Reasoning In Multiplication & Division

Instructor: Michelle Vannoy
Did you know that sometimes when you multiply two numbers, the product is smaller than the larger original number? And, sometimes, dividing two numbers results in a quotient greater than the original numbers being divided! This lesson will explain this unusual phenomenon.

When Multiplying Produces a Smaller Number

Most of the time when we think about multiplying, we think about the product of the two numbers being larger than either number. For example when we multiply 5 and 6, the product (result) is 30, which is larger than 5 or 6.

This is a normal circumstance. Multiplying numbers makes them larger. While this is the case most of the time, there are two instances where multiplying will make a number smaller:

  1. when you multiply a number by a positive fraction less than 1
  2. when you multiply a number by a positive decimal less than 1

Even though these are two separate instances, they are the same. A decimal is another way of writing a fraction. Therefore, any number that is between zero and one. Like 0.5, or 1/2, which are the same number.

Why does multiplying by a fraction or decimal less than 1 give me a smaller number? Let's reason through multiplying with fractions.

Multiplying With Fractions

Multiplying with fractions is a very simple thing to do. You just multiply across the top to get the new numerator and then multiply across the bottom to get the new denominator.

Multiplying with Fractions

Did you notice that when you multiply a fraction, you end up with a division problem? 6 / 2 is the same thing as 6 divided by 2, and the final answer is 3. Since a decimal and a fraction are the same things just written in two different forms, the same concept applies. This is why, when you multiply by a fraction or a decimal, your result is smaller than your original number.

When Division Produces a Larger Number

Just like multiplying by a fraction or a decimal produces a smaller number, dividing by a fraction or a decimal produces a larger number. Think about it like this; dividing is taking some objects and placing them into equal groups.

For example, say you have 12 objects, and you divide them into 3 equal groups so that each group has four objects in it. When dividing by a fraction we are asking ourselves how many jumps of that fraction would it take to get to the whole number on a number line.

Let's illustrate the division problem 3 divided by 1/3. With this problem, we are determining how many groups of 1/3 are in 3. There are nine 1/3 jumps between 0 and 3. In other words, there are nine thirds in 3.

When 3 is divided by 1/3, there are nine one third jumps to get from 0 to 3
Number line Thirds

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