proportionality.
That is \[x\propto 200\] and \[40\propto 2000\]
To remove the proportionality, we can divide both the expressions.
\[\Rightarrow \dfrac{x}{40}=\dfrac{200}{2000}\] --------------(2)
From equation (1) and (2), we find that the left-hand sides are equal.
\[\therefore \dfrac{x}{40}=\dfrac{2}{8}=\dfrac{200}{2000}\]
Now, to find the value of x, we have to use the rule ‘when\[\dfrac{a}{b}=\dfrac{c}{d}=\dfrac{e}{f}\], then \[\dfrac{a}{b}=\dfrac{c}{d}\times \dfrac{e}{f}\]’.
Here, a=x, b=40, c=2, d=8, e=200 and d=2000.
Therefore, we get
\[\dfrac{x}{40}=\dfrac{2}{8}\times \dfrac{200}{2000}\]
On further simplifications, we get
\[\dfrac{x}{40}=\dfrac{2}{8}\times \dfrac{200}{200\times 10}\]
We find that 200 are common in both the numerator and denominator.
Let us cancel 200 from the numerator and denominator.
\[\Rightarrow \dfrac{x}{40}=\dfrac{2}{8}\times \dfrac{1}{10}\]
We can write \[10=2\times 5\].
\[\Rightarrow \dfrac{x}{40}=\dfrac{2}{8}\times \dfrac{1}{2\times 5}\]
We find that 2 are common in both the numerator and denominator. Cancelling 2 from the expression, we get
\[\dfrac{x}{40}=\dfrac{1}{8}\times \dfrac{1}{5}\]
Since \[8\times 5=40\], we get
\[\dfrac{x}{40}=\dfrac{1}{40}\]
Now multiply the whole equation by 40.
\[\Rightarrow \dfrac{x}{40}\times 40=\dfrac{1}{40}\times 40\]
We find that 40 are common in the numerator and denominator of both the LHS and RHS.
 We get \[x=1\].
Therefore, 1 man will earn Rs.200 in 2 days.

Note:
Whenever we get such types of problems, we should not just equate both the number of days and number of workers. Instead we have to use proportionality. We should not directly cancel the proportionality constant which will lead to a wrong answer. Also, we should consider two cases, that is the earnings and total number of workers constant and equate then both.