Fractions can be written using two whole numbers separated by a bar. The number above the bar is called the ___________ and the number below the bar is called the ____________.

mixed number, improper fraction

Fractions Lesson: Adding, Subtracting, Multiplying, and Dividing

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Updated: Apr 3, 2025

Lesson

Fractions are a fundamental concept in mathematics, representing parts of a whole. They are used in everyday situations, from sharing food to measuring ingredients in a recipe. Understanding fractions is essential for working with numbers and solving various types of math problems.

In this lesson, we will explore fractions, their types, and how to perform operations such as addition and subtraction with fractions. You will also learn how to work with mixed numbers and improper fractions, and how to convert between them.

What is a Fraction?

A fraction is a way to represent part of a whole. It consists of two numbers:

Numerator (the top number): The number of parts you have.
Denominator (the bottom number): The total number of equal parts the whole is divided into.

For example, 1/2 means you have 1 out of 2 equal parts, while 3/4 means you have 3 out of 4 equal parts.

Types of Fractions:

Proper Fractions: The numerator is smaller than the denominator. Example: 3/4.
Improper Fractions: The numerator is equal to or larger than the denominator. Example: 7/4.
Mixed Numbers: A whole number combined with a fraction. Example: 1 1/2.

Mixed Numbers and Improper Fractions

A mixed number is a number that combines a whole number with a fraction. For example, 1 1/2 is a mixed number, where 1 is the whole number and 1/2 is the fraction.

An improper fraction is a fraction where the numerator is equal to or greater than the denominator. For example, 7/4 is an improper fraction because the numerator (7) is larger than the denominator (4).

Converting Improper Fractions to Mixed Numbers:

To convert an improper fraction to a mixed number, divide the numerator by the denominator. The quotient is the whole number, and the remainder becomes the numerator of the fraction part.

Example: 7/4

Divide 7 by 4: The quotient is 1, and the remainder is 3.
The mixed number is 1 3/4.

Converting Mixed Numbers to Improper Fractions:

To convert a mixed number back to an improper fraction, multiply the whole number by the denominator, then add the numerator.

Example: 1 1/2

Multiply the whole number (1) by the denominator (2): 1 × 2 = 2.
Add the numerator (1): 2 + 1 = 3.
The improper fraction is 3/2.

Take This Quiz:

Fraction Math Quiz

Adding Fractions

Adding fractions requires that the fractions have the same denominator. If the fractions have the same denominator, you can simply add the numerators and keep the denominator the same.

Example 1:

1/4 + 2/4
Since the denominators are the same, just add the numerators:
1 + 2 = 3, so 1/4 + 2/4 = 3/4.

Adding Fractions with Different Denominators:

If the fractions have different denominators, you need to find the least common denominator (LCD) before adding them. The LCD is the smallest number that both denominators can divide into.

Example 2:

1/3 + 1/4
The LCD of 3 and 4 is 12. Convert both fractions to have a denominator of 12:

1/3 = 4/12
1/4 = 3/12

Now, add the fractions:
4/12 + 3/12 = 7/12

Subtracting Fractions

Subtracting fractions is similar to adding fractions. If the fractions have the same denominator, subtract the numerators and keep the denominator the same.

Example 3:

5/6 - 2/6
Since the denominators are the same, just subtract the numerators:
5 - 2 = 3, so 5/6 - 2/6 = 3/6, which simplifies to 1/2.

Subtracting Fractions with Different Denominators:

To subtract fractions with different denominators, find the least common denominator (LCD) and convert the fractions to have the same denominator.

Example 4:

3/5 - 1/3
The LCD of 5 and 3 is 15. Convert both fractions to have a denominator of 15:

3/5 = 9/15
1/3 = 5/15

Now, subtract the fractions:
9/15 - 5/15 = 4/15

Multiplying Fractions

To multiply fractions, you simply multiply the numerators and the denominators.

Example 5:

2/3 × 3/4
Multiply the numerators: 2 × 3 = 6
Multiply the denominators: 3 × 4 = 12
So, 2/3 × 3/4 = 6/12, which simplifies to 1/2.

Dividing Fractions

To divide fractions, multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by swapping its numerator and denominator.

Example 6:

3/4 ÷ 2/5
To divide, multiply 3/4 by the reciprocal of 2/5, which is 5/2: 3/4 × 5/2 = 15/8

The result is 15/8, which is an improper fraction. You can convert it to a mixed number: 15/8 = 1 7/8.

Take This Quiz:

$Fractions - Quiz$

Fractions - Quiz

Word Problems Involving Fractions

Let's practice applying what you've learned to solve word problems:

Example 1:

Problem: Sarah has 8 apples, and she wants to give 1/4 of her apples to each of her friends. How many apples will each friend get?

Solution: 8 ÷ 4 = 2 apples per friend.

Example 2:

Problem: A pizza is cut into 8 slices. If John eats 3/8 of the pizza, how much pizza is left?

Solution: 8 - 3 = 5 slices are left. Therefore, 5/8 of the pizza remains.

Simplifying Fractions

Simplifying fractions is the process of reducing them to their lowest terms. To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD).