T.2 Multiply fractions by whole numbers
What does it mean to multiply a fraction by a whole number?
Multiplying a fraction by a whole number means taking the fraction a certain number of times. You can think of it as repeated addition of the same fraction. The whole number tells you how many times to add the fraction.
- 3 × 14 means add 14 + 14 + 14 = 34.
- 4 × 25 = 25 + 25 + 25 + 25 = 85.
Multiplication is faster than repeated addition. When you see “of” in a word problem, like “three-fourths of 20,” that also means multiplication: 34 × 20.
The rule for multiplying a fraction by a whole number
Rule: Multiply the numerator (top number) by the whole number. Keep the denominator (bottom number) the same. Then simplify (reduce) the fraction if needed.
Formula: whole number × numeratordenominator = whole number × numeratordenominator
- 5 × 38 = 5 × 38 = 158
- 2 × 49 = 2 × 49 = 89
- 7 × 23 = 143
You never multiply the denominators together when multiplying a fraction by a whole number. Only the numerator changes. The denominator stays exactly the same.
Writing the answer as a mixed number
When the numerator is larger than the denominator (an improper fraction), you should rewrite it as a mixed number. A mixed number has a whole number part and a fraction part.
- 158 → 15 ÷ 8 = 1 remainder 7 → 1 78
- 143 → 14 ÷ 3 = 4 remainder 2 → 4 23
- 94 → 9 ÷ 4 = 2 remainder 1 → 2 14
If a problem says “simplify” or “write in simplest form,” always change improper fractions to mixed numbers (or whole numbers) and reduce any fraction part if possible.
Simplifying the answer
Simplifying means writing the fraction in its lowest terms. Divide the numerator and denominator by the greatest common factor (GCF). If your answer is a mixed number, simplify only the fraction part.
- 4 × 310 = 1210 = 65 = 1 15
- 3 × 26 = 66 = 1 (whole number)
- 5 × 28 = 108 = 54 = 1 14
You can simplify before multiplying. Divide the whole number and the denominator by a common factor first. For example: 4 × 38 → divide 4 and 8 by 4 → 1 × 32 = 32 = 1 12.
Multiplying a whole number by a fraction (order does not matter)
Multiplication is commutative, meaning you can switch the order of the factors. The answer is the same whether you write whole number × fraction or fraction × whole number.
- 34 × 2 = 2 × 34 = 64 = 32 = 1 12
- 56 × 3 = 3 × 56 = 156 = 52 = 2 12
Always write the whole number as a fraction over 1 if it helps: 4 = 41. Then multiply numerators and denominators. That works for every problem!
Word problems with multiplying fractions by whole numbers
In real-life situations, you often need to find a fraction of a whole number. The word “of” usually means multiply. Read carefully to decide which number is the whole number and which is the fraction.
- Problem: A recipe needs 23 cup of sugar for one batch. How much sugar for 4 batches?
→ 4 × 23 = 83 = 2 23 cups. - Problem: Maria ran 34 of a mile each day for 5 days. How many miles total?
→ 5 × 34 = 154 = 3 34 miles. - Problem: What is 25 of 30?
→ 25 × 30 = 605 = 12.
Draw a picture or use a number line to check your thinking. If you multiply a fraction by a whole number, the answer can be less than, equal to, or greater than the whole number.
Common mistakes to avoid
Many students make simple errors when first learning this skill. Knowing these mistakes will help you get every answer correct.
- Mistake: Multiplying the denominator. Wrong: 3 × 14 = 312.
Fix: Keep denominator the same: 34. - Mistake: Forgetting to simplify. Wrong: 2 × 46 = 86 (not simplified).
Fix: 86 = 43 = 1 13. - Mistake: Writing an improper fraction when a mixed number is expected.
Fix: Always check if the numerator is larger than the denominator. If yes, convert to a mixed number.
Slow down and check each step: multiply numerator, keep denominator, simplify, then convert to mixed number if needed.
Practice strategy and mental math tips
You can solve many fraction multiplication problems mentally if you look for shortcuts. These strategies build number sense and save time.
- 6 × 13 = 6 ÷ 3 = 2 (because 6 × 13 means 6 thirds = 2 whole)
- 8 × 34 = (8 ÷ 4) × 3 = 2 × 3 = 6
- 10 × 25 = (10 ÷ 5) × 2 = 2 × 2 = 4
This shortcut works only when the whole number is divisible by the denominator. Otherwise, use the standard multiplication rule.
Review: steps to multiply any fraction by a whole number
Follow these four steps every time. You will get the correct answer without confusion.
Step 1: Write the problem (whole number × fraction).
Step 2: Multiply the whole number by the numerator.
Step 3: Write the product over the original denominator.
Step 4: Simplify and convert to a mixed number if necessary.
- Problem: 7 × 410
- Step 2: 7 × 4 = 28
- Step 3: 2810
- Step 4: Simplify: divide 28 and 10 by 2 → 145 → mixed number: 14 ÷ 5 = 2 remainder 4 → 2 45
- Final answer: 2 45
If the whole number and denominator share a common factor, you can simplify before multiplying. That often keeps numbers smaller and easier to work with.
Common Core alignment:
CCSS.MATH.CONTENT.4.NF.B.4 – Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
CCSS.MATH.CONTENT.4.NF.B.4.A – Understand a fraction a/b as a multiple of 1/b.
CCSS.MATH.CONTENT.4.NF.B.4.B – Multiply a fraction by a whole number.
Notes for teachers
This lesson is aligned with CCSS.MATH.CONTENT.4.NF.B.4, 4.NF.B.4.A, and 4.NF.B.4.B. All content is 100% free, original, and fact-checked. Use it for whole-class instruction, independent study, small group intervention, or homework. Encourage students to draw area models or number lines alongside these written rules.