Question

If the mean of 20 numbers is 9. If 3 is added to every number what will be the new mean?
A) 9
B) 12
C) 27
D) 15

Answer 1

Find the sum of the numbers with the help of the condition given i.e. the mean of 20 numbers is 9 and denote the sum with a single variable and add 3 times the total number and take the mean of the numbers will fetch you the new mean.

Complete step by step solution:
Given, Mean of 20 numbers is 9.
Assume the numbers be ${x_1},\,{x_2},.........{x_{20}}$and it is denoted by a single variable $x$.
i.e. ${x_1} + {x_2} + {x_3}.......... + {x_{20}} = x.........\left( 1 \right)$
According to the question, $\dfrac{{{x_1} + {x_2} + {x_3}.......... + {x_{20}}}}{{20}} = 9..........\left( 2 \right)$
Substitute (2) in (1), We get
$\Rightarrow \dfrac{{{x_1} + {x_2} + {x_3}.......... + {x_{20}}}}{{20}} = 9 \\\ \Rightarrow \dfrac{x}{{20}} = 9 \\\ \Rightarrow x = 180........\left( 3 \right) \\\$
Also given that If 3 is added to every number a new mean is obtained.
So, From (2) we get and assume the new mean to be $m$
$\Rightarrow \dfrac{{({x_1} + 3) + ({x_2} + 3) + ({x_3} + 3).......... + ({x_{20}} + 3)}}{{20}} = m \\\ \Rightarrow \dfrac{{x + 3\left( {20} \right)}}{{20}} = m \\\ \Rightarrow x + 60 = 20m \\\$
From (3) we get $x = 180$
$\Rightarrow x + 60 = 20m \\\ \Rightarrow 180 + 60 = 20m \\\ \Rightarrow 20m = 240 \\\ \Rightarrow m = 12 \\\$

So, the new mean is 12 and option B is correct.

Note:
A series of unknown numbers can be denoted by a single variable. If a number is added to every number then the sum is the product of that repeated number and number of numbers. Mean is a useful tool for data related works because sometimes they use to fill missing numbers with the mean of that category.