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How to Add, Subtract, and Multiply Fractions
Each fraction operation follows a specific rule. The reason these rules exist is that fractions represent parts of a whole, and you can only combine parts directly when they are measured in the same sized pieces. That is why addition and subtraction require a common denominator, while multiplication and division do not.
Adding fractions
You can only add fractions that share the same denominator, because the denominator tells you what size the pieces are. Adding 2/7 + 3/7 is straightforward: both fractions are measured in sevenths, so you add the numerators and get 5/7.
When the denominators differ, as in 1/4 + 2/3, you need to find a common denominator first. The least common denominator (LCD) of 4 and 3 is 12. Convert each fraction: 1/4 becomes 3/12 (multiply top and bottom by 3), and 2/3 becomes 8/12 (multiply top and bottom by 4). Now add: 3/12 + 8/12 = 11/12. The conversion does not change the value of either fraction; it just re-expresses them in the same unit so they can be combined. The fraction calculator handles this automatically for any pair of fractions.
Here is a second example with larger numbers. To add 5/8 + 7/12, the LCD of 8 and 12 is 24. Convert: 5/8 becomes 15/24 (multiply by 3), and 7/12 becomes 14/24 (multiply by 2). Add the numerators: 15 + 14 = 29. The result is 29/24, which equals 1 5/24 as a mixed number.
Subtracting fractions
Subtraction follows the same rule as addition. Get a common denominator, then subtract the numerators. For 5/6 - 1/4, the LCD of 6 and 4 is 12. Convert: 5/6 = 10/12, and 1/4 = 3/12. Subtract: 10/12 - 3/12 = 7/12.
If the denominators already match, subtract directly. 7/8 - 3/8 = 4/8, which simplifies to 1/2 (divide both by 4).
A common mistake with subtraction is forgetting to convert both fractions before subtracting. Subtracting 5/6 - 1/4 without finding a common denominator first (for instance, subtracting the numerators to get 4 and the denominators to get 2, yielding 4/2 = 2) gives a completely wrong answer. The denominators must match before you touch the numerators.
Multiplying fractions
Multiplication is simpler because you do not need a common denominator. Multiply the numerators together and the denominators together:
\[\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}\]For 2/3 x 4/5: numerators give 2 x 4 = 8, denominators give 3 x 5 = 15, result is 8/15.
This works because multiplying fractions means taking a fraction of a fraction. Two-thirds of four-fifths is a smaller piece, and the multiplication captures that directly. When the result can be simplified, divide both parts by their greatest common divisor (GCD). For 3/4 x 2/9: 6/36 simplifies to 1/6 (divide both by 6). A useful shortcut is to cancel common factors before multiplying: the 3 and the 9 share a factor of 3, so simplify first to get 1/4 x 2/3 = 2/12 = 1/6.
Dividing fractions
To divide fractions, flip the second fraction (take its reciprocal) and multiply:
\[\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}\]For 3/5 divided by 2/7: flip 2/7 to get 7/2, then multiply 3/5 x 7/2 = 21/10, which equals 2 1/10 as a mixed number. The reason this works is that dividing by a fraction is the same as asking “how many times does this fraction fit into that one,” and multiplying by the reciprocal answers that question.
Here is another example. To divide 4/9 by 2/3, flip 2/3 to get 3/2, then multiply: 4/9 x 3/2 = 12/18. Simplify by dividing both by 6: 2/3.
How to find the least common denominator
The LCD is the smallest number that both denominators divide into evenly. For small numbers, list multiples until you find a match: multiples of 4 are 4, 8, 12, 16…, multiples of 6 are 6, 12, 18…, so the LCD is 12. For larger numbers, use the formula:
\[\text{LCD} = \frac{a \times b}{\gcd(a, b)}\]For example, the LCD of 15 and 20: the GCD of 15 and 20 is 5, so the LCD is (15 x 20) / 5 = 60. This is faster than listing multiples for larger denominators.
For the connection between fractions and percentages, see the percentages, fractions, and ratios guide.
How to simplify a fraction
Divide both the numerator and denominator by their GCD. To simplify 18/24: the GCD of 18 and 24 is 6, so 18/6 = 3 and 24/6 = 4. Simplified: 3/4. A fraction is fully simplified when the numerator and denominator share no common factor other than 1.
When the GCD is not obvious, factor both numbers. The factors of 42 are 2, 3, 7, and the factors of 56 are 2, 2, 2, 7. The common factors are 2 and 7, so the GCD is 14. Therefore 42/56 simplifies to 3/4 (42 / 14 = 3, 56 / 14 = 4).
Working with mixed numbers
A mixed number combines a whole number and a fraction, like 3 1/2. To use mixed numbers in any operation, first convert them to improper fractions by multiplying the whole number by the denominator, adding the numerator, and placing the result over the original denominator. For 3 1/2: 3 x 2 + 1 = 7, so the improper fraction is 7/2. For 2 3/4: 2 x 4 + 3 = 11, so the improper fraction is 11/4.
Once both values are improper fractions, apply the standard rules. To multiply 3 1/2 x 2 3/4: convert to 7/2 x 11/4 = 77/8, which equals 9 5/8. To add 3 1/2 + 2 3/4: convert to 7/2 + 11/4, find the LCD of 2 and 4 (which is 4), convert 7/2 to 14/4, then add: 14/4 + 11/4 = 25/4 = 6 1/4.