How to work out the age of X person when the mean age of Y amount of people is given

Question

At a nursery, the mean age of 4 friends is 31 months. Katy joins the nursery, after which the mean age of all 5 children is 30 months.

How do I work out Kathy's age?

---

I tried assuming that if all four children were exactly 31 months, then all you have to do is figure how old you have to make Katy so that when they are all added together and divided you would get the age 30. This gave me 26, but I was thinking that there must be a formula to do it.

Answer 1

Call $S$ the sum of the ages of the 4 children, and $x$ the age of the 5th.
We have $\frac{S}{4}=31$ and $\frac{S+x}{5}=30$
So that $4 \times 31 = 5 \times 30 -x$
And finally $x=150-124=26$

Answer 2

let A B C D be the age of 4 kids

then $(A+B+C+D)/4=31/12$

or, $(A+B+C+D)=31/3$

let age of 5th kid be E

so $(A+B+C+D+E)/5=30/12$

or,$(A+B+C+D+E)=150/12$

so putting $31/3$ in place of $(A+B+C+D)$ we get $E=13/6$ year or $26$ months