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Add Fractions
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Subtract Fractions
×
Multiply Fractions
÷
Divide Fractions
Fraction Arithmetic Rules
Fractions represent parts of a whole. Every fraction has a numerator (top number) and denominator (bottom number). The four arithmetic operations each follow specific rules that ensure results are mathematically correct and expressed in lowest terms.
Adding Fractions
Find the LCD (Least Common Denominator), convert both fractions, add the numerators. Example: 1/2 + 1/3. LCD = 6. 3/6 + 2/6 = 5/6. Same-denominator fractions: just add numerators and keep denominator.
Subtracting Fractions
Same as addition: find LCD, convert, subtract numerators. Example: 3/4 − 1/3. LCD = 12. 9/12 − 4/12 = 5/12. Order matters — first fraction minus second.
Multiplying Fractions
Multiply numerators together, multiply denominators together. Example: 2/3 × 3/4 = 6/12 = 1/2. Can cross-simplify before multiplying to keep numbers smaller.
Dividing Fractions
Keep-Change-Flip: keep first fraction, change ÷ to ×, flip the second. Example: 5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 = 5/4. Dividing by a fraction = multiplying by its reciprocal.
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Fraction Rules Explained
How do you add fractions with different denominators?+
Find the LCD (Least Common Denominator), convert both fractions to use it, then add the numerators. Example: 1/2 + 1/3. LCD = 6. Convert: 3/6 + 2/6 = 5/6. Steps: multiply 1 × 3 and 2 × 3 to get 3/6, multiply 1 × 2 and 3 × 2 to get 2/6, then add: (3+2)/6 = 5/6. The LCD is the smallest number both denominators divide into evenly.
How do you multiply fractions?+
Multiply the numerators together and the denominators together, then simplify. Example: 2/3 × 3/4 = (2×3)/(3×4) = 6/12 = 1/2. You can also cross-simplify before multiplying — divide 2 and 4 by 2 (getting 1 and 2), divide 3 and 3 by 3 (getting 1 and 1), leaving 1/1 × 1/2 = 1/2. Cross-simplification avoids large intermediate numbers.
How do you divide fractions?+
Keep-Change-Flip (KCF): keep the first fraction, change division to multiplication, flip (invert) the second fraction. Example: 5/6 ÷ 2/3 = 5/6 × 3/2 = 15/12 = 5/4 = 1 and 1/4. The rule works because dividing by a fraction is mathematically identical to multiplying by its reciprocal. KCF works for all fractions including negative ones and improper fractions.
How do you simplify a fraction?+
Find the GCF (Greatest Common Factor) of the numerator and denominator, then divide both by it. Example: 12/18. GCF(12,18) = 6. 12/6 = 2, 18/6 = 3. Simplified: 2/3. A fraction is fully simplified when GCF(numerator, denominator) = 1. All results from this calculator are automatically simplified to lowest terms.
How do you convert a mixed number to an improper fraction?+
Multiply the whole number by the denominator, add the numerator, keep the same denominator. Example: 2 and 3/4 = (2×4 + 3)/4 = 11/4. To reverse (improper to mixed): divide numerator by denominator. 11 ÷ 4 = 2 remainder 3, so 11/4 = 2 and 3/4. Mixed numbers like 2 and 3/4 can be entered by converting to improper fractions first.
What is the Least Common Denominator (LCD)?+
The LCD is the Least Common Multiple (LCM) of the denominators. It is the smallest number that both denominators divide into evenly. Example: LCD(4, 6). LCM(4,6) = 12. To add 1/4 + 1/6: convert to 3/12 + 2/12 = 5/12. Using the LCD (rather than just multiplying denominators) keeps numbers smaller and reduces simplification needed at the end.
Can you add fractions with the same denominator?+
Yes — it is much simpler. When denominators are the same, just add (or subtract) the numerators and keep the denominator. Example: 3/8 + 2/8 = 5/8. Example: 7/10 − 3/10 = 4/10 = 2/5. No LCD calculation needed. This is why converting fractions to a common denominator is the key step in adding fractions with different denominators.
What is an improper fraction?+
An improper fraction has a numerator equal to or greater than the denominator. Examples: 7/4, 11/3, 5/5. Improper fractions represent values equal to or greater than 1. They are perfectly valid mathematically and are often easier to work with in calculations. Convert to mixed number: 7/4 = 1 and 3/4. Convert back: 1 and 3/4 = (1×4+3)/4 = 7/4.
How do you subtract a larger fraction from a smaller one?+
The result will be negative. Use the same process: find LCD, convert, subtract. Example: 1/4 − 3/4 = (1−3)/4 = −2/4 = −1/2. Or 1/3 − 1/2. LCD = 6. 2/6 − 3/6 = −1/6. Negative fractions follow the same rules as positive ones. A negative numerator indicates the result is negative.
Why do we need to find a common denominator to add fractions?+
Fractions with different denominators represent parts of different-sized wholes. You cannot directly add 1/2 and 1/3 without converting, just as you cannot add 50 cents and 33 cents (= 1/3 of a dollar) by adding 1+1=2. The common denominator converts both to the same "unit size." 1/2 = 3/6 and 1/3 = 2/6 — now both count sixths, so 3 sixths + 2 sixths = 5 sixths = 5/6.