Question

The average of 3 friends is 32 years. If the age of the fourth friend is added their average comes to 31 years. What is the age of the fourth friend?

Answer 1

First assume the age of four friends. Now substitute the value of mean and number of friends in the mean formula $\dfrac{{\sum {{x_i}} }}{n}$ to get the total age of three friends. After that add the age of the fourth friend to the total age of three friends to get the total age of 4 friends. Then, substitute the values in the mean formula $\bar x = \dfrac{{\sum {{x_i}} }}{n}$ to get the age of the fourth friend.

Complete step-by-step answer:
Let the age of four friends be ${x_1},{x_2},{x_3},{x_4}$.
The formula of mean is,
Mean $= \dfrac{{\sum {{x_i}} }}{n}$
Substitute the values,
$\Rightarrow 32 = \dfrac{{{x_1} + {x_2} + {x_3}}}{3}$
Cross-multiply the terms,
$\Rightarrow {x_1} + {x_2} + {x_3} = 96$..........….. (1)
Then the sum of the age of four friends will be,
$\Rightarrow \sum {{x_i}} = {x_1} + {x_2} + {x_3} + {x_4}$
Substitute the values from equation (1),
$\Rightarrow \sum {{x_i}} = 96 + {x_4}$
Substitute the values in the mean formula,
$\Rightarrow 31 = \dfrac{{96 + {x_4}}}{4}$
Cross-multiply the terms,
$\Rightarrow 96 + {x_4} = 124$
Move the constant part on the right side,
$\Rightarrow {x_4} = 124 - 96$
Subtract the terms,
$\therefore {x_4} = 28$

Hence, the age of the fourth friend is 28 years.

Note: Arithmetic mean represents a number that is obtained by dividing the sum of the elements of a set by the number of values in the set. So, you can use the layman term Average, or be a little fancier and use the word “Arithmetic mean“. Arithmetic means utilizing two basic mathematical operations, addition and division to find a central value for a set of values.