Question

The mean of eleven numbers is 47. If a number is added then the mean is increased by 2. Find the new number.
A) 51
B) 61
C) 71
D) 81

Answer 1

Hint: First assume the 11 numbers and the new number. After that find the sum of the numbers by substituting the value of the mean and total count of numbers by mean formula, $\dfrac{{\sum {{x_i}} }}{n}$. Then add the new number to the total. After that find the number by substituting the value of the mean and total count of numbers by mean formula, $\dfrac{{\sum {{x_i}} }}{n}$ and simplify.

Complete step-by-step answer:
The mean of eleven numbers is 47.
Let the 11 numbers are ${a_1},{a_2}, \ldots ,{a_{11}}$ and the new number be $x$.
We know that the general formula to find the mean value is,
Mean $ = \dfrac{{\sum {{x_i}} }}{n}$
Substitute the values in the above formula,
$ \Rightarrow 47 = \dfrac{{{a_1} + {a_2} + \ldots + {a_{11}}}}{{11}}$
Cross-multiply the terms,
$ \Rightarrow {a_1} + {a_2} + \ldots + {a_{11}} = 517$..............….. (1)
Now the new mean after adding a number is,
$ \Rightarrow 47 + 2 = 49$
Now, substitute the value in the mean formula,
$ \Rightarrow 49 = \dfrac{{{a_1} + {a_2} + \ldots + {a_{11}} + x}}{{12}}$
Cross-multiply the terms,
$ \Rightarrow {a_1} + {a_2} + \ldots + {a_{11}} + x = 49 \times 12$
Substitute the value from equation (1) and multiply the terms,
$ \Rightarrow 517 + x = 588$
Move the constant part on other sides and subtract,
$\therefore x = 71$

Hence, option (C) is correct.

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.