Question

A worker in a factory is paid rupees 2 per hour for normal work and double the rate for overtime work. Write down an expression for his one week’s earnings in which he worked for 40 hours out of which x hours were overtime. Also, find the number of hours of his normal work if he receives rupees 116 in all.

Answer 1

Hint: We are given the rupees paid to the worker for normal and overtime work and also we have been given the total work hours and hours of overtime work for a week. We can find the expression for his one week’s earnings by using the formula, rupees paid per hour $\times $ number of hours he did work in a week, for normal and overtime work hours separately and then adding them. For finding the number of hours of his normal work, we will equate 116 to the expression of total earnings of a week and get the value of x from there, and by substituting the value of x in expression (40 - x), we will get the number of hours of his normal work.

Complete step-by-step solution
In the above question, we are given that a worker in a factory is paid Rs. 2 per hour for normal work and double the rate for overtime work.
So, the worker will get $Rs.2\times 2=Rs.4$
Now, we are given that, in a week he worked for a total of 40 hours. And, also in a week, he did overtime for x hours.
Now, we can say that he did normal work is equal to the total hours he worked minus the hours he did overtime. So, we can write,
In a week, he did normal work for (40 - x) hours.
Now, we are also given that, the total amount received by the worker is Rs. 116.
Now, according to the above question, we have to find an expression for his one week’s earning and the number of hours of his normal work. Firstly, we will find an expression for his one week’s earnings.
We have the normal hours he worked and the overtime he worked for in a week. Also, we have the amount paid per hour for normal and overtime work.
Now, we know that the earnings of a week for the work done in normal work hours is given as,
Rupees paid per hour $\times $ number of hours he did work in normal work hours in a week. On substituting the respective values, we get,
$\begin{align}
  & =2\times \left( 40-x \right) \\
 & =\left( 80-2x \right) \\
\end{align}$
So, earnings of a week for the normal work = $\left( 80-2x \right).........(i)$
Now, we know that the earnings in a week for the work he does in overtime hours is given as,
Rupees paid per hour $\times $ number of hours he did work in overtime work hours in a week. On substituting the respective values, we get,
$\begin{align}
  & =4\times x \\
 & =4x \\
\end{align}$
So, earnings of a week for the overtime work = $4x.........(i)$
So, the total earnings of a week are equal to the sum of the earnings by normal work and the earnings by the overtime work. So, we can write,
Total earnings of a week = Earnings by normal work + Earnings by overtime work
$\begin{align}
  & \Rightarrow 80-2x+4x \\
 & =80+2x \\
\end{align}$
So, we have got the total earnings of a week = $80-2x.........(iii)$
Now, we have to find the number of hours of his normal work.
We are given that the total earnings that he receives is Rs. 116. So, on equating this value to equation (iii), we get,
$\begin{align}
  & 116=80+2x \\
 &\Rightarrow 2x=116-80 \\
 &\Rightarrow 2x=36 \\
\end{align}$
On dividing both the sides by 2, we get,
$\begin{align}
  & x=\dfrac{36}{2} \\
 &\Rightarrow x=18 \\
\end{align}$
We know that x denotes the hours for which he did overtime in a week.
Hence, we get the hours for which he did overtime work in a week as 18 hours.
Now, we have to find the hours for which he did the normal work in a week and we know that it is given by, (40 - x). So, on substituting the value of x in this, we get,
$\begin{align}
  & =40-18 \\
 & =22 \\
\end{align}$
Hence, we get the hours for which he did normal work in a week as 22 hours.
Therefore, we get the expression of the earning of the worker in one week as (80-2x). And the number of hours of his normal work is 22 hours.

Note: In the above question, students usually forget to find the normal work the worker did in a week and solve only one part of the question. Also, students must know how to calculate the total earnings, which is as follows, Rupees paid per hour $\times $ number of hours he did work in normal work hours in a week. Also, one of the most common mistakes that the students make while solving this question is that they write the expression for the hours the worker did normal work as (40 + x) instead of (40 - x) and this will lead to errors in the final answer. So, the students have to be careful while forming the expressions.