Big family

A certain number of people met at the family celebration - their average age was three times the number present. Then came my grandfather, who was 75 years old, and the average age was again three times the number present. How many people are celebrating right now?

Correct answer:

n =  13

Step-by-step explanation:

$$\begin{gathered} 3(n-1)=s/(n-1) \\ 3n=(s+75)/n \\ 3n^2-6n+3=s \\ 3n^2=s+75 \\ 3n^2=(3n^2-6n+3)+75 \\ 0=-6n+3+75 \\ 0=-6\cdot n+3+75 \\ 6n=78 \\ n=\frac{78}{6}=13 \\ n=13 \\ \text{ Verifying Solution: } \\ s=3\cdot n^2-75=3\cdot 13^2-75=432 \\ {a}_1=s/(n-1)=432/(13-1)=36 \\ {A}_1=3\cdot (n-1)=3\cdot (13-1)=36 \\ {a}_2=(s+75)/n=(432+75)/13=39 \\ {A}_2=3\cdot n=3\cdot 13=39 \end{gathered}$$

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