Question

The average (arithmetic mean) of a particular set of seven numbers is $12$. When one of the numbers is replaced by the number $6$, the average of the set increases to $15$. What is the number that was replaced?

• A. $-10$
• B. $-15$
• C. $-12$
• D. $0$
• E. $12$

Answer 1

Answer: (B)

$-15$

The average of a set of numbers is the sum of the numbers divided by the number of numbers:
i.e: average $=\frac{\text{sum of all}7\text{numbers}}{\text{No. of numbers}\left(N\right)}$
Sum of all $7$ numbers $=$average $\times N$
$=12\times 7=84$
Similarly, the sum of the new set of numbers is $=15\times 7=105$
Now, suppose that the seven numbers are $a,b,c,d,e,f,g$ and the $g$ gets replaced by $6$.
Then we have, $a+b+c+d+e+f+g=84$ ............$\left(1\right)$
$\Rightarrow a+b+c+d+e+f+6=105$
$\Rightarrow a+b+c+d+e+f=99$.............$\left(2\right)$
Plugging the vlaue from $\left(2\right)$ in $\left(1\right)$, we get
$99+g=84$
$g=-15$
Therefore, the number that was replaced was $-15$.
Hence, option $B$ is correct.