# Average of 7 numbers is 6.What is 8th no. to be added if the average increases by 10 ?

Mar 28, 2018

See a solution process below:

#### Explanation:

The formula for the average of a set of numbers is:

$a = \frac{s}{n}$ Where

• $a$ is the average of the numbers. We know from the problem the average of the first 7 numbers is $6$
• $s$ is the sum of the set of numbers. We need to solve for this to answer the questions.
• $n$ is the count of numbers in the set of number we are averaging. We know from the problem the count of numbers in the set is $7$

Substituting and solving for $s$ we can find the some of the numbers:

$6 = \frac{s}{7}$

$\textcolor{red}{7} \times 6 = \textcolor{red}{7} \times \frac{s}{7}$

$42 = \cancel{\textcolor{red}{7}} \times \frac{s}{\textcolor{red}{\cancel{\textcolor{b l a c k}{7}}}}$

$42 = s$

$s = 42$

If the average is to increase by 10, then the new average would be: $6 + 10 = 16$

The count of numbers would increase to $8$

And if we call the value of the new number added in $w$, then the sum of the set of numbers is: $42 + w$

Substituting this into the formula for average and solving for $w$ gives:

$16 = \frac{42 + w}{8}$

$\textcolor{red}{8} \times 16 = \textcolor{red}{8} \times \frac{42 + w}{8}$

$128 = \cancel{\textcolor{red}{8}} \times \frac{42 + w}{\textcolor{red}{\cancel{\textcolor{b l a c k}{8}}}}$

$128 = 42 + w$

$128 - \textcolor{red}{42} = 42 - \textcolor{red}{42} + w$

$86 = 0 + w$

$86 = w$

$w = 86$

The eight number added in to make an average of 16 would be 86

$a = \frac{42 + 86}{8}$

$a = \frac{128}{8}$

$a = 16$