Question

The average age of a group of persons going for a picnic is $16$ years. Twenty new persons with an average age of $15$ years join the group on the spot due to which their average becomes $15.5$ years. The number of persons initially going for the picnic is

• A. $5$
• B. $10$
• C. $20$
• D. $30$

Answer 1

Answer: (C)

$20$

Let the initial number of persons be $n$

Given that the average age of $n$ number of people is $16$ years

Therefore, $16=\frac{\text{sum of the ages of n people}}{n}$

$⟹\text{sum of the ages of n people}=16n$ -------(1)

Given that the average of the $20$ new people is $15$ years

Therefore, $15=\frac{\text{sum of the ages of 20 new people}}{20}$

$⟹\text{sum of the ages of 20 new people}=300$ -------(2)

Given that after adding $20$ new persons the total average becomes $15.5$ years

Therefore, $15.5=\frac{\text{sum of the ages of n people + sum of the ages of the 20 new people}}{20+n}$

$⟹\text{sum of the ages of n people + sum of the ages of the 20 new people}=15.5(20+n)$ -------(3)

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

$16n+300=310+15.5n$

$⟹16n-15.5n=310-300$

$⟹0.5n=10$

$⟹n=20$

Therefore number of persons initially going for the picnic is $20$