3-3 Divide Whole Numbers by Fractions

Property

Multiplicative Inverse (Reciprocal): For any non-zero real number $a$, there is a reciprocal, $\frac{1}{a}$, such that their product is one.

This property is essential for solving equations and simplifying expressions involving multiplication and division.

Property

If the product of two fractions is 1, the fractions are reciprocals. Another name for a reciprocal is a multiplicative inverse.

Examples

• The reciprocal of $\frac{7}{9}$ is $\frac{9}{7}$.
• The multiplicative inverse of 5 (which is $\frac{5}{1}$) is $\frac{1}{5}$.
• The number of $\frac{2}{5}$s in 1 is the reciprocal of $\frac{2}{5}$, which is $\frac{5}{2}$.

Explanation

Think of a reciprocal as a fraction's 'upside-down' twin! When you multiply a fraction by its reciprocal, they magically cancel out to equal 1. This 'flipping' action is the secret key to making division with fractions super easy. Mastering this move turns confusing division problems into simple multiplications you already know how to solve.

Property

Any whole number $n$ can be written as a fraction without changing its value by using $1$ as the denominator:

Examples