Lesson 9: Adding, Subtracting, and Multiplying Fractions

New Concept

When two numbers have a product of 1, they are reciprocals. This relationship is described by the Inverse Property of Multiplication.

Inverse Property of Multiplication

if $a$ is not 0.

What’s next

This lesson is the starting point. Next, you’ll work through examples on adding, subtracting, multiplying, and using reciprocals to solve for missing numbers.

Property

When adding fractions that have the same denominators, we add the numerators and write the sum over the common denominator.

Examples

To add $\frac{1}{7} + \frac{2}{7} + \frac{3}{7}$, we add the numerators: $1+2+3=6$. The sum is $\frac{6}{7}$.
Adding mixed numbers works the same way: $1 \frac{1}{8} + 1 \frac{1}{8} = (1+1) + (\frac{1}{8}+\frac{1}{8}) = 2 \frac{2}{8}$.
How much is $\frac{1}{4}$ of a dollar plus $\frac{3}{4}$ of a dollar? $\frac{1}{4} + \frac{3}{4} = \frac{4}{4}$, which equals one whole dollar!

Explanation

Imagine adding pizza slices! If your fractions share a denominator, it means all the slices are the same size. You just have to count up the numerators to find out how many slices you have altogether. The size of the slice, the denominator, stays the same. Simple as that!

Property

To multiply fractions, multiply the numerators to find the product's numerator, and multiply the denominators to find the product's denominator. The word "of" often means to multiply: $\frac{1}{2} \text{ of } \frac{1}{2} = \frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$.

Examples

What is $\frac{1}{2}$ of $\frac{1}{3}$? We multiply straight across: $\frac{1 \times 1}{2 \times 3} = \frac{1}{6}$.
To find $\frac{1}{2} \cdot \frac{3}{4} \cdot \frac{1}{5}$, multiply all numerators and all denominators: $\frac{1 \cdot 3 \cdot 1}{2 \cdot 4 \cdot 5} = \frac{3}{40}$.
If you shade $\frac{1}{2}$ of a rectangle, then shade $\frac{1}{4}$ of that shaded part, you just found $\frac{1}{2} \times \frac{1}{4} = \frac{1}{8}$.

Explanation

Multiplying fractions is like finding a piece of a piece. If you take half of a cookie, and then take half of that piece, you end up with a quarter of the original cookie. You're making the piece smaller! Just multiply the tops together and the bottoms together. Top times top, bottom times bottom!

Property

When you invert a fraction by switching its numerator and denominator, you create its reciprocal. The product of a number and its reciprocal is always 1.

Examples

The reciprocal of $\frac{3}{5}$ is $\frac{5}{3}$. Let's check: $\frac{3}{5} \cdot \frac{5}{3} = \frac{15}{15} = 1$.
The reciprocal of the whole number 3 (which is $\frac{3}{1}$) is $\frac{1}{3}$. And their product is $3 \cdot \frac{1}{3} = 1$.
How many $\frac{3}{4}$s are in 1? The answer is the reciprocal of $\frac{3}{4}$, which is $\frac{4}{3}$.

Explanation

Every fraction has an upside-down twin called a reciprocal! It's like a secret identity. The coolest part is when a number and its reciprocal are multiplied, they magically turn into 1. This "canceling out" effect is a superhero power in algebra, helping you solve for missing numbers in equations. It’s a trick worth remembering!