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Fraction Basics and Operations
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Fractions represent parts of a whole. Despite being taught in elementary school, many adults feel uncertain about fraction arithmetic. This guide covers the four operations with concrete examples.
Key Terms
In the fraction 3/4:
• 3 is the numerator (top number)
• 4 is the denominator (bottom number)
A proper fraction has a numerator smaller than the denominator (3/4). An improper fraction has a numerator equal to or larger than the denominator (7/4). A mixed number combines a whole number and a fraction (1¾).
Adding and Subtracting
Fractions must have the same denominator.
Same denominator: 3/8 + 1/8 = 4/8 = 1/2
Different denominators: 1/3 + 1/4
1. Find the LCD (Least Common Denominator): LCD of 3 and 4 = 12
2. Convert: 1/3 = 4/12 and 1/4 = 3/12
3. Add: 4/12 + 3/12 = 7/12
Multiplying
Multiply numerators together and denominators together.
2/3 × 3/5 = (2×3)/(3×5) = 6/15 = 2/5
Simplify before multiplying when possible: 4/9 × 3/8. Notice that 3 and 9 share a factor of 3, and 4 and 8 share a factor of 4. Simplify first: (4÷4)/(9÷3) × (3÷3)/(8÷4) = 1/3 × 1/2 = 1/6.
Dividing
Flip the second fraction (take its reciprocal) and multiply.
2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6
Memory aid: "Keep, Change, Flip" — Keep the first fraction, Change ÷ to ×, Flip the second fraction.
Simplifying (Reducing) Fractions
Divide numerator and denominator by their Greatest Common Factor (GCF).
12/18: GCF of 12 and 18 is 6. → 12÷6 / 18÷6 = 2/3
A fraction is fully simplified when the only common factor is 1.
Try the Fraction Calculator.
Frequently Asked Questions
Is 1/3 greater or less than 3/8?
1/3 ≈ 0.333 and 3/8 = 0.375. So 3/8 is slightly larger. Convert to decimals or find a common denominator to compare fractions: 1/3 = 8/24, 3/8 = 9/24. Since 9/24 > 8/24, we confirm 3/8 > 1/3.
How do I convert a fraction to a decimal?
Divide the numerator by the denominator. 3/8 = 3 ÷ 8 = 0.375. To convert back, write the decimal as a fraction over the appropriate power of 10 and simplify: 0.375 = 375/1000 = 3/8.