Adding and Subtracting Fractions

Key Terms

Same Denominator — The Simple Case

Why Common Denominators Are Required

Finding a Common Denominator

Converting to Common Denominator

Adding Mixed Numbers

Subtracting Mixed Numbers

Adding and Subtracting with Whole Numbers

Common Mistakes

Adding and Subtracting Fractions at a Glance

Adding and Subtracting Fractions FAQ

Finding Common Ground

Adding and subtracting fractions requires a common denominator. Unlike multiplication, where numerators and denominators combine directly, addition and subtraction can only proceed when the fractions represent same-sized pieces. This page covers operations with same and different denominators, working with mixed numbers, and avoiding common errors.

Key Terms

Common Denominator— the shared denominator needed before combining fractions

Equivalent Fractions— used to rewrite fractions with matching denominators

Mixed Number— adding and subtracting mixed numbers requires special handling

See All Arithmetic Definitions →

Same Denominator — The Simple Case

When fractions share a denominator, add or subtract the numerators and keep the denominator unchanged.

\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}

\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}

For example,

\frac{2}{7} + \frac{3}{7} = \frac{5}{7}

and

\frac{5}{8} - \frac{2}{8} = \frac{3}{8}

.

This works because the denominator names the size of each piece. Adding two-sevenths and three-sevenths means combining five pieces that are each one-seventh in size. The result may need simplifying —

\frac{4}{8} + \frac{2}{8} = \frac{6}{8} = \frac{3}{4}

after reducing to simplest form.

Adding sevenths

All three bars are cut into sevenths, so every part has the same size. Two parts and three more parts make five parts: 2/7 + 3/7 = 5/7. The denominator names the size of the parts and does not change; only the number of parts is added.

Subtraction works the same way: 5/7 − 3/7 removes 3 parts and leaves 2/7.

Why Common Denominators Are Required

Fractions with different denominators represent pieces of different sizes. Adding

\frac{1}{3}

and

\frac{1}{4}

directly makes no sense because thirds and fourths are not the same unit.

Consider the analogy of adding 2 feet and 5 inches. The numbers cannot be combined until both are expressed in the same unit — either all inches or all feet. Similarly,

\frac{1}{3}

and

\frac{1}{4}

must be rewritten as equivalent fractions with matching denominators before the numerators can be added.

Visually, a bar divided into thirds and a bar divided into fourths have different segment sizes. Only by subdividing both bars into twelfths — a common unit — can the shaded regions be meaningfully combined.

Finding a Common Denominator

A common denominator is any number that both original denominators divide into evenly. For

\frac{1}{3}

and

\frac{1}{4}

, common denominators include 12, 24, 36, and infinitely many others.

The least common denominator (LCD) is the smallest such number, keeping arithmetic manageable. The LCD of 3 and 4 is 12. The LCD of 6 and 8 is 24.

Finding the LCD uses the least common multiple. List multiples of each denominator until a match appears, or use prime factorization for larger numbers. For denominators 4 and 6: multiples of 4 are 4, 8, 12, 16... and multiples of 6 are 6, 12, 18... The first common multiple is 12.

Converting to Common Denominator

Once the common denominator is identified, convert each fraction by multiplying its numerator and denominator by the same factor.

To add

\frac{2}{3} + \frac{1}{4}

with LCD 12:

\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}

\frac{1}{4} = \frac{1 \times 3}{4 \times 3} = \frac{3}{12}

\frac{8}{12} + \frac{3}{12} = \frac{11}{12}

For subtraction, the process is identical. To compute

\frac{5}{6} - \frac{1}{4}

with LCD 12: convert to

\frac{10}{12} - \frac{3}{12} = \frac{7}{12}

Adding 2/3 and 1/4 through twelfths

Thirds and quarters are different sizes, so 2/3 and 1/4 cannot be added part by part. Cutting both bars into twelfths, the least common denominator, turns 2/3 into 8/12 and 1/4 into 3/12. Now the parts match and the counts add: 8/12 + 3/12 = 11/12, shown on the last bar.

11/12 is already in simplest form, since 11 and 12 share no factor.

Adding Mixed Numbers

Two methods handle addition of mixed numbers. The first adds whole parts and fractional parts separately. The second converts everything to improper fractions.

Using the first method for

2\frac{1}{3} + 4\frac{1}{4}

: add wholes

2 + 4 = 6

, then add fractions

\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}

. The result is

6\frac{7}{12}

.

When the fractional sum exceeds 1, regroup. For

3\frac{3}{4} + 2\frac{1}{2}

: wholes give

3 + 2 = 5

, fractions give

\frac{3}{4} + \frac{2}{4} = \frac{5}{4} = 1\frac{1}{4}

. Combine:

5 + 1\frac{1}{4} = 6\frac{1}{4}

Subtracting Mixed Numbers

Subtraction of mixed numbers also works by handling parts separately, but borrowing may be necessary when the first fractional part is smaller than the second.

For

5\frac{3}{4} - 2\frac{1}{4}

: subtract wholes

5 - 2 = 3

, subtract fractions

\frac{3}{4} - \frac{1}{4} = \frac{2}{4} = \frac{1}{2}

. The result is

3\frac{1}{2}

.

For

5\frac{1}{4} - 2\frac{3}{4}

: the fraction

\frac{1}{4}

is smaller than

\frac{3}{4}

, so borrow 1 from the 5. Rewrite as

4\frac{5}{4} - 2\frac{3}{4}

. Now subtract:

4 - 2 = 2

and

\frac{5}{4} - \frac{3}{4} = \frac{2}{4} = \frac{1}{2}

. The result is

2\frac{1}{2}

Case	Procedure	Example
Add — no carry	add wholes; add fractions; fractional sum stays proper	2 1⁄3 + 4 1⁄4 → 6 + 7⁄12 = 6 7⁄12
Add — with carry	add wholes; add fractions; if fractional sum ≥ 1, carry the extra whole	3 3⁄4 + 2 1⁄2 → 5 + 5⁄4 = 5 + 1 1⁄4 = 6 1⁄4
Subtract — no borrow	subtract wholes; subtract fractions; first fractional part ≥ second	5 3⁄4 − 2 1⁄4 → 3 + 2⁄4 = 3 1⁄2
Subtract — with borrow	first fractional part < second; borrow 1 from the whole as denom⁄denom, then subtract	5 1⁄4 − 2 3⁄4 → 4 5⁄4 − 2 3⁄4 = 2 2⁄4 = 2 1⁄2

Adding and Subtracting with Whole Numbers

A whole number can be treated as a fraction with denominator 1, or the whole part can be handled separately.

For

5 + \frac{3}{4}

: the result is simply

5\frac{3}{4}

.

For

5 - \frac{3}{4}

: rewrite 5 as

4\frac{4}{4}

, then subtract

4\frac{4}{4} - \frac{3}{4} = 4\frac{1}{4}

.

Alternatively, express 5 as

\frac{5}{1}

, find a common denominator with

\frac{3}{4}

, and compute:

\frac{20}{4} - \frac{3}{4} = \frac{17}{4} = 4\frac{1}{4}

.

To check a sum or difference that mixes whole numbers, mixed numbers and fractions, enter it in the fraction calculator, which shows each step.

Common Mistakes

Adding denominators is the most frequent error. The sum

\frac{1}{2} + \frac{1}{3}

does not equal

\frac{2}{5}

. Only numerators are added after finding a common denominator:

\frac{3}{6} + \frac{2}{6} = \frac{5}{6}

.

Forgetting to convert all fractions to the LCD leads to incorrect results. Each fraction must be rewritten before combining.

Failing to simplify the final answer leaves work incomplete. The sum

\frac{2}{8} + \frac{2}{8} = \frac{4}{8}

should be reduced to

\frac{1}{2}

.

Improper borrowing with mixed numbers causes errors in subtraction. When borrowing, the borrowed 1 becomes a fraction with the same denominator as the fractional part, not simply 1.

Why 1/2 + 1/3 is not 2/5

Adding tops and bottoms gives 2/5, but the 2/5 bar is shorter than the 1/2 bar on its own, and adding a positive amount cannot make something smaller. Converting to sixths gives the true sum: 1/2 = 3/6 and 1/3 = 2/6, so 1/2 + 1/3 = 5/6, the amber bar.

A quick check: a sum must be at least as large as each fraction in it.

Adding and Subtracting Fractions at a Glance

The page covered same-denominator combination, why common denominators are required, finding and applying the LCD, the four mixed-number cases, working with whole numbers, and the most common errors. The table below collects the rules and identities worth keeping at hand.

Concept	Statement	Example
Same-denominator rule	combine numerators, keep the denominator unchanged	2⁄7 + 3⁄7 = 5⁄7
Why common denom is required	different denominators name different-size pieces, like feet vs inches — convert before combining	1⁄3 + 1⁄4 must become twelfths
LCD = LCM of denominators	least common denominator is the least common multiple of the denominators	LCD(3, 4) = 12
Conversion step	multiply numerator and denominator by the factor that reaches the LCD; this produces an equivalent fraction	2⁄3 → ×4⁄4 → 8⁄12
After conversion	add or subtract the numerators; the common denominator carries through	8⁄12 + 3⁄12 = 11⁄12
Whole + fraction	place them side by side as a mixed number — no arithmetic needed	5 + 3⁄4 = 5 3⁄4
Whole − fraction	rewrite the whole as (n − 1) + denom⁄denom, then subtract the fraction	5 − 3⁄4 = 4 4⁄4 − 3⁄4 = 4 1⁄4
Borrow uses denom⁄denom	when borrowing 1 from the whole, it becomes a fraction with the current denominator, not the literal 1	5 1⁄4 → 4 5⁄4 (the borrowed 1 is 4⁄4)
Never add denominators	denominators name piece size and do not change when combining; only numerators combine	1⁄2 + 1⁄3 ≠ 2⁄5 (it is 5⁄6)
Reduce the final result	simplify the sum or difference to lowest terms	4⁄8 → 1⁄2

Adding and Subtracting Fractions FAQ

How do you add fractions with different denominators?

Find a common denominator, rewrite each fraction with it, then add the numerators and keep that denominator. For 2/3 + 1/4 the LCD is 12, so 2/3 becomes 8/12 and 1/4 becomes 3/12, giving 11/12. Subtraction works identically: 5/6 − 1/4 converts to 10/12 − 3/12 = 7/12. Simplify the result if it is not already in lowest terms.Read more →

Do you add denominators when adding fractions?

No, never. The denominator names the size of each piece, and combining pieces does not change how big they are — only how many you have. So 1/2 + 1/3 is not 2/5. Convert to a common denominator first, then add only the numerators: 3/6 + 2/6 = 5/6. Adding denominators would make the answer smaller than one of the parts.Read more →

How do you subtract mixed numbers with borrowing?

Borrow when the first fractional part is smaller than the second. For 5¼ − 2¾, the ¼ cannot cover ¾, so take 1 from the 5 and add it to the fraction as 4/4: 5¼ becomes 4⁵⁄₄. Now subtract in parts: 4 − 2 = 2 and 5/4 − 3/4 = 1/2, giving 2½. The borrowed 1 always becomes a fraction with the denominator already in use.Read more →

Should you simplify after adding fractions?

Yes — a sum is not considered complete until it is in lowest terms. Adding 2/8 + 2/8 gives 4/8, which reduces to 1/2. Divide the numerator and denominator by their greatest common factor to reduce in one step. Using the least common denominator rather than a larger one keeps the numbers small and often leaves nothing to simplify.Read more →