What would the bowler have to average for the rest of the season to bring his 90-game average score up to 170?
I've been working on this problem for at least 3 hours and I can't seem to figure it out. I tried using sigma notation and then get stumped from there. Am I supposed to be using sigma notation for this problem, or is it simply just an average/mean problem?
The 'standard' way to think about this is to say: the bowler needs an average of $170$ over $90$ games, so the bowler needs $90 \cdot 170 = 15300$ points total. So far, the bowler has managed $21 \cdot 164 = 3444$ points, so the bowler still needs $15300-3444=11856$ points, meaning that for the remaining $90-21=69$ games, the bowler needs $\frac{11856}{69} \approx 171.8$ points on average per game.
But here is another way to think about this:
For those first $21$ games the bowler was $6$ points behind the goal of getting $170$ points. So, the bowler 'lost' $6 \cdot 21 = 126$ points that the bowler needs to make up in the remaining $90-21=69$ games. That is, the bowler needs to make up $\frac{125}{69} \approx 1.8$ points per game by getting that many points over $170$ on average. So that would mean the bowler needs to bowl $\approx 170+\frac{125}{69} \approx 171.8$ points per game for the remaining season on average.
I not only like this approach of 'losing ground' and 'gaining ground' conceptually, but as you can see it sometimes simplifies the math in that you don't work with as big of numbers ... Indeed, I have found that there are many instances where the math is simple enough with this method that you can do this in your head, and fairly quickly too.
For example, if it would have been an $84$-game season, then there would have been $63$ games remaining, which is $3$ times the number of past games, so with an average 'loss' of $6$, the bowler should make an average 'gain' of $2$, so the bowler should average $172$ for the remaining games ... indeed, even though the numbers with $90$ total games do not work out to nice whole numbers, I was immediately able to tell that the bowler should average a little under $172$ just eye-balling the numbers. So this is a powerful method!