Question

Find the mean of first $10$ multiple of $3$.
A. $14.5$
B. $13$
C. $16.5$
D. $23$

Answer 1

Hint: According to the question given in the question we have to determine the mean of first $10$ multiple of $3$. So, first of all we have to determine the first multiple of 10 which means we have to determine the numbers that are easily divisible by 3.
Now, we have to determine the mean for all the 10 numbers obtained which are divisible by 3 but before that we have to understand about the mean which is as explained below:
Mean: The mean or we can say that the average of a data set which can be found by adding all the given numbers in the data set and then dividing by the number of the values in the set.
Now, we have to find the sum of all the 10 numbers obtained which are the multiple of 3 with the help of the formula as mentioned below:
$ \Rightarrow $Sum\[ = \dfrac{{a + l}}{2}...............(A)\]
Where, a is the first number and l is the last number of the obtained numbers basically we find sum with the help of the A.P because all the numbers obtained are in A.P.
Now, we have to find the total numbers as we know that the total numbers are 10 so, to find the mean we just have to divide the obtained sum with the total number of terms.

Complete step-by-step solution:
Step 1: First of all we have to determine the first multiple of 10 which means we have to determine the numbers that are easily divisible by 3 as mentioned in the solution hint. Hence,
$ = 3,6,9,12,15,.............,30$
Step 2: Now, we have to find the sum of all the 10 numbers obtained which are the multiple of 3 with the help of the formula (A) as mentioned in the solution hint.
$
   = \dfrac{{10}}{2}(3 + 30) \\
   = 5(33) \\
   = 165
 $
Step 3: Now, as we have already known that the total number of terms are 10 and we have determined the sum hence, to determine the mean we just have to divide the obtained sum with the total number of terms as mentioned in the solution hint.
$ \Rightarrow $Mean:
$
   = \dfrac{{165}}{{10}} \\
   = 16.5
 $
Hence, with the help of the formula (A) as mentioned in the solution hint we have determined the mean which is 16.5.
Therefore option (C) is correct.

Note: The numbers which are the multiple of 3 are in A.P because if we find the common difference for the series of 10 numbers obtained is same so, to find all the 10 numbers easily we can apply the formula to find the sum of an A.P series.
The average of a data set which can be found by adding all the given numbers in the data set and then dividing by the number of the values in the set.