An Average Bowl

Relevant wiki: Mean (Average)

James’ score of $199$ is $199 - 177 = 22$ points above his previous average. James raised his average $1$ point. Therefore, his latest game with the $199$ score must have been his $22^{nd}$ game. To raise his average $2$ points in his $23^{rd}$ game he must bowl $2 \times 23 = 46$ points above his $178$ average. He must bowl $178 + 46 = 224$ in his next game.

$\therefore$ James must bowl $\boxed{224}$ in his next game to move his average from $178$ to $180$ .

Suppose James scored a total of $S$ points in $n$ games, and obtained an average of $177$ points. Therefore, his total score is given by the following expression:

$S = 177n$ .

Suppose that in his next game he scored $199$ points and raises his average to $178$ points. An expression for his original total score $S$ in terms of $n$ is given by:

$S + 199 = 178(n+1)$ .

Rearranging the previous equation gives:

$S = 178n - 21$

Equating this with the first equation yields:

$177n = 178n - 21$

Thus, $n = 21$ is the solution to this equation (this will be handy later on).

Let $x$ be the number of points scored in the following game such that the average after $n+2$ games is $180$ . His original total score, $S$ , is expressed in terms of $n$ as follows:

$S + 199 + x = 180(n+2)$

Since $S=177n$ , it is observed that:

$177n + 199 + x = 180n + 360$

Rearranging this equation and substituting the result of $n=21$ into the previous equation:

$x = 3n + 161 = 63 + 161 = 224$

as required.