Ok, this one was bugging the crap outta me.. so as I usually do I did research until I understood the problem. Math needs to make sense, and it seems like this wasn't doing that. The question really is why does the division of fractions produce a larger number?
What we normally think about is that when you divide, your answer should be smaller. Twenty-four divided by six is four. Four is lower than twenty-four.
But when you divide fractions, like 1/4 divided by 1/4, the answer is a
larger number than 1/4. Why is this? I try telling myself that fractions are less than 1 and therefore the math may be different. But in math the answer always must make sense. This was simply not making sense
Then I found an answer that makes sense. It's important for our math to make sense. If our number sense disagrees with the answer, either the answer or our sense has to be fixed. If we leave them both as they are, we'll be in trouble. In this case, what seems natural and obvious when we're used to working with whole numbers has to be modified when we introduce fractions.
Let's think about why dividing by a whole number produces a smaller
result. Dividing 24 by 4 means I want to find out how many 4's it
takes to make 24. If I have 24 sticks
||||||||||||||||||||||||(24)
And I divide 24 by 4 by counting out 6 groups of 4 sticks:
||||(1) ||||(2) ||||(3) ||||(4) ||||(5) ||||(6)
Since each group contains more than one stick, there are fewer groups
than sticks.
Now I divide 6 by 1/4. That means I want to find out how many 1/4's it takes to make 6. If I have 6 sticks
||||||(6)
and I break each of them into 4 pieces:
||||(1) ||||(2) ||||(3) ||||(4) ||||(5) ||||(6)
I find that there are 24 quarter-sticks. Since each stick has been
turned into 4 pieces, there are more pieces than sticks - the result
of the division is greater than the original number of sticks.
(By the way, this also helps to explain why dividing by a fraction is the same as multiplying by the reciprocal.)
You're exactly right... when you divide by a number less than one,
things turn upside down, and the quotient is now larger than the
dividend.
In fact, you're multiplying by the reciprocal, which is a number greater
than one, and you therefore increase the number.
Similarly, multiplying by a number greater than one increases a number, and multiplying by a number less than one decreases it:
24 / 4 = 6 smaller.......6 x 4 = 24 bigger
6 / 1/4 = 24 bigger...... 24 x 1/4 = 6 smaller
So there you have it.. It's actually very simple once you understand the concept behind it... now it makes sense. You see, you can teach an old dog new tricks.
Yes RKC, I was wrong. It happens from time to time. But sometimes it's only by being wrong that you learn why something is right.
