Question

The arithmetic mean of five consecutive odd numbers is 63. Then the fourth number in the series is:
A. 63
B. 65
C. 67
D. 69

Answer 1

Hint: Here, we have given that the arithmetic mean of 5 consecutive odd numbers is 63 and we have to find the fourth number. For this, we will first assume the first number to be ‘a’. Then, as there is always a difference of 2 between successive odd numbers, we will get all the 5 numbers in terms of a. Now, since we have already been given the value of the arithmetic mean of the 5 numbers, we will put the value of it and the assumed numbers in the formula for the arithmetic mean of ‘n’ numbers given as \[\text{Arithmetic mean=}\dfrac{\text{sum of n numbers}}{\ \text{n}}\]. Then we will solve the equation then form and obtain the value of a. With the help of that, we will find the value of the fourth number and hence we will get our answer.

Complete step by step answer:
Now, we have been given that the arithmetic mean of 5 consecutive odd numbers is 63 and we have to find the fourth number.
For thus, let us first assume the first number to be ‘a’. Now, we know that in every successive consecutive number, there is a difference of 2. Hence, the 5 consecutive odd numbers will come out as:
a, a+2, a+4, a+6, a+8
Now, we know that the arithmetic mean of ‘n’ numbers is given as:
\[\text{Arithmetic mean=}\dfrac{\text{sum of n numbers}}{\ \text{n}}\]
Here, we have:
Arithmetic mean=63
n=5
Thus, putting these values into the formula for arithmetic mean we get:
\[\begin{align}
  & \text{Arithmetic mean=}\dfrac{\text{sum of n numbers}}{\ \text{n}}\\
 & \Rightarrow 63=\dfrac{\left( a \right)+\left( a+2 \right)+\left( a+4 \right)+\left( a+6 \right)+\left( a+8 \right)}{5} \\
\end{align}\]
Now, solving this we get:
\[\begin{align}
  & 63=\dfrac{\left( a \right)+\left( a+2 \right)+\left( a+4 \right)+\left( a+6 \right)+\left( a+8 \right)}{5} \\
 & \Rightarrow 315=5a+20 \\
 & \Rightarrow 5a=295 \\
 & \Rightarrow a=\dfrac{295}{5} \\
 & \Rightarrow a=59 \\
\end{align}\]
Hence, the value of ‘a’ is 59. Thus, the first odd number is 59.
Now, we have already established above that the fourth number is given by a+6.
Hence, the fourth number is given as:
$\begin{align}
  & a+6 \\
 & \Rightarrow 59+6 \\
 & \therefore 65 \\
\end{align}$
Thus, the fourth number of the series is 65.

So, the correct answer is “Option B”.

Note: Here, we have been given the arithmetic mean of 5 numbers. Remember, whenever we have been given the arithmetic mean of an odd number of numbers which form a series, the arithmetic mean is always the middle number of the given series. Here, the middle number is the third number hence it is equal to the given mean. Thus, the third number of the series is 63. So, the fourth number will become 63+2=65.