Introduction to Fractions

What You’ll Learn

In this lesson you’ll learn what a fraction is, how to read and write one, and the different types of fractions you’ll encounter.

What Is a Fraction?

A fraction represents a part of a whole. Think of cutting a pizza into equal slices. If you cut it into 4 equal slices and eat 1, you’ve eaten $\frac{1}{4}$ of the pizza.

Every fraction has two parts:

• The numerator (top number) - how many parts you have
• The denominator (bottom number) - how many equal parts the whole is divided into

$$\frac{\text{numerator}}{\text{denominator}} = \frac{\text{parts you have}}{\text{total equal parts}}$$

Some examples:

• $\frac{1}{2}$ - one out of two equal parts (half)
• $\frac{3}{4}$ - three out of four equal parts (three quarters)
• $\frac{2}{3}$ - two out of three equal parts
• $\frac{5}{5}$ - five out of five parts, which is the whole thing, so $\frac{5}{5} = 1$

Types of Fractions

A few things worth knowing:

• If the numerator is smaller than the denominator, the fraction is less than 1 (like $\frac{3}{8}$). These are called proper fractions.
• If the numerator equals the denominator, the fraction equals 1 (like $\frac{4}{4}$).
• If the numerator is larger than the denominator, the fraction is greater than 1 (like $\frac{7}{4}$). These are called improper fractions and can be written as mixed numbers: $\frac{7}{4} = 1\frac{3}{4}$.

Mixed numbers combine a whole number with a fraction. To convert an improper fraction to a mixed number, divide the numerator by the denominator:

• $\frac{7}{4}$: $7 \div 4 = 1$ remainder $3$, so $\frac{7}{4} = 1\frac{3}{4}$
• $\frac{11}{3}$: $11 \div 3 = 3$ remainder $2$, so $\frac{11}{3} = 3\frac{2}{3}$

To go the other way (mixed number to improper fraction), multiply the whole number by the denominator and add the numerator:

• $2\frac{1}{5} = \frac{2 \times 5 + 1}{5} = \frac{11}{5}$

Equivalent Fractions

Equivalent fractions are different ways of writing the same amount. $\frac{1}{2}$, $\frac{2}{4}$, and $\frac{3}{6}$ all represent the same value: half.

You get equivalent fractions by multiplying (or dividing) both the numerator and denominator by the same number:

$$\frac{1}{2} = \frac{1 \times 2}{2 \times 2} = \frac{2}{4} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}$$

Simplifying a fraction means dividing both parts by their greatest common factor (GCF). For example, $\frac{6}{8}$: the GCF of 6 and 8 is 2, so $\frac{6}{8} = \frac{6 \div 2}{8 \div 2} = \frac{3}{4}$.

Comparing Fractions

To compare fractions with the same denominator, just compare the numerators: $\frac{3}{7} < \frac{5}{7}$ because 3 is less than 5.

For different denominators, find a common denominator first. Which is bigger, $\frac{2}{3}$ or $\frac{3}{5}$?

• Convert both to fifteenths: $\frac{2}{3} = \frac{10}{15}$ and $\frac{3}{5} = \frac{9}{15}$
• $\frac{10}{15} > \frac{9}{15}$, so $\frac{2}{3} > \frac{3}{5}$

Real-World Applications

Fractions are everywhere in daily life. Recipes call for $\frac{1}{2}$ cup or $\frac{3}{4}$ teaspoon. A sale might be $\frac{1}{3}$ off. You might work $\frac{3}{4}$ of a shift or split a bill into $\frac{1}{4}$ shares. Understanding fractions means you can handle measurements, portions, and splits without second-guessing.

You’ve Got This

Fractions can feel intimidating at first, but they’re just a way of describing parts of something. The numerator counts, the denominator names the size of the parts. Once that clicks, everything else (adding, comparing, simplifying) builds naturally from there. Take it one piece at a time.

Quiz

In the fraction 3/8, what does the 8 represent?

• A.How many parts you have
• B.The whole number
• C.How many equal parts the whole is divided into
• D.The remainder

Which of these is an improper fraction?

• A.2/5
• B.3/3
• C.7/4
• D.1/2

What is 2 1/3 as an improper fraction?

• A.3/3
• B.7/3
• C.5/3
• D.8/3

Which fraction is equivalent to 1/2?

• A.2/3
• B.3/6
• C.2/5
• D.1/3

Simplify the fraction 8/12.

• A.2/3
• B.4/6
• C.3/4
• D.1/2