Question

The A.M. of a set of 50 numbers is 38. If two numbers of the set, namely 55 and 45 are discarded, the A.M. of the remaining set of numbers is:
A) 36
B) $36.5$
C) $37.5$
D) $38.5$

Answer 1

Here we will find the A.M (Arithmetic Mean) of the remaining numbers by using the data given to us. We will find the sum of all the numbers by using the A.M formula and then subtract the two numbers which are discarded from it. Then we will divide the new sum by the remaining numbers to get the new A.M.

Formula used:
A.M$= \dfrac{1}{n}\sum\limits_{i = 1}^n {{a_i}}$, where, A.M is Arithmetic Mean, $n$ is number of values and ${a_i}$ is the data set of values

Complete step by step solution:
We know that A.M of 50 given numbers is 38.
Let the sum of 50 numbers be $x$.
Therefore, using the formula of arithmetic mean, we get
A.M $= \dfrac{x}{{50}}$
\Rightarrow 38 = \dfrac{x}{{50}}$$$$$$ On cross multiplication, we get \Rightarrow x = 38 \times 50$$ \Rightarrow x = 1900$So, sum of 50 numbers is 1900. Now, it is given that two numbers 55, 45 are discarded so the sum of number after removing them is: New sum$ = 1900 - \left( {55 + 45} \right)$Adding the terms inside the bracket, we get$ \Rightarrow $New sum$ = 1900 - 100$Subtracting the terms, we get$ \Rightarrow $New sum$ = 1800$So, our new sum of 48 numbers is 1800. Therefore, A.M of the remaining 48 numbers is given by dividing new sum by remaining numbers. Now using the formula of arithmetic number, we get A.M$ = \dfrac{{1800}}{{48}}$Dividing the terms, we get$ \Rightarrow $A.M$ = 37.5$So, A.M of the remaining number is$37.5$$.

Hence, option (C) is correct.

Note:
Arithmetic Mean is also known as the average of a set of numbers. We can use a direct method by subtracting 55 and 45 by the product of the original A.M and the original set of numbers and then divide it by 48 as two numbers are being removed.
So, we get our equation as:
A.M = \dfrac{{\left[ {50 \times 38} \right] - 55 - 45}}{{50 - 2}}$$$$$$ Simplifying the expression, we get \Rightarrow $A.M$ = \dfrac{{1800}}{{48}}$Dividing the terms, we get$ \Rightarrow $A.M$ = 37.5$So, A.M of the remaining number is$37.5$$.