The line in a fraction means divide. 3/4 means 3 divided by 4. The number on top is called the numerator — it's what's being divided. The number on bottom is the denominator — it's what you're dividing by.
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A fraction is a division problem that hasn't been solved yet. That's it.
The line in a fraction means divide. 3/4 means 3 divided by 4. The number on top is called the numerator — it's what's being divided. The number on bottom is the denominator — it's what you're dividing by.
The denominator tells you how many equal pieces the whole has been cut into. The numerator tells you how many of those pieces you have. 3/4 means: cut something into 4 equal pieces, take 3 of them.
Why does the denominator go on the bottom? Because it's describing the size of the pieces. A bigger denominator means smaller pieces — 1/8 of a pizza is smaller than 1/4, even though 8 is bigger than 4. The bottom number divides, so bigger bottom = smaller pieces.
The line means divide. Always. 1/2 = 1 ÷ 2 = 0.5. This is why fractions and decimals are the same thing in different clothes. Whenever you're confused, just do the division and see what number comes out.
Key concept
1/2 and 2/4 and 4/8 are all the same amount. They look different but they're equal. How? Because multiplying or dividing both the top and bottom by the same number doesn't change the value — it just changes how the fraction is written.
Think of it like a ratio. 1:2 and 2:4 describe the same relationship. As long as you treat the top and bottom the same way, the fraction doesn't change its value.
This is the foundation for everything else. Adding fractions, simplifying fractions, comparing fractions — all of it depends on understanding that you can rewrite a fraction without changing its value by multiplying top and bottom by the same thing.
The four operations
You can only add fractions with the same denominator. Why? Because you can only add things that are the same size. You can't add thirds and fourths directly — they're different sized pieces.
Step 1: Find the LCD — the Least Common Denominator. The smallest number both denominators divide into evenly. For 1/3 + 1/4, the LCD is 12.
Step 2: Convert both fractions to equivalent fractions with that denominator. 1/3 = 4/12. 1/4 = 3/12.
Step 3: Add the numerators. Keep the denominator. 4/12 + 3/12 = 7/12.
Common mistake: Adding the denominators. 1/3 + 1/4 ≠ 2/7. The denominator tells you the size of the pieces. You're not adding the sizes — you're adding how many pieces you have once they're the same size.
Multiplying fractions is the easiest operation. Multiply top times top, bottom times bottom. Done.
Why does this work without a common denominator? Because "of" means multiply. 3/4 × 2/5 means "3/4 of 2/5." You're finding a fraction of a fraction — shrinking it — and that's exactly what multiplying by a number less than 1 does.
Simplify after multiplying by dividing both top and bottom by their greatest common factor. 6/20 — both divisible by 2 — simplifies to 3/10.
Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal just means flipping it — 3/4 becomes 4/3.
The method: Keep the first fraction as-is. Change the division sign to multiplication. Flip the second fraction. Then multiply normally.
Why does flipping work? Because dividing by a number is the same as multiplying by 1 over that number. 6 ÷ 2 = 6 × 1/2 = 3. Same logic, same result. Flipping a fraction gives you its reciprocal — which is the same as dividing by 1.
A fraction is simplified when the top and bottom share no common factors other than 1. Find the Greatest Common Factor (GCF) of the numerator and denominator, then divide both by it.
12/18 — GCF is 6. Divide both by 6: 2/3. That's the simplified form. You can check: 2 and 3 share no common factors other than 1. Done.
Quick check: If both numbers are even, divide by 2. Keep dividing by 2 until one of them is odd. Then check for other common factors. It's faster than hunting for the GCF immediately.