Fractions Explained
Fractions confuse a lot of people, and most of that confusion traces back to learning the rules without understanding what fractions actually represent. Once the concept clicks, the rules stop feeling arbitrary.
A fraction like 3/4 means you cut something into 4 equal pieces and you have 3 of them. The bottom number (denominator) is the size of each piece. The top number (numerator) is how many pieces you have. To add fractions, the pieces need to be the same size , same denominator. To multiply, just multiply straight across.
What a fraction actually is
Think of a pizza cut into 8 equal slices. You eat 3 slices. You've eaten 3/8 of the pizza. The 8 tells you the pizza was cut into 8 equal parts. The 3 tells you how many of those parts you consumed.
Change either number and you change the fraction. If the pizza were cut into 4 slices instead and you ate 3, you'd have eaten 3/4 , much more pizza. The denominator controls the size of each piece; the numerator counts how many you have.
Equivalent fractions represent the same amount with different numbers. 1/2 = 2/4 = 3/6 = 4/8. You're just describing the same portion with differently sized pieces. Multiply or divide both parts of a fraction by the same number and you get an equivalent fraction , because you're really multiplying by 1 in disguise (2/2 = 1, 3/3 = 1, etc.).
Simplifying fractions
A fraction is fully simplified when the numerator and denominator share no common factor other than 1. To simplify, find the GCF (greatest common factor) of both numbers and divide.
Adding and subtracting fractions
You can only combine fractions that have the same denominator , the same size pieces. If the denominators differ, you need to convert both fractions to a common denominator first.
Why? Because adding 1/3 + 1/4 is like trying to add apples and oranges. One third is a different size piece than one quarter. You need to convert both into the same unit before you can add them.
1/3 + 1/4 is not 2/7. This is one of the most common fraction mistakes. The denominator tells you the size of the pieces , it doesn't change when you combine pieces. Only the numerators get added, and only after the denominators match.
Multiplying fractions
This is where fractions get easier, not harder. Multiply the numerators together and multiply the denominators together. No common denominator needed.
Something worth noticing: multiplying fractions gives you a smaller result than either fraction. 1/2 x 1/2 = 1/4. Half of a half is a quarter. Multiplication doesn't always make things bigger , only when the numbers are greater than 1.
Dividing fractions
Keep the first fraction, flip the second, then multiply. "Keep, change, flip" is the phrase most people learn. The formal reason: dividing by a number is the same as multiplying by its reciprocal.
The "flip and multiply" rule trips people up because they forget which fraction gets flipped. It's always the one you're dividing by , the second one. The first fraction stays exactly as it is.