Question

The mean of 5 numbers is 30. If one number is excluded then their mean becomes 28. What is the excluded number?

Answer 1

Hint: Now we are given the mean of 5 numbers is 30. Mean of data is given by $\dfrac{\text{sum of data}}{\text{number of terms}}$ , hence using this we get the first equation. Now we are given that if one number is excluded then their mean becomes 28. Hence we will exclude one term and form a new equation again by using mean = $\dfrac{\text{sum of data}}{\text{number of terms}}$ . Now we will subtract the two equations obtained to find the excluded term.

Complete step by step answer:
Now we let the five numbers be \[{{x}_{1}},{{x}_{2}},{{x}_{3}},{{x}_{4}}\] and ${{x}_{5}}$ .
Now we know that mean is nothing but the average of numbers.
Mean of data is given by $\dfrac{\text{sum of data}}{\text{number of terms}}$ .
Now we know that we are given with 5 numbers whose sum is ${{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}+{{x}_{5}}$
Hence mean of the terms is $\dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}+{{x}_{5}}}{5}$ .
Now we are given that the mean of these 5 terms is equal to 30. Hence we have,
$\dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}+{{x}_{5}}}{5}=30$
Multiplying the whole equation by 5 we get,
$\begin{align}
  & \Rightarrow {{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}+{{x}_{5}}=30\times 5 \\
 & \Rightarrow {{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}+{{x}_{5}}=150.......................\left( 1 \right) \\
\end{align}$
Now again we are given that if one number is excluded then their mean becomes 28.
Let us say we exclude the term ${{x}_{5}}$ . Then again using the formula for mean we get $\dfrac{{{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}}{4}=28$
Multiplying the equation by 4 we get,
$\begin{align}
  & {{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}=28\times 4 \\
 & \Rightarrow {{x}_{1}}+{{x}_{2}}+{{x}_{3}}+{{x}_{4}}=112...................\left( 2 \right) \\
\end{align}$
Now let us subtract equation (2) from equation (1). Hence we get,
$\begin{align}
  & {{x}_{5}}=150-112 \\
 & \Rightarrow {{x}_{5}}=38 \\
\end{align}$

Hence the number excluded was 38.

Note: Note that mean, median and mode are all types of averages. Mean is the usual average of and is given by $\dfrac{\text{sum of data}}{\text{number of terms}}$ . Median of data is the middle term of the term in the list when arranged ascending order, while Mode is the number that appears most in the list.