## Sunday, February 17, 2013

### Batting Title 2012 Problem One.

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.  How many additional hits would have resulted in Andrew McCutchen winning the National League batting crown based on his 593 at bats?

Answer:  In order to have a batting average of 0.336 in 593 at bats Andrew McCutchen would have needed 199.2 hits.  (0.336x593=199.2.) Of course, it is impossible to get 0.2 of a hit, so the minimum number of hits that would result in Andrew McCutchen obtaining a batting average that was higher than Buster Posey’s average was 200 hits.  Andrew McCutchen already had 193 hits; therefore, 7 more hits would have given him the batting crown.

Another way to solve this problem is through an algebraic equation generated by the batting average formula.  Define X as the additional number of hits needed for Andrew McCutchen to have tied Buster Posey for the batting title at 0.336.  Plug X into the batting average formula

0.336=(193+X)/593

Simplify

193+X=0.336*593=199.2

Or

X=199.2-193=6.2.

Again, there is no such thing as 0.2 of a hit.

Andrew McCutchen needed 7 more hits to win the batting crown.

This analysis is based on 593 at bats.  There are of course other ways Andrew McCutchen might have won the title.  He could have had fewer at bats if walked more frequently.  This will be considered in tomorrow’s problem.