**Inverse Relationship between Multiplication and Division**

The inverse relationship is a relationship between two numbers in which an increase in the value of one number results in a decrease in the value of the other number. The inverse relationship is also known as negative correlation in regression analysis; this means that when one variable increases, the other variable decreases, and vice versa.

**Multiplication and Division Relationship**

There is an inverse relationship between multiplication and division just like there was between addition and subtraction.

The equation 3 * 7 = 21 has the inverse relationships:

- 21 ÷ 3 = 7
- 21 ÷ 7 = 3

Similar relationships exist for the division. The equation 45 ÷ 5 = 9 has the inverse relationships:

- 5 * 9 = 45
- 9 * 5 = 45

**Explanation**

A multiplicative inverse is reciprocal. What is a reciprocal? A reciprocal is one of a pair of numbers that when multiplied by another number equals the number 1. For example, if we have the number 7, the multiplicative inverse, or reciprocal, would be 1/7 because when you multiply 7 and 1/7 together, you get 1!

**Examples**

Let’s look at a couple examples before proceeding with the lesson.

Example 1: What is the multiplicative inverse of 15? In other words, which number, when multiplied with 15, would give us the number 1 as a result? Let’s solve this in an algebraic way, with x being the unknown multiplicative inverse.

15 * x = 1

x = 1/15

That’s it! It was really that simple! The multiplicative inverse of a number is that number as the denominator and 1 as the numerator. When we multiply 15 and 1/15, we get 1.

Example 2: What is the multiplicative inverse of 1/4? Now, this example is a little different because we are beginning with a fraction. Let’s again solve this algebraically, with x being the unknown multiplicative inverse of 1/4.

1/4 * x = 1

x = 1 / (1/4)

(1/1) / (1/4) = (1/1) * (4/1) = 4

Remember that when you divide fractions, you must flip the numerator and denominator of the second fraction and then multiply. We got 4 as the multiplicative inverse of 1/4. Makes sense, right?

So, the conclusion that we can draw from these two examples is that when you have a whole number, the multiplicative inverse of that number will be that number in fraction form with the whole number as the denominator and 1 as the numerator. When you have a fraction with 1 as the numerator, the multiplicative inverse of that fraction will simply be the denominator of the fraction.

Information Source: