Adding and Subtracting Fractions with Unlike Denominators

You cannot add the tops and bottoms of unlike fractions independently. The denominators tell you the size of the pieces, so the pieces must match before combining them.

Find a common denominator

For 1/4 + 2/3, a common denominator is 12. Rewrite 1/4 as 3/12 and 2/3 as 8/12. The fractions now describe twelfths, so add the numerators: 3/12 + 8/12 = 11/12.

Use the least common denominator

The least common denominator (LCD) is the least common multiple of the denominators. For 5/6 − 1/4, the LCD is 12. Convert 5/6 to 10/12 and 1/4 to 3/12, then subtract to get 7/12. Using the LCD often reduces later simplification, though a larger common denominator still gives an equivalent answer.

Convert correctly

To make an equivalent fraction, multiply the numerator and denominator by the same nonzero number. If 2/5 needs denominator 20, multiply both by 4 to get 8/20. Changing only the denominator changes the value and is a common error.

Mixed numbers and simplification

For mixed numbers, either convert to improper fractions before finding a common denominator or combine whole-number and fractional parts carefully. After adding or subtracting, simplify by dividing numerator and denominator by their greatest common factor. In a word problem, check that the result is reasonable and state the requested units.

Choose a common denominator efficiently

The least common denominator is the least common multiple of the denominators, but it is not necessary to find it by listing every multiple. Factor the denominators and use each prime factor at its greatest required power. For 1/6 + 1/8, the denominators are 2×3 and 2³, so the LCD is 2³×3 = 24. Then 1/6 = 4/24 and 1/8 = 3/24, giving 7/24. Any shared multiple works, but the LCD usually keeps the numbers smaller.

Subtract carefully, including mixed numbers

For 3/4 − 2/3, rewrite both fractions with denominator 12: 9/12 − 8/12 = 1/12. The order of the numerators follows the original subtraction; do not reorder them just because one is larger. For a mixed-number example, 2 1/4 − 1 2/3 can be written as 9/4 − 5/3 = 27/12 − 20/12 = 7/12. Converting to improper fractions first avoids errors when borrowing from the whole-number part.

Work with negative fractions

The sign belongs to the value of the fraction. For −1/3 + 1/4, use denominator 12: −4/12 + 3/12 = −1/12. Parentheses can make subtraction clearer: 2/5 − (−1/10) means add 1/10, while 2/5 − 1/10 subtracts it. Convert each term without changing its sign, then combine the numerators over the common denominator.

Simplify and interpret the result

Simplifying before the final answer means dividing numerator and denominator by their greatest common factor; it does not change the fraction's value. For a word problem involving lengths or portions, retain the unit and check whether the answer is smaller or larger than the quantities being combined. If fractions describe parts of one whole, an answer greater than 1 may be valid when the question combines multiple wholes, but it may indicate a setup error if the context allows only one whole.

Common denominator mistakes

• Adding denominators after changing the fractions: once denominators match, keep that denominator and combine only numerators.
• Multiplying only the denominator during conversion, which changes the fraction's value.
• Using the numerator instead of the denominator to identify a common multiple.
• Rounding a repeating decimal too early rather than keeping the exact fraction.
• Confusing addition and multiplication: multiplying fractions does not require a common denominator.

Check with equivalent values

A fraction can be checked by converting it to a decimal or by comparing it with familiar benchmark fractions such as 1/2. For example, 7/24 is slightly less than 1/3 because 1/3 = 8/24. The decimal is only a check; the exact fraction is more precise and avoids rounding discrepancies. Ensure the final answer is reduced if the question expects simplest form.

Find a common denominator before adding

Fractions can be added or subtracted directly only when they have the same denominator. For unlike denominators, find a common denominator, preferably the least common multiple (LCM). To add 1/4 + 1/6, the LCM of 4 and 6 is 12, so rewrite as 3/12 + 2/12 = 5/12. The denominator names the equal-sized parts; it is not added.

You may use the product of denominators as a common denominator, but it may not be the smallest. For 1/4 + 1/6, denominator 24 also works: 6/24 + 4/24 = 10/24 = 5/12. Reducing the final result gives the simplest equivalent fraction. Using the LCM usually keeps the arithmetic smaller.

Subtract carefully and regroup when needed

For 5/8 − 1/3, use denominator 24: 15/24 − 8/24 = 7/24. If the first numerator is smaller after conversion, the result is negative unless the expression includes a whole-number part that can be regrouped. Do not subtract denominators or subtract numerators before the fractions have a common denominator.

For a mixed number such as 2 1/4 − 5/6, convert to improper fractions: 9/4 − 5/6. Use denominator 12 to get 27/12 − 10/12 = 17/12 = 1 5/12. Converting first prevents losing a whole unit during subtraction. Alternatively, regroup 2 1/4 as 1 + 1 1/4 and subtract, but keep each step explicit.

Estimate and reduce the result

Before calculating, compare magnitudes. Since 1/4 is less than 1/3, the difference 1/3 − 1/4 should be a small positive fraction; the exact result is 1/12. After adding or subtracting, reduce numerator and denominator by their greatest common factor. A fraction with numerator larger than denominator may be an improper fraction; convert to a mixed number only when useful or requested.

A fraction bar groups the entire numerator and denominator. When an expression has several terms, use parentheses or rewrite it clearly. Keep negative signs attached to the correct numerator and use equivalent fractions rather than rounding to decimals unless the question requests an approximation.

• Find a common denominator, often the LCM.
• Rewrite each fraction with an equivalent numerator and denominator.
• Add or subtract numerators only; keep the shared denominator.
• Reduce the result and convert to a mixed number if appropriate.
• Estimate the sign and size before calculating to catch errors.

Exam takeaway

Match the denominators first, then combine numerators and simplify. Keep the fraction's value unchanged when converting by multiplying top and bottom by the same factor.