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January 31st, 2007, 04:32 AM   #1
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Fraction addition-where is the hole?

One of my students asked me why, when adding fractions, we can't
just add the numerators and denominators, and he used this as his
example: "In the first game of a double-header, Joe gets 1 hit
in 3 at bats. In the second game, he gets 2 hits in 5 at bats.
That's 3 hits in 8 at bats, and 1/3 + 2/5 = 3/8."
I know this isn't right, but I'm not sure how to explain why.
Help!
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January 31st, 2007, 07:41 AM   #2
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If you're a teacher and can't explain why we add two simple fractions in a certain fashion , you've got some problems.

1/3 + 2/5 = 11/15

The reason is simple:

1/3=5/15
2/5=6/15

You can't add two fractions that don't have the same denominator, for the reason that they aren't the same thing. 1+2 just doesn't ever equal 4.

Just cut up a pizza to represent this problem, and the reason becomes clear. 1/3 + 2/5 can never equal 3/8, it's impossible. The rules for adding fractions just reflect this.
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January 31st, 2007, 11:09 AM   #3
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Simply adding the numerators and denominators would lead to, for example, 1/2 + 0/2 = 1/4. Clearly, such a method has major limitations, even if not termed "addition".
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January 31st, 2007, 02:03 PM   #4
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Here's why

Let's change the numbers. (I know nothing about baseball, so forgive me if the numbers are unrealistic.)
In the first game Joe gets 1 hit in 2 at bats.
In the second game the same happens.
So the hit rate is 1/2 in the first game, in the second game, and over all.
We see that the over all hit rate is not the same as the sum of the two,
since the sum of two halves is 1.
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January 31st, 2007, 03:55 PM   #5
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See http://en.wikipedia.org/wiki/Farey_sequence for an example where fractions are combined as described above. It's not called addition, of course, but the mathematics is entertaining.
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January 31st, 2007, 07:07 PM   #6
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---------clarification---------

come on guys ... i know all those silly fraction addition rules

wat i am askin is that, wat is wrong in my student's interpretation of
1/3 + 2/5 = 3/8 in the given example??

I am askin y is that 3/8 seems to b correct here since he has totally hit 3 shots out of 8 which is indeed correct!!

Also wat is the significance of 3/8 in this context (and not in pizza problems to understand addition) and also wat is the meaning of 11/15 in this context.

Hope u guys understand me.. else will try to elobarate stil ... thx
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January 31st, 2007, 08:02 PM   #7
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It's not that your student is wrong; rather, it's that his way of combining fractions doesn't deserve to be called addition. The fact that it involves adding doesn't imply that the overall effect is simple addition of fractions, which has already been defined in a different way.

Mathematics allows you to define your own way of combining values to suit the problem at hand. When you want to combine scores of the type you mentioned, the appropriate operation isn't simple addition.
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January 31st, 2007, 08:24 PM   #8
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---skip jack-------

hi skipjack,

i too fully agree and feel that the way of combinin/addin of the scores in that particular way isn't correct ... then i wan to know wat does that operation mean in the given context !
bcoz even in all kinds of paradox,fallacies there wil b a mistake(like divide be 0 etc.) , then wat is the mistake if any here ??

hope u get me ..

arun
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February 1st, 2007, 04:41 AM   #9
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In each game, the fraction gives the proportion of Joe's attempts that are hits. Such a proportion could be called a success rate. Joe can use these proportions to calculate the correct corresponding proportion for the two games combined, provided that he retains the original numerators and denominators and separately adds the numerators and denominators as described. The method works, but there is no reason to call it addition.

When, in contrast, one simply adds proportions using normal addition, one is always trying to calculate a total quantity rather than an overall proportion and the proportions added must relate to equal quantities.

If you knew only that in the first game 1/3 of Joe's attempts were hits and in the second game 2/5 of Joe's attempts were hits, but didn't know how many attempts Joe made in each game, you wouldn't have enough information to be able to find either Joe's overall success rate or his total number of hits, so any method of combining the proportions would be incorrect.
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February 1st, 2007, 04:47 AM   #10
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Quote:
come on guys ... i know all those silly fraction addition rules
Oh, I'm sure you probably know all those fraction addition rules better than I do, but I just couldn't resist ...


Here's the difference between your fraction addition method and your student's addition method, rephrased to make it easier to understand than the ball problem:

1 out of 3 pieces of a pie were eaten and then 2 out of 5 pieces of another pie were eaten. What fraction of a pie was eaten?

1/3 + 2/5 = 11/15

1 out of 3 pieces of a pie were eaten and then 2 out of 5 pieces of another pie were eaten. What fraction of the number of pieces of pie were eaten?

(1+2)/(3 + 5)= 3/8


Hope this helps.
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