Question

The mean of 10 numbers is 24. If one more number is included, the new mean is 25. Find the included number.
A. 28
B. 32
C. 35
D. 40

Answer 1

Hint:First, we need to apply the formula of mean for the first 10 numbers. Next, apply the formula for the mean along with the included number, hence find the included number. The mean of numbers is also termed as the average of the numbers.

Complete step-by-step answer:
Let \[{x_1},{x_2},{x_3}, \ldots {x_{10}}\] be the numbers.
Now, since the mean of \[{x_1},{x_2},{x_3}, \ldots {x_{10}}\] is 24, therefore,
\[
  \,\,\,\,\,\,\dfrac{{{x_1} + {x_2} + {x_3} + \ldots + {x_{10}}}}{{10}} = 24 \\
   \Rightarrow {x_1} + {x_2} + {x_3} + \ldots + {x_{10}} = 24 \times 10 \\
   \Rightarrow {x_1} + {x_2} + {x_3} + \ldots + {x_{10}} = 240\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,......\left( 1 \right) \\
\]
Let another number be \[{x_{11}}\].
Now, if \[{x_{11}}\] is added with \[{x_1},{x_2},{x_3}, \ldots {x_{10}}\], the mean becomes 25. Therefore,
\[
  \,\,\,\,\,\,\dfrac{{{x_1} + {x_2} + {x_3} + \ldots + {x_{10}} + {x_{11}}}}{{11}} = 25 \\
   \Rightarrow {x_1} + {x_2} + {x_3} + \ldots + {x_{10}} + {x_{11}} = 25 \times 11 \\
   \Rightarrow {x_1} + {x_2} + {x_3} + \ldots + {x_{10}} + {x_{11}} = 275 \\
   \Rightarrow {x_{11}} = 275 - \left( {{x_1} + {x_2} + {x_3} + \ldots + {x_{10}}} \right)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,......\left( 2 \right) \\
\]
Now, from equation (1), substitute 240 for \[{x_1} + {x_2} + {x_3} + \ldots + {x_{10}}\] in equation (2) to obtain the value of \[{x_{11}}\].
\[
  \,\,\,\,\,\,{x_{11}} = 275 - \left( {240} \right) \\
   \Rightarrow {x_{11}} = 275 - 240 \\
   \Rightarrow {x_{11}} = 35 \\
\]
Thus, the included number would be 35, hence option (C) is the correct answer.

Note: The mean or average of the numbers is defined as the ratio of the sum of the numbers to the total numbers. To obtain the mean of 10 numbers, add all the 10 numbers and divide it by 10.