Multiply & Divide Fractions

5.E.1Fraction × whole01/04

Multiply a fraction by a whole number.

Step01of 04

Multiply numerator by the whole number; keep the denominator.

5 \times \frac{3}{8} = \frac{15}{8} = 1 \frac{7}{8}

5.E.1 — Multiply a fraction by a whole number.

Step 01 of 04

Multiply numerator by the whole number; keep the denominator.

5 \times \frac{3}{8} = \frac{15}{8} = 1 \frac{7}{8}

Step 02 of 04

Visualize. 5 × 3/8 = five copies of 3/8 = 15/8. Same answer as repeated addition: 3/8 + 3/8 + 3/8 + 3/8 + 3/8.

Step 03 of 04

If the result is improper, convert to a mixed number when asked.

Step 04 of 04

Word problem. Each glass holds 3/4 cup. How much for 6 glasses?

6 \times \frac{3}{4} = \frac{18}{4} = 4 \frac{2}{4} = 4 \frac{1}{2} cups

Key insight

Multiply numerator by the whole number; denominator stays. Result might be improper — convert to mixed if asked.

5.E.2 — Multiply two fractions.

Step 01 of 04

Multiply numerators together. Multiply denominators together.

\frac{a}{b} \times \frac{c}{d} = \frac{a c}{b d}

Step 02 of 04

Worked example.

\frac{2}{3} \times \frac{4}{5} = \frac{2 \cdot 4}{3 \cdot 5} = \frac{8}{15}

Step 03 of 04

Cancel before multiplying for cleaner numbers.

\frac{3}{4} \times \frac{8}{9}

The 3 and 9 share a factor of 3; the 4 and 8 share a factor of 4. Cancel: $\frac{1}{1} \times \frac{2}{3} = \frac{2}{3}$ .

Step 04 of 04

Visualize. 1/2 × 1/3 = "half of one-third." Splitting 1/3 into halves gives 1/6 — matches the formula.

Key insight

Multiply numerators, multiply denominators. Cancel common factors first to keep numbers small. Result is "of": 2/3 of 4/5 means take 2/3 of a 4/5 chunk.

5.E.3 — Multiply mixed numbers.

Step 01 of 04

Convert each mixed number to an improper fraction first. Then multiply normally.

Step 02 of 04

Worked example. $2 \frac{1}{4} \times 1 \frac{2}{3}$ .

2 \frac{1}{4} = \frac{9}{4}, 1 \frac{2}{3} = \frac{5}{3}

Step 03 of 04

Multiply:

\frac{9}{4} \times \frac{5}{3} = \frac{45}{12} = \frac{15}{4} = 3 \frac{3}{4}

Step 04 of 04

Don't try to "multiply parts." 2 1/4 × 1 2/3 ≠ 2 × 1 + 1/4 × 2/3. Always convert to improper first.

Key insight

Mixed → improper → multiply → simplify → mixed (if asked). Never multiply whole and fraction parts separately — that gives wrong answers.

5.E.4 — Multiplication as scaling.

Step 01 of 04

Multiplying by a number tells you how the result COMPARES to the original — bigger, smaller, or same.

Multiplier > 1	→result is BIGGER than the original
Multiplier = 1	→result equals the original
Multiplier between 0 and 1	→result is SMALLER than the original

Step 02 of 04

Examples.

8 × 3	→24 (bigger)
8 × 1	→8 (same)
8 × 1/2	→4 (smaller — half)
8 × 1/4	→2 (smaller — quarter)

Step 03 of 04

This is sometimes counterintuitive: multiplying by a FRACTION less than 1 makes the answer SMALLER. (Compare to multiplying whole numbers, which always makes the answer bigger or equal.)

Step 04 of 04

Without computing, decide. "Will $12 \times 5/8$ be bigger or smaller than 12?" Smaller — 5/8 < 1.

Key insight

Multiplying by a fraction less than 1 SHRINKS the number. By exactly 1 keeps it the same. By more than 1 grows it. Useful sanity check before computing.

5.E.5 — Divide a unit fraction by a whole number.

Step 01 of 04

Dividing a fraction by a whole number SHRINKS the fraction. Multiply the denominator by the whole number; keep the numerator.

\frac{1}{n} \div k = \frac{1}{nk}

Step 02 of 04

Worked example. 1/3 ÷ 4.

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

Picture: take 1/3 of a pie and split it into 4 equal pieces. Each piece is 1/12 of the original whole.

Step 03 of 04

Equivalent rule: "divide by k = multiply by 1/k."

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

Step 04 of 04

Word problem. A cup of ice cream is shared equally among 4 friends, but only 1/3 cup is in the carton. How much per friend?

\frac{1}{3} \div 4 = \frac{1}{12} cup per friend

Key insight

Dividing a fraction by a whole shrinks it. Multiply denominator by the whole number, OR multiply by the reciprocal — same answer.

5.E.6 — Divide a whole number by a unit fraction.

Step 01 of 04

Dividing a whole number by a unit fraction GROWS the answer. Multiply the whole by the denominator.

k \div \frac{1}{n} = k \times n

Step 02 of 04

Worked example. 6 ÷ 1/4.

6 \div \frac{1}{4} = 6 \times 4 = 24

Picture: how many quarter-pieces fit into 6 pies? Each pie holds 4 quarters; 6 × 4 = 24 quarters total.

Step 03 of 04

Same rule: multiply by the reciprocal.

5 \div \frac{1}{3} = 5 \times 3 = 15

Step 04 of 04

Word problem. A 5-yard ribbon is cut into 1/2-yard pieces. How many?

5 \div \frac{1}{2} = 5 \times 2 = 10 pieces

Key insight

Dividing a whole by 1/n is the same as multiplying by n. Bigger denominator → more pieces fit. The "÷ = × reciprocal" rule covers both directions of fraction division.

5.E.7 — Fraction word problems (multiply and divide).

Step 01 of 04

Multiplication often appears as "of" in word problems. Division appears as "how many groups" or "shared equally."

Step 02 of 04

Multiplication. A garden has 24 sq ft. 3/4 is planted with vegetables. How many sq ft of vegetables?

24 \times \frac{3}{4} = 18 sq ft

Step 03 of 04

Division. A 4-cup carton of yogurt is split into 1/3-cup servings. How many servings?

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

Step 04 of 04

Mixed example. A board is 5 1/4 ft long. You need 7/8-ft pieces for shelves. How many pieces?

5 \frac{1}{4} \div \frac{7}{8} = \frac{21}{4} \times \frac{8}{7} = \frac{168}{28} = 6 pieces

Key insight

"Of" means multiply. "How many groups / shared equally" often means divide. Convert mixed to improper before dividing or multiplying. Always answer with units.

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