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## Backwards Batting Average

An American League baseball player noticed that after he got his 50th career hit against the Yankees, his career batting average rose exactly 0.0005. What was his most likely batting average before this last hit?

Note: In this instance, no round-off is involved, either before or after the hit.

Source: Eugene Sard's "Quickie A877," Mathematics Magazine, April 1998, pp. 143, 150-151

Hint: One approach is guess-and-check...or even a systematic spreadsheet routine....but, think about the relationships. Can you represent them by an equation that can be manipulated?

Solution Commentary: Author suggests this solution, with sprinkles of my commentary.

Prior to the hit, let H = number of hits and A = number of official at bats. Then, (H+1)/(A+1)=(H/A)+(1/2000). Solving, we get H = [A(1999-A)]/2000.

Since H>0, the factor 16 divides either A or 1999-A and 125 divides the other. (NOTE: Think about why this is necessary before continuing.) Thus, write A and 1999-A as 16M and 125N (the order irrelevant).

Since 1999 = 16M + 125N, we know that both 0 < N < 16 and 125N ≡ 1999 (mod 16) [or N ≡ -5(mod 16) by congruence rules]. Thus, N = 11 and M = 39, which means there are two possibilities for A and H:

• If A = 1375 and H =429, then the batting average was 0.312
• If A = 624 and H = 429, then the batting average was 0.6875
• The first option seems more reasonable, given the number of bats that have occurred (i.e. 50th career hit against one team).