Question

when you divide a fraction by a whole number greater than one, how does the quotient compare to the fraction

Answer 1

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 I can 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 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 |||||| 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 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