Fraction Rules- Operations and Simplification Tips

What a Fraction Actually Is

A fraction represents a part of a whole. The top number (numerator) tells you how many parts you have. The bottom number (denominator) tells you how many equal parts make up the whole thing.

So 3/4 means three parts out of four equal pieces. That's it. No more confusion about what these things represent.

Simplifying Fractions: Cut the Bullshit

Simplifying means reducing a fraction to its lowest terms. You find the largest number that divides evenly into both the top and bottom.

How to Simplify

Find the GCF (greatest common factor) of numerator and denominator
Divide both numbers by that factor
Keep going until only 1 divides into both

Example: 12/18

GCF of 12 and 18 is 6
12 ÷ 6 = 2
18 ÷ 6 = 3
Simplified: 2/3

You can also divide incrementally if you can't spot the GCF immediately. Keep dividing by small primes like 2, 3, or 5 until you can't go further.

Adding Fractions

This is where people screw up. You cannot add fractions by adding numerators and denominators directly. That's wrong. Here's the correct approach:

Same Denominator? Easy

When denominators match, just add the numerators and keep the denominator.

1/5 + 2/5 = 3/5

That's it. Nothing else needed.

Different Denominators? Find the LCD

You need the least common denominator. That's the smallest number both denominators divide into evenly.

Example: 1/4 + 1/6

Multiples of 4: 4, 8, 12, 16, 20...
Multiples of 6: 6, 12, 18, 24...
LCD is 12
Convert: 1/4 = 3/12, 1/6 = 2/12
Add: 3/12 + 2/12 = 5/12

Always simplify your final answer.

Subtracting Fractions

Same process as addition. Subtract the numerators, keep the denominator.

5/8 - 3/8 = 2/8 = 1/4

For different denominators, find the LCD first, then subtract.

Multiplying Fractions

This one's simpler than addition. Multiply numerators together, then multiply denominators together.

2/3 × 4/5 = (2×4)/(3×5) = 8/15

The Cross-Cancel Trick

Before multiplying, you can cancel diagonally to keep numbers smaller.

4/9 × 3/8

Cancel: 4 and 8 share a factor of 4 → 4÷4=1, 8÷4=2
Cancel: 3 and 9 share a factor of 3 → 3÷3=1, 9÷3=3
Now multiply: 1/3 × 1/2 = 1/6

Compare to doing it blind: 4/9 × 3/8 = 12/72 = 1/6. Same result, messier math.

Dividing Fractions

Flip the second fraction (take its reciprocal), then multiply.

1/2 ÷ 3/4

Flip 3/4 to get 4/3
Multiply: 1/2 × 4/3 = 4/6 = 2/3

The reciprocal just means swapping the numerator and denominator. Keep the first fraction as-is.

Quick Reference Table

Operation	Rule
Simplify	Divide numerator and denominator by GCF
Add (same denominator)	Add numerators, keep denominator
Add (different denominators)	Find LCD, convert, then add numerators
Subtract	Same as addition, but subtract numerators
Multiply	Multiply numerators × numerators, denominators × denominators
Divide	Multiply by reciprocal of second fraction

Common Mistakes to Avoid

Adding denominators: 1/2 + 1/3 ≠ 2/5. Find the LCD instead.
Forgetting to simplify: 4/8 is correct but 1/2 is better
Cross-canceling when adding: This trick only works for multiplication
Flipping the wrong fraction when dividing: Always flip the one after the division sign

Getting Started: Practice Problems

Work through these to lock in the rules:

Simplify 24/36
Calculate 2/5 + 1/4
Solve 7/9 - 1/3
Multiply 3/7 × 14/15
Divide 2/5 ÷ 4/9

Answers:

2/3 (GCF is 12)
13/20 (LCD is 20: 8/20 + 5/20)
4/9 (LCD is 9: 7/9 - 3/9)
2/5 (cross-cancel: 3/7 × 14/15 → 3/1 × 2/15 = 6/15 = 2/5)
9/10 (2/5 × 9/4 = 18/20 = 9/10)

Mixed Numbers and Improper Fractions

A mixed number combines a whole number with a fraction (like 2 1/3). An improper fraction has a larger numerator than denominator (like 7/3).

To convert mixed to improper: multiply the whole number by the denominator, add the numerator, keep the denominator.

2 1/3 = (2×3 + 1)/3 = 7/3

To convert improper to mixed: divide numerator by denominator. The quotient is the whole number, the remainder goes over the original denominator.

7/3 = 7 ÷ 3 = 2 remainder 1 → 2 1/3