Expert: Paul Klarreich - 9/24/2008

Question
QUESTION: How do I solve this word problem:\

Player A has a higher batting average than player B for the first half of the baseball season.  Player A also has a higher battng average than player B for the second half of the season.  Is it necessarily true that player A has a higher batting average than player B for the entire season?

ANSWER: Hi, Sarah,

I was unsure of this, so I consulted a well-known mathematician. [I know her very well indeed, having been a part of her life since she was born.]   

Here is our conversation: (It is up to you to figure out who Jeter and A-rod are.)


Grandpa Paul Klarreich wrote:
> Hi, Erica,
>  
> A 'student' asked: If Jeter has a higher batting average than A-rod in
> the first half of the season, and also in the second half, must he
> have a higher average over the whole season?
>  
> Looks obvious, but is this one of those paradoxes?  
>  
> Love,
>  
> Daddy
>  
>  
Dear Daddy,

It is one of those "paradoxes". Suppose for example the records are:

Jeter:
1st half: 100 hits in 300 at bats --  .333
2nd half: 15 hits in 30 at bats   --  .500

A-rod:
1st half: 9 hits in 30 at bats    --  .300
2nd half: 149 hits in 300 at bats --  .497

Then A-rod's batting average is higher even though Jeter beats him in each half. You see the principle.

>> Yes.  Jeter is  115 for 330, A-rod is 158 for 300.

By the way, I had heard the story that when graduate students leave mathematics to study physics, it increases the average IQ in both departments.

Love,
Eri
------------------
A note for the future (and the present):  Normally I don't answer 'private' questions, but this one was interesting enough.  Why don't you resubmit it so everyone can see the answer.


---------- FOLLOW-UP ----------

QUESTION: Thanks for your help, but how do I do this problem if I had to use variables rather than an example?

Answer
Questioner:  Sarah
Private: no
Subject:   

 
Question:  
QUESTION: How do I solve this word problem:\

Player A has a higher batting average than player B for the first half of the baseball season.  Player A also has a higher battng average than player B for the second half of the season.  Is it necessarily true that player A has a higher batting average than player B for the entire season?

ANSWER: Hi, Sarah,

I was unsure of this, so I consulted a well-known mathematician. [I know her very well indeed, having been a part of her life since she was born.]   

Here is our conversation: (It is up to you to figure out who Jeter and A-rod are.)


Grandpa Paul Klarreich wrote:
> Hi, Erica,
>  
> A 'student' asked: If Jeter has a higher batting average than A-rod in
> the first half of the season, and also in the second half, must he
> have a higher average over the whole season?
>  
> Looks obvious, but is this one of those paradoxes?  
>  
> Love,
>  
> Daddy
>  
>  
Dear Daddy,

It is one of those "paradoxes". Suppose for example the records are:

Jeter:
1st half: 100 hits in 300 at bats --  .333
2nd half: 15 hits in 30 at bats   --  .500

A-rod:
1st half: 9 hits in 30 at bats    --  .300
2nd half: 149 hits in 300 at bats --  .497

Then A-rod's batting average is higher even though Jeter beats him in each half. You see the principle.

>> Indeed.  Jeter is  115 for 330, A-rod is 158 for 330, for the season.  What we have is four 'half-season' averages -- in descending order, they are J2, A2, J1, A1.  But A2 is close to J2 and over a much larger number of at-bats, i.e. carring a heavier weight in the calculations.

By the way, I had heard the story that when graduate students leave mathematics to study physics, it increases the average IQ in both departments.

Love,
Eri

---------- FOLLOW-UP ----------

QUESTION: Thanks for your help, but how do I do this problem if I had to use variables rather than an example?

...................................................
Since you are being asked "Is such-and-such always true?" and you are going to answer NO, it is quite appropriate to give a 'counter-example'.  

Did you get this problem with some other conditions attached?