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Mathematics › 🌉 Grade 5 › Adding & Subtracting Mixed Numbers: Same-Size Pieces, Please!
🌉 Grade 5 · Lesson 10 of 11

Adding & Subtracting Mixed Numbers: Same-Size Pieces, Please!

Whole bars and fraction slices together — line up the slice sizes and you can combine anything.

Grade 5Elementary
Adding & Subtracting Mixed Numbers: Same-Size Pieces, Please! — illustration
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The big idea: A mixed number is a whole number plus a fraction. To add or subtract mixed numbers, both fractions must first be re-sliced to the same denominator so you are always combining equal-sized pieces. For subtraction, when the fraction part is too small, you unwrap one whole into extra slices before subtracting.
🎯 By the end, you'll be able to
  • Convert fractions to a common denominator before adding mixed numbers
  • Add and subtract the whole-number parts and fraction parts separately
  • Unwrap one whole into equivalent fraction slices when the fraction part is too small to subtract
  • Verify a result by converting mixed numbers to improper fractions
📎 You should already know
  • Finding the least common denominator (LCD)
  • What a mixed number is

Two and a third plus one and three-quarters — how much apple?

You have 2⅓ apples and your friend has 1¾ apples. Together you clearly have more than 3 apples — but how much exactly?

A common mistake is to add the numerators and denominators separately: ⅓ + ¾ ≠ 4/7. To see why this is wrong, try ½ + ½. Common sense says the answer is 1. But adding tops and bottoms gives 2/4 = ½ — half an apple instead of a whole one. That’s clearly wrong!

The fix: before adding any fractions, re-slice both into the same size pieces. Then the numerators can be combined safely.

⚠️ NEVER add numerators and denominators as separate numbers!

Writing ½ + ⅔ = 3/5 is a very common error. But ½ is already bigger than 3/5, so the “sum” would be smaller than one of the addends — that cannot be right.

Fractions are ratios. Their denominators describe slice sizes. You can only count slices together when all the slices are the same size — that’s why you need a common denominator first.

Finding the LCD and re-slicing

To add 2⅓ + 1¾, you need both fractions to share a denominator.

  1. Find the LCM of 3 and 4. Multiples of 3: 3, 6, 9, 12. Multiples of 4: 4, 8, 12. LCD = 12.
  2. Re-slice ⅓: multiply top and bottom by 4 → ⅓ = 4/12.
  3. Re-slice ¾: multiply top and bottom by 3 → ¾ = 9/12.
  4. Rewrite both mixed numbers: 2⁴⁄₁₂ + 1⁹⁄₁₂.
  5. Add fractions: 4/12 + 9/12 = 13/12.
  6. Add whole parts: 2 + 1 = 3.
  7. Simplify the improper fraction: 13/12 = 1¹⁄₁₂, so 3 + 1¹⁄₁₂ = 4¹⁄₁₂.
🎮 Mixed Number Fraction Bars LIVE
Watch the whole bars and partial bars animate to a common denominator, then combine. The 'unwrap' animation shows how to borrow a whole when subtracting.
📝 Worked example: Find 2⅓ + 1¾.
  1. Find LCD of 3 and 4: LCM = 12.
  2. Re-slice ⅓: multiply top & bottom by 4 → 4/12.
  3. Re-slice ¾: multiply top & bottom by 3 → 9/12.
  4. Rewrite: 2⁴⁄₁₂ + 1⁹⁄₁₂.
  5. Add fraction parts: 4/12 + 9/12 = 13/12.
  6. Add whole-number parts: 2 + 1 = 3.
  7. Combine: 3¹³⁄₁₂. Since 13/12 > 1, simplify: 3 + 1¹⁄₁₂ = 4¹⁄₁₂.
✓ 2⅓ + 1¾ = <strong>4¹⁄₁₂</strong>.
✨ Unwrapping a whole for subtraction

For 3¼ − 1¾, notice that ¼ < ¾ — the top fraction is smaller, so we cannot subtract yet.

Solution: unwrap one whole from the whole-number part.

  • 3¼ = 2 + 1 + ¼ = 2 + 4/4 + ¼ = 2⁵⁄₄.
  • Now 5/4 > ¾, so subtraction works.
  • Fraction part: 5/4 − 3/4 = 2/4 = ½.
  • Whole-number part: 2 − 1 = 1.
  • Answer: 1½.
📝 Worked example: Find 3¼ − 1¾ using the unwrap method.
  1. Check: both fractions already share denominator 4, but ¼ < ¾. Unwrap needed.
  2. Unwrap 1 whole from 3: 3¼ = 2 + 4/4 + ¼ = 2⁵⁄₄.
  3. Subtract fraction parts: 5/4 − 3/4 = 2/4.
  4. Subtract whole-number parts: 2 − 1 = 1.
  5. Combine: 1²⁄₄. Simplify: 2/4 = ½.
✓ 3¼ − 1¾ = <strong>1½</strong>.
📝 Worked example: Find 1⁵⁄₆ + 2⅔.
  1. Find LCD of 6 and 3: LCM = 6 (since 6 is already a multiple of 3).
  2. Re-slice ⅔: multiply top & bottom by 2 → ⅔ = 4/6. (5/6 stays as 5/6.)
  3. Rewrite: 1⁵⁄₆ + 2⁴⁄₆.
  4. Add fraction parts: 5/6 + 4/6 = 9/6.
  5. Add whole-number parts: 1 + 2 = 3.
  6. Combine: 3⁹⁄₆. Simplify: 9/6 = 1½. So 3 + 1½ = 4½.
✓ 1⁵⁄₆ + 2⅔ = <strong>4½</strong>.

Check your understanding

1. What is the LCD of 4 and 6?
Multiples of 4: 4, 8, 12. Multiples of 6: 6, 12. The smallest number they share is 12.
2. What is 1½ + 2⅓?
LCD of 2 and 3 is 6. ½ = 3/6, ⅓ = 2/6. Add: 1³⁄₆ + 2²⁄₆ = 3⁵⁄₆.
3. For 4¼ − 1¾, why must we unwrap a whole from 4?
The denominators are already the same (both 4), so no LCD step is needed. But ¼ < ¾, meaning the top fraction is too small. Unwrapping one whole gives 3⁵⁄₄, and now 5/4 > ¾.
4. A student added ⅓ + ½ and got 2/5. What mistake did they make?
Adding tops (1+1=2) and bottoms (3+2=5) is never a valid method for fractions. Correct method: LCD=6, ⅓=2/6, ½=3/6, sum = 5/6.
5. A recipe needs 2½ cups of flour. You already have 1¼ cups. How many more cups do you need?
2½ − 1¼. LCD = 4. ½ = 2/4. Fraction: 2/4 − 1/4 = 1/4. Whole: 2 − 1 = 1. Answer: 1¼ cups.
✅ Key takeaways
  • A mixed number is a whole number plus a fraction; add or subtract them part by part.
  • Always find a common denominator before adding or subtracting the fraction parts.
  • Never add numerators and denominators separately — that always gives a wrong answer.
  • When the fraction part of the top number is too small to subtract, unwrap 1 whole into equivalent fraction slices.
  • After adding fraction parts, simplify any improper fraction result by converting the excess back into whole numbers.
  • Verify answers by converting mixed numbers to improper fractions and checking the arithmetic.