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College Preparatory Mathematics: Help and Review21 chapters | 175 lessons

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

In this lesson, we will look at how multiplication and division are intricately related. We will use this fact to illustrate what repeated subtraction is and how to use it to perform division.

As you probably know, multiplication and division are intricately related. Their relationship can be described by the rule stating that if *a* x *b* = *c*, then *c* / *a* = *b* and *c* / *b* = *a*. Of course, this rule does not apply when *a* or *b* is equal to zero.

To illustrate this, let's consider an example. We know that 7 x 5 = 35. From this, we can deduce that 35 / 7 = 5 and 35 / 5 = 7. We see that in a way, division can be thought of as the opposite of multiplication and vice versa.

Another well-known fact is that multiplication can be thought of as **repeated addition**. That is, *a* x *b* is the same as adding *b* copies of *a* together or adding *a* copies of *b* together. This is illustrated in the image.

For example, consider our example of 7 x 5. We know that 7 x 5 = 35, so based on the rule of repeated addition, it should be the case that adding together 7 copies of 5 or adding together 5 copies of 7 should give us 35.

7 x 5 = 7 + 7 + 7 + 7 + 7 = 35

7 x 5 = 5 + 5 + 5 + 5 + 5 + 5 + 5 = 35

We see that multiplication is the same as repeated addition.

Now let's put our two facts together. We know that multiplication is repeated addition and that division and multiplication are opposites of each other. These two facts along with the fact that subtraction and addition are opposites of each other may have you deducing that division might be thought of as **repeated subtraction**. Well, you are exactly right! Repeated subtraction is a method of performing division the same way that repeated addition is a method of performing multiplication.

Let's explore this concept further. Division has four different components. The number we are dividing by is called the **divisor**. The number we are dividing into is called the **dividend**. The number of times the divisor fits into the dividend is called the **quotient**. Lastly, whatever is left over is called the **remainder**.

We can use repeated subtraction when faced with a division problem to find the quotient and the remainder. Suppose you have 33 pieces of candy to pass out to 5 people. This represents the division problem 33 / 5. You want each person to get an equal amount of candy, so you decide to give 5 pieces of candy at a time (one piece each), until you run out or you don't have enough to give each of the 5 people another piece. Notice that each time you give out 5 pieces of candy, you have to subtract 5 from the total left over, and each person gets one piece of candy. Let's see what happens.

1.) 33 - 5 = 28 Each person now has 1 piece of candy, and there are 28 more to pass out.

2.) 28 - 5 = 23 Each person now has 2 pieces of candy, and there are 23 more to pass out.

3.) 23 - 5 = 18 Each person now has 3 pieces of candy, and there are 18 more to pass out.

4.) 18 - 5 = 13 Each person now has 4 pieces of candy, and there are 13 more to pass out.

5.) 13 - 5 = 8 Each person now has 5 pieces of candy, and there are 8 more to pass out.

6.) 8 - 5 = 3 Each person now has 6 pieces of candy, and there are 3 more to pass out.

At this point, you don't have enough to pass out another piece to each of the people, so you stop there. Each person got 6 pieces of candy, and there are 3 left over. This tells us that when we divide 33 / 5, we get a quotient of 6 and a remainder of 3.

This is how we use repeated subtraction to perform division. This can be rewritten in a more compact form.

33 - 5 - 5 - 5 - 5 - 5 - 5 = 3

We see we subtracted 5 six times from 33 to get 3. Thus, the quotient is 6 and the remainder is 3. In general, if we are dividing *a* / *b*, the number of times we subtract *b* from *a* to get to zero or a number between zero and *b* is the quotient, and the number leftover is the remainder. This is illustrated in the image.

Consider one more quick example. Suppose we wanted to divide 52 / 13. To perform the division using repeated subtraction, we simply subtract copies of 13 from 52 until we get zero or a number between zero and 13.

52 - 13 - 13 - 13 - 13 = 0

We see that when we subtract 4 copies of 13 from 52 we get zero. Therefore, 52 / 13 = 4 with a remainder of 0.

Multiplication and division go hand in hand. They can be thought of as opposites of each other. Multiplication can be thought of as **repeated addition**, and division can be thought of as **repeated subtraction**. In a division problem, there is a divisor, a dividend, a quotient, and a remainder. When we use repeated subtraction to perform the division *a* / *b*, *a* is the dividend, *b* is the divisor, the number of times we subtract *b* from *a* to get to zero or a number between zero and *b* is the quotient, and what's leftover is the remainder. Repeated subtraction is another way to perform division, so it is great to add this process to our math toolbox.

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College Preparatory Mathematics: Help and Review21 chapters | 175 lessons

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