How can i solve this problem about averages?

Question

The average (arithmetic mean) bowling score of $n$ bowlers is $160$. The average of these $n$ scores together with a score of $170$ is $161$. What is the number of bowlers, $n$?

I tried this:

$$X'*n = 160*n = x_1 + \dots + x_n$$

$$X'*n = 161*n = x_1 + \dots + x_n + 170$$

$$\frac{161*n}{160*n} = \frac{x_1 + \dots + x_n + 170}{x_1 + \dots + x_n}$$

Answer 1

Assume the total score of the $n$ bowlers is $S$, then the average is

$$A = \frac{S}{n} = 160\,,\quad (1) $$

The second statement is telling you that the average of these $n$ scores adding to them the score (170) of another player (you will have $n+1$ bowlers) is 161. That translates to

$$ 161 = \frac{ S+170 }{n+1}\,,\quad (2) \,. $$

Now, solve the two equations to get $n$. Solution $(n=9)$

Answer 2

Hint: what is the total score of the original $n$ bowlers? If you add another bowler who hits $170$, what is the total? How many bowlers are there now?