Question

A factory is operating in two shifts, day and night, with 70 and 30 workers respectively. If per day mean wage of the day shift workers is Rs 54 and per day mean wage of all the workers is Rs60, then per day mean wage of the night shift workers (in Rs) is :
A. 66
B. 74
C. 75
D. 69

Answer 1

Hint: First find the total number of workers. After that, find the total wages of all workers by the mean formula, $\bar x = \dfrac{{\sum {{x_i}} }}{n}$. Then find the total wages of day shift workers by the mean formula. Now subtract the total wages of day shift workers from the total wages of all workers to get the total wages of night shift workers. After that use the mean formula to get the mean wages to the night shift workers.

Formula used: The mean is calculated by dividing the sum of observations by the number of observations. The formula is given by,
$\bar x = \dfrac{{\sum {{x_i}} }}{n}$
where $\sum {{x_i}} = $ Sum of the observations.
n = total number of observations.

Complete step-by-step solution:
Let the total wages of all workers be $x$, total wages of day shift workers be $y$ and the total wages of night workers be $z$.
The total number of workers is the sum of the workers of day and night shifts,
$ \Rightarrow 70 + 30 = 100$
Now, the mean wage of all the workers is Rs. 60 and the total workers is 100.
Substitute the values in the mean formula to get the total wages of all workers,
$ \Rightarrow 60 = \dfrac{x}{{100}}$
Cross-multiply the terms,
$ \Rightarrow x = 6000$ ………………….. (1)
Now, the mean wage of the day shift workers is Rs. 54 and the total number of day shift workers is 70.
Substitute the values in the mean formula to get the total wages of day shift workers,
$ \Rightarrow 54 = \dfrac{y}{{70}}$
Cross-multiply the terms,
$ \Rightarrow y = 3780$.............….. (2)
Subtract equation (2) from equation (1) to get the total wages of night shift workers,
$ \Rightarrow z = 6000 - 3780$
Subtract the values,
$ \Rightarrow z = 2220$
Now use the mean formula to find the mean wages of night shift workers,
$ \Rightarrow \bar x = \dfrac{{2220}}{{30}}$
Divide numerator by the denominator,
$\therefore \bar x = 74$
Thus, the mean wage of the night shift workers is Rs. 74.
Hence, option (B) is the correct answer.

Note: Whenever we face such types of problems the key point that we need to recall is that Mean or Arithmetic mean is the average of given numbers. It is found by calculating the sum of the given numbers and dividing it by how many numbers there are. These types of questions are based only on this definition and can be easily solved using this.