How to Work with Fractions
Learn how to add, subtract, multiply, and divide fractions with step-by-step examples. Master simplifying fractions, finding common denominators, and converting between mixed numbers and improper fractions.
Understanding Fractions
A fraction represents a part of a whole and consists of a numerator (top number) and a denominator (bottom number). The denominator tells how many equal parts the whole is divided into, and the numerator tells how many of those parts are being considered. For example, 3/8 means 3 out of 8 equal parts.
Simplifying Fractions
To simplify a fraction, divide both the numerator and denominator by their Greatest Common Factor (GCF). For 12/18, the GCF is 6, so 12/18 = 2/3. A fraction is in its simplest form (lowest terms) when the GCF of the numerator and denominator is 1.
Adding and Subtracting Fractions
Fractions must have a common denominator before adding or subtracting. If denominators differ, find the Least Common Denominator (LCD) and convert each fraction. For 1/4 + 2/6, the LCD is 12: convert to 3/12 + 4/12 = 7/12. Once denominators match, add or subtract the numerators and keep the denominator.
Multiplying Fractions
To multiply fractions, multiply the numerators together and multiply the denominators together: (a/b) × (c/d) = (a×c)/(b×d). For example, (2/3) × (4/5) = 8/15. No common denominator is needed. You can also simplify diagonally (cross-cancel) before multiplying to keep numbers smaller.
Dividing Fractions
To divide by a fraction, multiply by its reciprocal (flip the second fraction). The rule is: (a/b) ÷ (c/d) = (a/b) × (d/c) = (a×d)/(b×c). For (3/4) ÷ (2/5), multiply by the reciprocal: (3/4) × (5/2) = 15/8 = 1 7/8. This "keep, change, flip" method always works for fraction division.
Mixed Numbers and Improper Fractions
A mixed number (like 2 3/4) combines a whole number and a fraction. To convert to an improper fraction, multiply the whole number by the denominator and add the numerator: 2 3/4 = (2×4 + 3)/4 = 11/4. To convert an improper fraction back, divide the numerator by the denominator and express the remainder as a fraction.
Comparing Fractions
To compare two fractions, convert them to a common denominator and compare numerators. Alternatively, cross-multiply: for a/b vs. c/d, compare a×d to b×c. For 3/5 vs. 4/7, compare 3×7 = 21 and 5×4 = 20, so 3/5 > 4/7. Another approach is to convert both to decimals for quick comparison.
Try These Calculators
Put what you learned into practice with these free calculators.
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