Question

The mean of the 5 numbers is 32. If one of the numbers is excluded, then the mean is reduced by 4. Find the excluded number?

Answer 1

Hint: We start solving the problem by assigning the variables for the excluded number and the remaining 4 numbers. We then recall the fact that the mean of the ‘n’ numbers ${{a}_{1}}$, ${{a}_{2}}$, ……, ${{a}_{n}}$ is defined as $\dfrac{{{a}_{1}}+{{a}_{2}}+......+{{a}_{n}}}{n}$. We then apply this formula of mean to the assumed 5 numbers to get our first relation. We then apply the mean for the remaining 4 numbers to get our second relation. We then solve the obtained relations to get the value of the excluded number.

Complete step-by-step solution
According to the problem, we are given that the mean of the 5 numbers is 32 and if one of the numbers is excluded, then the mean is reduced by 4. We need to find the excluded number.
Let us assume the excluded number be ‘x’ and the remaining numbers be a, b, c, and d.
We know that the mean of the ‘n’ numbers ${{a}_{1}}$, ${{a}_{2}}$, ……, ${{a}_{n}}$ is defined as $\dfrac{{{a}_{1}}+{{a}_{2}}+......+{{a}_{n}}}{n}$.
So, we get $\dfrac{a+b+c+d+x}{5}=32$.
$\Rightarrow a+b+c+d+x=160$.
$\Rightarrow a+b+c+d=160-x$ ---(1).
Now, we are given that the mean of the numbers is reduced by 4 if the number ‘x’ is excluded from them.
So, we get $\dfrac{a+b+c+d}{4}=32-4$.
$\Rightarrow \dfrac{a+b+c+d}{4}=28$.
$\Rightarrow a+b+c+d=112$ ---(2).
Let us substitute equation (2) in equation (1).
$\Rightarrow 160-x=112$.
$\Rightarrow x=160-112$.
$\Rightarrow x=48$.
$\therefore$ We have found the value of the excluded number as 48.

Note: Whenever we get this type of problem, we first assign variables to the unknowns to avoid confusion while solving the problem. We should not take the mean of the remaining four numbers like 4 as it is the most common mistake done by students. We should not make calculation mistakes while solving this problem. Similarly, we can expect problems to find the remaining numbers if the remaining four numbers are in Arithmetic progression.