Question

The average of $5$ numbers is $6$. The average of the three numbers is $8$. What is the average of the remaining two numbers?

Answer 1

Hint: The given problem requires us to find the average of two of the total five numbers under some conditions. We should know the formula for finding the average of given numbers as ${\text{Average = }}\dfrac{{{\text{Sum of numbers}}}}{{{\text{Total number}}}}$. So, we will first find the sum of all the five numbers with the help of the average given to us. Then, we will find the sum of the three numbers using the same formula. Then, we will get the sum of the last two numbers and find their average.

Complete step by step answer:
So, we are given $5$ numbers.
The average of five numbers is $6$.
Substituting the known values to find the sum of these five numbers, we get,
$ \Rightarrow {\text{6 = }}\dfrac{{{\text{Sum of 5 numbers}}}}{{\text{5}}}$
Using the method of transposition to find the value of unknown quantity, we multiply both sides of equation by five,
$ \Rightarrow {\text{Sum of 5}}\,{\text{numbers}} = 6 \times 5 = 30$
Now, we are also given the average of three numbers amongst the five numbers separately.
So, we have, ${\text{8 = }}\dfrac{{{\text{Sum of 3 numbers}}}}{{\text{3}}}$
Multiplying both sides by three,
$ \Rightarrow {\text{Sum of 3}}\,{\text{numbers}} = 24$
So, we can find out the sum of the remaining two numbers by subtracting the sum of three numbers from the sum of five numbers. So, we get,
${\text{Sum of 2}}\,{\text{numbers}} = \left( {{\text{Sum of 5}}\,{\text{numbers}}} \right) - \left( {{\text{Sum of 3}}\,{\text{numbers}}} \right)$
Simplifying the expression,
$ \Rightarrow {\text{Sum of 2}}\,{\text{numbers}} = \left( {{\text{30}}} \right) - \left( {{\text{24}}} \right)$
$ \Rightarrow {\text{Sum of 2}}\,{\text{numbers}} = 6$
Hence, the average of two numbers can be calculated using the formula ${\text{Average = }}\dfrac{{{\text{Sum of numbers}}}}{{{\text{Total number}}}}$.
So, ${\text{Average of 2 numbers = }}\dfrac{{{\text{Sum of the numbers}}}}{{{\text{Total number}}}}$
$ \Rightarrow {\text{Average of 2 numbers = }}\dfrac{{\text{6}}}{2} = 3$
So, the average of the two numbers is $3$.

Note:
We must know the formula for the mean and average of some given numbers to solve the problem. Average of n numbers can be computed using the formula: ${\text{Average}}\,{\text{ = }}\dfrac{{{\text{Sum of n numbers}}}}{{\text{n}}}$ . The value of any of the parameters in the formula can be obtained by substituting the values of remaining ones.
Sometimes students think to assume variables in these types of questions but if we analyse carefully there is no need to do that.
Average and mean are the two terms which are often used interchangeably. Mean is the average of values present in the data set. The central value which is called as average in mathematics the same value is called as mean in statistics.