Question

Average of 6 numbers is 8, what number should be added to be it to make the average $9$?

Answer 1

Hint: In this question, we have to find the number which makes the average of six numbers eight to nine. First, we have to understand the given data, and using the average formula we can solve this given problem easily. Understanding the given data is important to solve the problem.
The formula used in the problem;
 In this problem, we used the average formula.
Average $ = \dfrac{{1 + 2 + ....... + n}}{n}$
Add all the terms and divide the total sum by the number of terms.

Complete step-by-step solution:
In this problem, we are going to find the unknown number which should be changed to an average of six numbers.
First, we want to understand the data.
The given data in the problem is given below.
The average of six numbers is eight.
After one number is added the new average is nine.
Let be the six numbers are, \[{x_{1,}}{x_{2,}}{x_{3,}}{x_{4,}}{x_{5,}}{x_6}\]
The average six is eight.
Next we want to know about the term average.
The average is defined as the number we get when adding the two or more numbers together and then dividing the total sum by the number of terms we added.
Therefore the average of six numbers is eight.
We can write this as,
Average of six numbers $
   = \dfrac{{{x_1} + {x_2} + {x_3} + {x_4} + {x_5} + {x_6}}}{6} = 8 \\
   = {x_1} + {x_2} + {x_3} + {x_4} + {x_5} + {x_6} = 8 \times 6 \\
 $
 Now let the unknown term be ${x_7}$
After adding the unknown term, we get changes in the average.
After adding the term we get the average is nine.
We can write this as,
$ = \dfrac{{{x_1} + {x_2} + {x_3} + {x_4} + {x_5} + {x_6} + {x_7}}}{7} = 9$
Now we can write the first equation,
Take cross multiplication,
$\Rightarrow {x_1} + {x_2} + {x_3} + {x_4} + {x_5} + {x_6} = 8 \times 6$
\[\Rightarrow {x_1} + {x_2} + {x_3} + {x_4} + {x_5} + {x_6} = 48\] .......................................\[\left( 1 \right)\]
That like that we can write the second equation as,
\[{x_1} + {x_2} + {x_3} + {x_4} + {x_5} + {x_6} + {x_7} = 63\]……………………………………………………… \[\left( 2 \right)\]
Now solve the equations we get,
\[48 + {x_7} = 63\]
On solving this, we get,
\[\Rightarrow {x_7} = 63 - 48\]
\[\Rightarrow {x_7} = 15\]
Hence the unknown number is fifteen.
Therefore the solution is \[{x_7} = 15\].

Note: An average is a number calculated by adding quantities together and then dividing the total sum of the number by the number of quantities. Formally, it is called mathematics and statistics is mean. In other words, we can say this as, calculating the central value of the set numbers.