Fractions Explained: A Step-By-Step Beginner's Guide
Half a pizza. Three-quarters of a tank of gas. Fractions describe the world. This lesson builds your understanding from scratch — no prior fraction knowledge needed.
Learning Objectives
Define numerator and denominator and explain what each represents
Compare fractions to determine which is larger
Add and subtract fractions with the same denominator and with different denominators
Simplify fractions to their lowest terms using the greatest common factor
Convert between improper fractions and mixed numbers
Prerequisites
Before this lesson, you should be comfortable with:
Basic division — fractions and division are closely related. If you need a refresher, try our Long Division lesson first.
Multiplication tables — you need them to find common denominators.
Greatest Common Factor (GCF) — understanding what factors are will help when simplifying.
What is a Fraction?
A fraction represents part of a whole. Imagine you cut a pizza into 8 equal slices and eat 3 of them. You ate 3 out of 8 equal pieces — written as the fraction 3/8.
Every fraction has two parts separated by a horizontal line (called a vinculum):
Numerator (top number) — how many parts you have
Denominator (bottom number) — how many equal parts the whole is divided into
Think of the denominator as the name of the fraction type: eighths, thirds, fifths. The numerator just counts how many of that type you have. So 5/8 means "five eighths" — five pieces out of a whole divided into eight equal pieces.
Types of Fractions
Proper fraction — numerator < denominator. The value is less than 1. Examples: 1/2, 3/4, 7/10.
Improper fraction — numerator ≥ denominator. The value is 1 or greater. Examples: 5/4, 9/3, 7/7.
Mixed number — a whole number plus a proper fraction. Example: 2 and 3/4 (written as 2¾). Every improper fraction can be converted to a mixed number.
Comparing Fractions
To compare fractions, it helps to have the same denominator. Once denominators match, the fraction with the larger numerator is larger.
Same denominator — easy comparison
Which is larger: 3/8 or 5/8? Both have denominator 8 (same-size slices). Five slices is more than three slices, so 5/8 > 3/8.
Different denominators — find the LCD first
Worked Example: Compare 2/3 and 3/4
Step 1 — Find the Least Common Denominator (LCD):
Multiples of 3: 3, 6, 9, 12, 15 ...
Multiples of 4: 4, 8, 12, 16 ...
LCD = 12
Step 2 — Convert each fraction to twelfths:
2/3 = 8/12 (multiply top and bottom by 4)
3/4 = 9/12 (multiply top and bottom by 3)
Step 3 — Compare numerators:
8/12 vs 9/12 → 9/12 is larger
Conclusion: 3/4 > 2/3
Adding and Subtracting Fractions
Same denominator
Add (or subtract) the numerators. The denominator stays the same.
Example: 2/7 + 3/7
= (2 + 3) / 7
= 5/7
Different denominators
Find the LCD of the two denominators.
Convert each fraction to an equivalent fraction with the LCD as the denominator.
Add or subtract the numerators. Keep the denominator.
Simplify the result if possible.
Example: 1/3 + 1/4
LCD of 3 and 4 = 12
1/3 = 4/12 (×4 on top and bottom)
1/4 = 3/12 (×3 on top and bottom)
4/12 + 3/12 = 7/12
7/12 is already in lowest terms (7 is prime, does not divide 12).
Answer: 7/12
Simplifying Fractions
A fraction is in its simplest form (also called lowest terms) when the numerator and denominator share no common factors other than 1. To simplify, divide both numbers by their Greatest Common Factor (GCF).
The quotient becomes the whole-number part of the mixed number.
The remainder becomes the new numerator. The denominator stays the same.
Example: Convert 17/5 to a mixed number
17 ÷ 5 = 3 remainder 2
Mixed number: 3 and 2/5 (three and two-fifths)
Check: 3 × 5 + 2 = 17 ✓
Practice Problems
Try each problem before peeking at the answer.
Simplify 12/16.
GCF of 12 and 16 = 4. 12÷4=3, 16÷4=4. Answer: 3/4
Add 2/5 + 1/3.
LCD=15. 2/5=6/15, 1/3=5/15. 6+5=11. Answer: 11/15
Which is larger: 5/6 or 7/9?
LCD=18. 5/6=15/18, 7/9=14/18. 15>14 so 5/6 is larger
Convert 23/7 to a mixed number.
23÷7=3 R2. Answer: 3 and 2/7
Subtract 3/4 − 1/6.
LCD=12. 3/4=9/12, 1/6=2/12. 9−2=7. Answer: 7/12
Common Mistakes to Avoid
Adding denominators instead of finding the LCD. Never do 1/3 + 1/4 = 2/7. The denominator is the size of each piece, not a count to add.
Forgetting to simplify your answer. 6/8 is correct but 3/4 is the expected simplified form. Always check whether the GCF is greater than 1.
Multiplying only the numerator when converting to an equivalent fraction. When you find the LCD, you must multiply both the numerator and denominator by the same number. Multiplying only one changes the fraction's value.
Confusing numerator and denominator roles. The denominator names the fraction type (eighths, thirds). The numerator counts. Keep this distinction clear and most fraction errors disappear.
Free MIT math resources that build on fraction concepts.
Frequently Asked Questions
What is a fraction?
A fraction represents a part of a whole. The denominator tells you how many equal parts the whole is split into; the numerator tells you how many of those parts you have.
How do you add fractions with different denominators?
Find the least common denominator (LCD), convert each fraction to that denominator, then add the numerators. Keep the denominator the same.
What is a proper vs improper fraction?
A proper fraction (e.g., 3/4) has its numerator smaller than the denominator. An improper fraction (e.g., 7/4) has its numerator equal to or larger than the denominator and can be written as a mixed number.
How do you simplify a fraction?
Divide both the numerator and denominator by their greatest common factor. For 8/12: GCF=4, so 8÷4 / 12÷4 = 2/3.
What comes after fractions?
Decimals (fractions in base-10 notation) and then percentages are the natural next topics.