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I am helping my son with his math homework. Neither of us can make sense of this problem but we can solve all the others in the chapter. Is the answer key wrong?

The winning pitcher faced 32 batters during the game. If the opposing team had only 9 players, how many of those players did the pitcher face 4 times?

The answer key says 5.

How??

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  • 9*3 = 27, so all 9 batters faced the pitcher at least 3 times. 32-27 = 5, so the first 5 batters faced the pitcher one additional time. Thus, 5 batters faced the pitcher 4 times. – Henry Apr 14 '21 at 23:24
  • The pitcher will face each player the players in the order of player1, player2, player3, ... player 9, then wrap around start back again at the front of the line-up. This is, of course, ignoring pinch-hitters and pitcher changes. So... the pitcher faces some number of the opposing team $n$ times, and the remaining $n+1$ times each. You will find that this corresponds to $3\cdot 9 + 5$... that is, he pitched to $5$ people four times each and the remaining $4$ people three times each. – JMoravitz Apr 14 '21 at 23:25
  • How did we find $3$ and $4$? That would be $\left\lfloor \frac{32}{9}\right\rfloor$ and $\left\lceil\frac{32}{9}\right\rceil$ – JMoravitz Apr 14 '21 at 23:27

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The opponents batted around three times. That accounts for $27=3(9)$ batters faced. But $32-27=5$, so only those five got a fourth turn at bat.

Is this really a "ratio problem"? I think it is. Mathematically, we perform a "division with remainder", which in some places is written

$$32\,\div\,9\,\,\,=\,\,\,3\,\,R\,\,5.$$

This can also be viewed as a clock-type arithmetic, such as shown here: What is 8 times 9 in 12-hour clock arithmetic?

Mathematicians would also talk about finite cyclic groups, link below. https://en.wikipedia.org/wiki/Cyclic_group

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