Fractions: Add, Subtract, Multiply

4.G.1Decompose into unit fractions01/04

Decompose a fraction into unit fractions.

Step01of 04

A unit fraction has 1 as numerator: $1/2, 1/3, 1/4, 1/5, \dots$ . Every fraction is a SUM of unit fractions.

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

Three copies of $1/5$ . The numerator counts how many.

4.G.1 — Decompose a fraction into unit fractions.

Step 01 of 04

A unit fraction has 1 as numerator: $1/2, 1/3, 1/4, 1/5, \dots$ . Every fraction is a SUM of unit fractions.

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

Three copies of $1/5$ . The numerator counts how many.

Step 02 of 04

Use this to understand fraction addition.

\frac{2}{7} + \frac{4}{7} = \frac{6}{7}

Two copies of $1/7$ plus four copies of $1/7$ = six copies of $1/7$ . Same denominator, just count up the unit pieces.

Step 03 of 04

Mixed numbers as decompositions.

2 \frac{1}{4} = 1 + 1 + \frac{1}{4} = \frac{4}{4} + \frac{4}{4} + \frac{1}{4} = \frac{9}{4}

Convert mixed → improper: multiply whole × denominator + numerator, keep denominator.

Step 04 of 04

Why this matters. Decomposing into unit fractions makes adding, subtracting, and visualizing fractions transparent — instead of memorizing rules, you're literally counting pieces.

Key insight

Numerator = how many unit pieces. Same denominator → just count the numerators. Mixed number = wholes + leftover unit pieces.

4.G.2 — Add fractions with like denominators.

Step 01 of 04

When fractions have the SAME denominator, add the numerators and keep the denominator.

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

Two pieces + three pieces = five pieces. Pieces are the same size, so the denominator doesn't change.

Step 02 of 04

Don't add the denominators!

\frac{2}{7} + \frac{3}{7} \neq = \frac{5}{14}

The denominator names the SIZE of pieces. Adding it would shrink the pieces — but we still have the same-sized pieces, just more of them.

Step 03 of 04

Result > 1. The result might be improper (numerator ≥ denominator). That's fine — convert to mixed if asked.

\frac{4}{5} + \frac{3}{5} = \frac{7}{5} = 1 \frac{2}{5}

Step 04 of 04

Three or more. Same approach.

\frac{1}{8} + \frac{3}{8} + \frac{2}{8} = \frac{6}{8} = \frac{3}{4}

Add all numerators, keep the same denominator, simplify if possible.

Key insight

Like denominators: add numerators, keep denominator. Don't touch the denominator (it names the size of the pieces). Convert improper to mixed and simplify if asked.

4.G.3 — Subtract fractions with like denominators.

Step 01 of 04

Same idea as addition, but with subtraction.

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

Five pieces minus two pieces = three pieces. Pieces are the same size.

Step 02 of 04

From a mixed or whole number. If the whole-number part has 0 pieces of the right size, "borrow" 1 whole as denominator-sized pieces.

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

Rewrote 1 as $5/5$ (using the same denominator as the other fraction).

Step 03 of 04

Mixed-number subtraction.

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

Borrow 1 from the 3 to add to $1/4$ : $3 \frac{1}{4} = 2 \frac{5}{4}$ . Now subtract:

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

Step 04 of 04

Word problem. A jar has 7/8 cup of flour. You use 3/8. How much is left?

\frac{7}{8} - \frac{3}{8} = \frac{4}{8} = \frac{1}{2} cup

Key insight

Like denominators: subtract numerators. From a whole, rewrite the whole as denominator-over-denominator. Mixed numbers may need borrowing — same idea as borrowing in whole-number subtraction.

4.G.4 — Add and subtract mixed numbers (like denominators).

Step 01 of 04

Two ways to handle mixed-number arithmetic: (1) add/subtract whole and fraction parts SEPARATELY, or (2) convert each to improper fractions and operate.

Step 02 of 04

Method 1 — separate parts.

2 \frac{1}{6} + 3 \frac{2}{6}

Whole parts: $2 + 3 = 5$ . Fraction parts: $1/6 + 2/6 = 3/6 = 1/2$ .

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

Step 03 of 04

If the fraction parts add to MORE than 1, carry to the whole.

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

Whole parts: 3. Fractions: $4/5 + 3/5 = 7/5 = 1 \frac{2}{5}$ . Combine: $3 + 1 \frac{2}{5} = 4 \frac{2}{5}$ .

Step 04 of 04

Subtraction with borrowing.

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

Can't subtract 3/4 from 1/4. Borrow 1 from the 5: $5 \frac{1}{4} = 4 \frac{5}{4}$ .

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

Key insight

Add or subtract whole and fraction parts separately. Carry / borrow between them when the fraction parts cross a whole. Or convert everything to improper fractions if you prefer.

4.G.5 — Multiply a fraction by a whole number.

Step 01 of 04

Multiplying a fraction by a whole number is repeated addition of the fraction.

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

Multiply the whole number by the numerator. Keep the denominator.

Step 02 of 04

Visual model. 4 × 1/3 = four pieces, each 1/3 of the whole. Total = 4/3 = 1 1/3. Fits the pattern.

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

Step 03 of 04

Worked example. 6 × 5/8.

6 \times \frac{5}{8} = \frac{30}{8} = 3 \frac{6}{8} = 3 \frac{3}{4}

Step 04 of 04

Word problem. A recipe needs 3/4 cup of flour per batch. How much for 5 batches?

5 \times \frac{3}{4} = \frac{15}{4} = 3 \frac{3}{4} cups

Key insight

Multiply the whole number by the numerator; keep the denominator. Same as repeated addition. Convert improper results to mixed if asked.

4.G.6 — Fraction word problems.

Step 01 of 04

Read carefully to identify the operation.

"Combine"	→add
"How much remains"	→subtract
"How much in all (with same-size parts)"	→multiply
"X times as much"	→multiply

Step 02 of 04

Add example. A pitcher has 3/8 gal and another has 1/8 gal. How much total?

\frac{3}{8} + \frac{1}{8} = \frac{4}{8} = \frac{1}{2} gal

Step 03 of 04

Subtract example. A board is 5 ft long. You cut off 2 1/4 ft. How much remains?

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

Step 04 of 04

Multiply example. Each glass holds 2/3 cup. How much in 5 glasses?

5 \times \frac{2}{3} = \frac{10}{3} = 3 \frac{1}{3} cups

Key insight

Translate the question to an operation, do the arithmetic carefully (especially borrowing for subtraction), then state the answer with UNITS.

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