Question

In a city, the total income of all people with a salary below Rs. 10000 per annum is less than the total income of all people with a salary above Rs. 10000 per annum. If the salaries of people in the first group decreased by 5% and the salaries of people in the second group decreased by 5%, then the average income of all people
A) Increases
B) Decreases
C) Remains the same
D) Cannot be determined from the data

Answer 1

In this question, first assume the number of people with a salary below and above Rs. 10000. Also, assume the salary be A and B. After that find the average of all people with a salary below Rs. 10000 after a 5% decrease. Also, find the average of all people with a salary above Rs. 10000 after a 5% decrease. After that find the average of all people and compare them with the average before deduction.

Complete step-by-step answer:
Let the number of people with a salary below Rs. 10000 be $x$ and the number of people with a salary above Rs. 10000 be $y$.
Let the salary below Rs. 10000 be $A$ and the salary above Rs. 10000 be $B$.
The total salary of all people will be the sum of the total salary of people below Rs. 10000 and total salary of people above Rs. 10000. So,
$\Rightarrow$ Total Salary $= Ax + By$
The total number of people will be the sum of total people below Rs. 10000 and total people above Rs. 10000. So,
$\Rightarrow$ Total People $= x + y$
Then, the average income of all people is,
$\Rightarrow$ Average $= \dfrac{{Ax + By}}{{x + y}}$
Now, the new salary of people is 5% less than the original.
The total salary of all people will be the sum of the total salary of people below Rs. 10000 and total salary of people above Rs. 10000. So,
$\Rightarrow$ Total Salary $= A\left( {1 - \dfrac{5}{{100}}} \right)x + B\left( {1 - \dfrac{5}{{100}}} \right)y$
Simplify the terms,
$\Rightarrow$ Total Salary $= \dfrac{{95}}{{100}}Ax + \dfrac{{95}}{{100}}By$
Cancel out the terms,
$\Rightarrow$ Total Salary $= \dfrac{{19}}{{20}}Ax + \dfrac{{19}}{{20}}By$
Take common from both terms,
$\Rightarrow$ Total Salary $= \dfrac{{19}}{{20}}\left( {Ax + By} \right)$
Then, the new average income of all people is,
$\Rightarrow$ New Average $= \dfrac{{\dfrac{{19}}{{20}}\left( {Ax + By} \right)}}{{x + y}}$
Compare with equation (1),
$\therefore$ New Average = $\dfrac{{19}}{{20}} \times$Average.
So, the new average is less than the original.

Hence, option (B) is correct.

Note: The average is defined as the sum of given numbers divided by the total number of numbers being averaged.
Average = (Sum of given number)/ (Total count of number)
An average is a single number taken as representative of a list of numbers. Often, Average refers to the arithmetic mean.