Add & Subtract Fractions

5.D.1Common denominators01/04

Equivalent fractions and common denominators.

Step01of 04

To add or subtract fractions with DIFFERENT denominators, you first need to convert them to fractions with the SAME denominator. The shared one is called a common denominator.

5.D.1 — Equivalent fractions and common denominators.

Step 01 of 04

To add or subtract fractions with DIFFERENT denominators, you first need to convert them to fractions with the SAME denominator. The shared one is called a common denominator.

Step 02 of 04

Find a common denominator. The simplest is the LEAST common multiple (LCM) of the two denominators.

Denominators 4 and 6	→LCM = 12
Denominators 3 and 5	→LCM = 15 (their product, since they share no factors)
Denominators 2 and 8	→LCM = 8 (since 8 is already a multiple of 2)

Step 03 of 04

Convert each fraction to the LCM denominator. Multiply top AND bottom by whatever takes the original denominator to the LCM.

\frac{3}{4} = \frac{9}{12} \frac{1}{6} = \frac{2}{12}

Step 04 of 04

Once both fractions share a denominator, you can compare, add, or subtract them. The "like denominators" rules from grade 4 apply.

Key insight

Find the LCM of the denominators. Convert each fraction to use that LCM. Multiply top AND bottom — same fraction, different form. Then proceed normally.

5.D.2 — Add fractions with unlike denominators.

Step 01 of 04

The recipe: find LCM of denominators → convert each fraction to that denominator → add numerators → simplify.

Step 02 of 04

Worked example. $\frac{3}{4} + \frac{1}{6}$ .

LCM = 12. Convert:

\frac{3}{4} = \frac{9}{12}, \frac{1}{6} = \frac{2}{12}

Step 03 of 04

Add:

\frac{9}{12} + \frac{2}{12} = \frac{11}{12}

11/12 is already in simplest form (11 is prime).

Step 04 of 04

Another example with simplification. $1/3 + 1/6 = 2/6 + 1/6 = 3/6 = 1/2$ . Always check if the answer simplifies.

Key insight

LCM of denominators → convert each → add numerators → simplify. Don't just add tops and bottoms — that's wrong.

5.D.3 — Subtract fractions with unlike denominators.

Step 01 of 04

Same procedure as addition: LCM denominator, convert, then subtract numerators.

Step 02 of 04

Worked example. $\frac{5}{6} - \frac{1}{4}$ .

LCM = 12.

\frac{5}{6} = \frac{10}{12}, \frac{1}{4} = \frac{3}{12}

Step 03 of 04

Subtract:

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

Step 04 of 04

Subtract from a whole. Rewrite the whole using the needed denominator.

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

Key insight

Same recipe as addition. From a whole, rewrite the whole as denominator/denominator. Always simplify the result.

5.D.4 — Add and subtract mixed numbers.

Step 01 of 04

Two methods: (1) work with whole and fraction parts SEPARATELY, or (2) convert each mixed number to an improper fraction first.

Step 02 of 04

Worked example. $2 \frac{3}{4} + 1 \frac{5}{6}$ .

Method 1 — separate parts. Whole: 2 + 1 = 3. Fractions: $3/4 + 5/6$ . LCM = 12: $9/12 + 10/12 = 19/12 = 1 \frac{7}{12}$ . Combine: $3 + 1 \frac{7}{12} = 4 \frac{7}{12}$ .

Step 03 of 04

Subtraction with borrowing. $5 \frac{1}{4} - 2 \frac{2}{3}$ .

LCM 12: $5 \frac{3}{12} - 2 \frac{8}{12}$ . Can't subtract 8/12 from 3/12 — borrow. $5 \frac{3}{12} = 4 \frac{15}{12}$ . Now: $4 \frac{15}{12} - 2 \frac{8}{12} = 2 \frac{7}{12}$ .

Step 04 of 04

Improper fractions method. Convert both to improper, add or subtract, convert back. Both methods give the same answer; pick whichever you prefer.

Key insight

Add or subtract whole and fraction parts separately, with carrying or borrowing as needed. Or convert to improper fractions first if you prefer that path.

5.D.5 — Fraction word problems (add and subtract).

Step 01 of 04

Translate the problem into an addition or subtraction. Find a common denominator, do the arithmetic, simplify.

Step 02 of 04

Add example. A recipe needs 2/3 cup flour and 1/4 cup sugar. Total dry ingredients?

\frac{2}{3} + \frac{1}{4} = \frac{8}{12} + \frac{3}{12} = \frac{11}{12} cup

Step 03 of 04

Subtract example. A pitcher had 3 quarts of juice. After serving, 1 1/2 quarts remain. How much was served?

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

Step 04 of 04

Multi-step example. A board is 7 ft. You cut off 2 1/4 ft, then another 1 1/2 ft. How much remains?

2 \frac{1}{4} + 1 \frac{1}{2} = 2 \frac{1}{4} + 1 \frac{2}{4} = 3 \frac{3}{4} ft cut

7 - 3 \frac{3}{4} = 6 \frac{4}{4} - 3 \frac{3}{4} = 3 \frac{1}{4} ft remaining

Key insight

Identify the operation. Find common denominators. Use borrowing for subtraction with mixed numbers. Always answer with UNITS.

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