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.
A) Rupees $2\left( {x + 40} \right)$ and $22{\text{ hrs}}$
B) Rupees $2\left( {x - 40} \right)$ and $12{\text{ hrs}}$
C) Rupees $\left( {x - 40} \right)$ and ${\text{16 hrs}}$
D) Rupees $\left( {x + 40} \right)$ and ${\text{13 hrs}}$

Answer 1

Hint:In the above question we are given the rupees paid per hour for normal work and overtime. And we are also given the total work hours and hours for overtime for a week, from which we can find the hours of normal work for a week. Now we have to find the expression for his one week’s earnings in which he worked, which we can find easily using $ = {\text{rupees paid per hour}} \times {\text{number of hours he did work in a week}}$, for normal work and overtime separately and then adding the answers. Now we have to find the number of hours of his normal work, which we can obtain easily from the expression for his one week’s earnings in which he worked and by using the given information about the total earnings of the person.

Complete step-by-step answer:
In this question we given that-
Rupees paid per hour for normal work is $2$ ……………..(1)
Now we are given that rupees paid for overtime is double of normal work, so we can write,
Rupees paid per hour for over time is $2\left( 2 \right) = 4$ ……………..(2)
Now we are given that
In week the person worked for total $40{\text{ hrs}}$ ……………..(3)
And in one week the person did over time for $x{\text{ hrs}}$ ……………..(4)
We can say that-
The person did normal work is equal to the total hours person worked minus the hour’s person did over time.
So, we can write using (3) and (4),
In one week, the person did normal work for $\left( {40 - x} \right){\text{ hrs}}$ ……………..(5)
Now we are also given that
The total rupees person received is $116$ ……………..(6)
Now according to the question, we have to find an expression for his one week’s earnings and the number of hours of his normal work.
Firstly, we will find an expression for his one week’s earnings-
From (4) and (5), we have hours of overtime and normal work he did in one week. And from (1) and (2), we have rupees paid per hour for normal work and for overtime.
Now we know that the earning of the week by normal work is
$ = {\text{rupees paid per hour}} \times {\text{number of hours he did normal work in a week}}$
Now substituting values from (1) and (5), we get
$ = 2 \times \left( {40 - x} \right)$
The earning of the week by normal work is $ = \left( {80 - 2x} \right)$ ……………..(7)
Now we know that the earning of the week by overtime is
$ = {\text{rupees paid per hour}} \times {\text{number of hours he did overtime in a week}}$
Now substituting values from (2) and (4), we get
$ = 4 \times \left( x \right)$
The earning of the week by overtime is $ = 4x$ ……………..(8)
The one week’s earning is equal to the sum of earning by normal work and overtime in a week.
Now substituting values from (7) and (8), we get
The one week’s earning $ = 80 - 2x + 4x$
The one week’s earning $ = 80 + 2x$
We can also write it as
The one week’s earning $ = 2\left( {40 + x} \right)$ ……………..(9)
So, (9) is the required expression.
Now we have to find the number of hours of his normal work.
Now from (6), we have the total earning of the person in week is $116$.
Also, we can clearly see that (9) is also the total earning of the person for a week.
So, substituting value from (6) in (9), we get
$116 = 2\left( {40 + x} \right)$
Now solving it further we get,
$116 = 80 + 2x$
So, we get
$
  2x = 36 \\
  x = 18 $ ……………..(10)
Now from (4), we know that $x$ is the hours for which a person did overtime in a week.
So, we get
In one week, the person did over time for ${\text{x = 18 hrs}}$.
Now to get the hours of normal work, we will substitute value of $x$ from (10) in (5), we get,
In one week, the person did normal work for $ = \left( {40 - 18} \right){\text{ hrs}}$
In one week, the person did normal work for $ = 22{\text{ hrs}}$ ……………..(11)
So, (11) is the required answer.
Hence from (9) and (11), we get,
The one week’s earning $ = 2\left( {40 + x} \right)$ and
In one week, the person did normal work for $ = 22{\text{ hrs}}$

So, the correct answer is “Option A”.

Note:In the above question students generally forgot to find the normal work person did in a week, which made them unable to reach the answer.
Also, we must know the formula for calculating total earnings, that is,
 $ = {\text{rupees paid per hour}} \times {\text{number of hours he did work in a week}}$
So, by keeping these small things in mind we can easily solve the above question.