Question

The average of $11$ numbers is $32$. If the average of the first six numbers is $34$ and that of the last six is $33$, then the sixth number is

• A. $34$
• B. $25$
• C. $50$
• D. $36$

Answer 1

The correct option is C $50$

Average is the sum of the data values divided by the total number of data values.

Let the $11$ numbers be $x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9,x_{10},x_{11}$

Given that average of first $6$ numbers is $34$

$⟹34=\frac{x_1+x_2+x_3+x_4+x_5+x_6}{6}$

$⟹x_1+x_2+x_3+x_4+x_5+x_6=34\times 6=204$ ------(1)

Given that average of last $6$ numbers is $33$

$⟹33=\frac{x_6+x_7+x_8+x_9+x_{10}+x_{11}}{6}$

$⟹x_6+x_7+x_8+x_9+x_{10}+x_{11}=33\times 6=198$

$⟹x_7+x_8+x_9+x_{10}+x_{11}=198-x_6$------(2)

Given that average of all the $11$ numbers is $32$

$⟹32=\frac{x_1+x_2+x_3+x_4+x_5+x_6+x_7+x_8+x_9+x_{10}+x_{11}}{11}$ ------(3)

substituting (1) and (2) in (3) we get

$32=\frac{204+198-x_6}{11}$

$⟹402-x_6=352$

$⟹x_6=402-352=50$

Therefore, the sixth number is $50$