Fractions Explained: From Addition to Simplification

The One Math Skill That Shows Up Everywhere

A recipe calls for 2/3 cup of flour, but you want to make half a batch. A contractor quotes a job at 3/4 of a day's rate. A nurse calculates a drug dosage as 1 1/2 tablets. Fractions appear in every domain that involves dividing something that doesn't divide evenly. Yet surveys of adult numeracy show that nearly 1 in 4 adults cannot correctly add 1/2 and 1/3. The reason is almost always the same: they never understood why the rules work, only that the rules exist.

This guide builds fractions from the ground up. You'll see what a numerator and denominator represent, why finding a common denominator is not a ritual but a logical necessity, and why multiplying fractions is so much simpler than adding them. Each operation gets a step-by-step walkthrough with numbers small enough to follow mentally.

By the end, you'll work through adding mixed numbers, simplifying with the greatest common factor, and placing fractions on a number line, the three skills that trip up most learners after the basics.

The Basics: What a Fraction Represents

A fraction is a number that represents a part of a whole. It has two parts separated by a line. The numerator (top number) tells you how many parts you have. The denominator (bottom number) tells you how many equal parts the whole gets divided into.

Take 3/4. The denominator 4 means the whole splits into 4 equal pieces. The numerator 3 means you hold 3 of those pieces. A pizza cut into 4 slices where you ate 3 slices, that's 3/4 of the pizza.

Fractions come in three types. A proper fraction has a numerator smaller than the denominator, like 3/4 or 5/8. Its value is between 0 and 1. An improper fraction has a numerator equal to or larger than the denominator, like 7/4 or 9/3. Its value is 1 or greater. A mixed number combines a whole number with a proper fraction, like 1 3/4. Mixed numbers and improper fractions represent the same amounts, 7/4 equals 1 3/4 exactly.

To convert an improper fraction to a mixed number, divide the numerator by the denominator. 7 ÷ 4 = 1 remainder 3, so 7/4 = 1 3/4. To go the other way, multiply the whole number by the denominator and add the numerator: 1 × 4 + 3 = 7, giving 7/4.

The Four Operations, Step by Step

Addition and subtraction require a common denominator. You can only add or subtract fractions when they express parts of the same-sized whole. 1/4 + 1/4 = 2/4 works because both pieces are the same size. But 1/2 + 1/3 requires you to rewrite both fractions with the same denominator first.

Find the least common denominator of 2 and 3, which is 6. Rewrite 1/2 as 3/6 (multiply top and bottom by 3) and 1/3 as 2/6 (multiply top and bottom by 2). Now add: 3/6 + 2/6 = 5/6. The denominator stays 6; only the numerators add.

Subtraction follows the same process. 5/6 - 1/4: LCD of 6 and 4 is 12. Rewrite as 10/12 - 3/12 = 7/12.

Multiplication is the simplest operation. Multiply numerator by numerator, denominator by denominator. 2/3 × 3/5 = (2×3)/(3×5) = 6/15 = 2/5 after simplifying. No common denominator needed.

Division: flip the second fraction and multiply. To compute 3/4 ÷ 2/5, rewrite as 3/4 × 5/2 = 15/8 = 1 7/8. The flip (taking the reciprocal) converts division into multiplication, which is the one operation fractions handle without extra steps.

Simplifying a fraction means dividing both numerator and denominator by their greatest common factor. 12/18: the GCF of 12 and 18 is 6. Divide both by 6: 12/6 = 2, 18/6 = 3. Result: 2/3. You can simplify before multiplying too, cancel a numerator factor with a denominator factor across any two fractions in the problem to keep numbers small.

Common Misconceptions

• You can add numerators and denominators separately. 1/2 + 1/3 does not equal 2/5. Adding straight across gives the wrong answer because the pieces are different sizes. Always find the common denominator first.
• Bigger denominators mean bigger fractions. 1/10 is smaller than 1/2, not larger. A larger denominator means the whole splits into more pieces, so each piece is smaller.
• You must simplify before multiplying. Simplifying before multiplying is a shortcut, not a requirement. 2/3 × 3/4 = 6/12 = 1/2 whether you simplify before or after. Simplifying before keeps the arithmetic easier.
• Mixed numbers add the same way as whole numbers. 1 2/3 + 2 3/4 does not equal 3 5/7. You still need a common denominator for the fraction parts: 2/3 + 3/4 = 8/12 + 9/12 = 17/12 = 1 5/12. The total is 3 + 1 5/12 = 4 5/12.