Question

how does multiplying a fraction get a small amount of number and dividing a fraction get a greatest amount of number

Answer 1

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

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24

I can divide 24 by 4 by counting out 6 groups of 4 sticks:

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1 2 3 4 5 6

Since each group contains more than one stick, there are fewer groups than sticks.

Now let's 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

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6

and I break each of them into 4 pieces:

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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 that 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 * 4 = 24 bigger

6 / 1/4 = 24 bigger 24 * 1/4 = 6 smaller