Batting title and walks

Question:  Buster Posey had the best batting average in the National League in 2012, 0.336, 178 hits in 530 at bats.  Andrew McCutchen had the second best batting average in the league – 0.327, 194 hits in 593 at bats.  Had Andrew McCutchen received more walks and made fewer outs he would have had a higher batting average.  How many more walks or fewer outs would have resulted in Andrew McCutchen winning the batting title?

 Answer:  A batting average is hits divided by at bats.  A walk is not at bat.  If Andrew McCutchen had walked more and had fewer at bats his batting average would have been higher.

 

Andrew McCutchen had 194 hits.  Solve this problem by using the definition of batting average to solve for the number of at bats consistent with 194 hits and a 0.336 average.

0.336=(194/AB)

Or

AB=194/0.336

 Or

 AB=577.38

577.38 is the breakeven point.  However, there is no such thing as a fractional at bat.  At AB=577, Andrew McCutchen has a higher batting average than Buster Posey.

Andrew McCutchen actually had 593 at bats.  An additional 16 walks (593-577) would have given Andrew McCutchen the batting title.

This problem could also be solved with algebra.  

X is the number of additional walks

593-X is the number of at bats after X additional walks

From the batting average definition:

0.336=194/(593-X)

or

193=0.336*593-0.336*X

Or

0.336*X=0.336*593-194

or

X=(199.248-194)/0.336

X=15.6

Again, there is no such thing as a fractional walk   Had Andrew McCutchen walked an additional 16 times he would have won the batting title.