Question

If the average of 5 numbers is 10, and the fifth number is 6, find the average of the first four numbers.
A. 9
B. 10
C. 11
D. 12
E. 13

Answer 1

Hint: First we’ll find the sum of the 5 numbers using the average formula, after getting the difference of that sum to the fifth number we’ll have the sum of the first four numbers. After getting the sum of the first four numbers we can easily find the average of those 4 numbers.

Complete step by step answer:

Given data: Average of 5 numbers=10
Fifth number=6
Now, ${\text{average = }}\dfrac{{{\text{sum of all elements}}}}{{{\text{number of elements}}}}$
According to the given data, we can conclude that,
Average of 5 numbers${\text{ = }}\dfrac{{{\text{sum of 5 numbers}}}}{{\text{5}}}$
$
   \Rightarrow {\text{10 = }}\dfrac{{{\text{sum of 5 numbers}}}}{{\text{5}}} \\
   \Rightarrow {\text{sum of 5 numbers = 10(5)}} \\
  {\text{ = 50}} \\
 $
Therefore the sum of 5 numbers=50
But the fifth number=6
Now, we can say that the sum of the first 4 numbers will be equal to the difference of the sum of 5 numbers and the fifth number i.e.
the sum of the first four numbers=sum of five numbers-fifth number
\[
  {\text{the sum of the first four numbers = 50 - 6}} \\
  {\text{ = 44}} \\
 \]
Therefore, according to the formula of average, the average of the first four numbers will be
\[
  \therefore {\text{average of the first four numbers = }}\dfrac{{{\text{sum of first four numbers}}}}{{\text{4}}} \\
  {\text{ = }}\dfrac{{{\text{44}}}}{{\text{4}}} \\
  {\text{ = 11}} \\
 \]
Option (C) 11 is the correct option

Note: An alternative way to find the solution can be,
Since it is well known that,
\[
  {\text{average of the first four numbers = }}\dfrac{{{\text{sum of first four numbers}}}}{{\text{4}}} \\
   \Rightarrow {\text{sum of first four numbers = 4}}\left( {{\text{average of the first four numbers}}} \right) \\
 \]
And,
$
  {\text{Average of 5 numbers = }}\dfrac{{{\text{sum of 5 numbers}}}}{{\text{5}}} \\
   \Rightarrow {\text{10 = }}\dfrac{{{\text{sum of first 4 numbers + fifth number}}}}{{\text{5}}} \\
   \Rightarrow {\text{10 = }}\dfrac{{{\text{sum of first 4 numbers + 6}}}}{{\text{5}}} \\
 $
From the above equations, we conclude that
$
  {\text{10 = }}\dfrac{{{\text{4}}\left( {{\text{average of the first four numbers}}} \right){\text{ + 6}}}}{{\text{5}}} \\
   \Rightarrow {\text{4}}\left( {{\text{average of the first four numbers}}} \right){\text{ + 6 = 5(10)}} \\
   \Rightarrow {\text{4}}\left( {{\text{average of the first four numbers}}} \right){\text{ = 50 - 6}} \\
   \Rightarrow {\text{average of the first four numbers = }}\dfrac{{{\text{44}}}}{{\text{4}}} \\
  {\text{ = 11}} \\
 $