The average age of a group of persons going for a picnic is $16.75$ years. $20$ new persons with an average age of $13.25$ years join the group on the spot due to which the average of the group becomes $15$ years. Find the number of persons initially going for the picnic.
I assumed number of people in initial group to be $x$ and $$avg.age(x group)=16.75 years$$ number of people in addition to be $y$ which will be $20$ and $$avg.age(y group)=13.25 years$$
Now the groups are added let that be $x+y$ and $avg.age(x+y)=15years$ We do know that sum of all the quantities in a list by number of quantities will give us the average. I took the sum of average of both the groups which is 15
$$(avg.age(x)+avg.age(y))/(20+x)=15$$ then $$x=18$$ which is a wrong answer.
Please provide me with right solution and explanation.
Let the original amount of people be $x$, so there are $x+20$ people in the end. Then the total age of all the people will be $16.75x+13.25\times20$, so the average of ages of all the people is
\begin{align} \frac{16.75x+265}{x+20}&=15\\ 16.75x+265&=15(x+20)\\ 16.75x+265&=15x+300\\ 1.75x&=35\\ 0.25x&=5\\ x&=20 \end{align}
Therefore, there are originally $20$ people at the picnic.