Add and Subtract Mixed Numbers | Fifth Grade

Complete Notes & Formulas

1. Estimate Sums and Differences of Mixed Numbers

Definition: Estimate the sum or difference of mixed numbers by rounding each mixed number to the nearest whole number before calculating.

📝 Steps to Estimate:

Step 1: Round each mixed number to the nearest whole number
Step 2: Add or subtract the rounded whole numbers
Step 3: State your estimate

🔑 Rounding Rules for Mixed Numbers:

If fraction part < 1/2 → Round DOWN to the whole number
If fraction part ≥ 1/2 → Round UP to the next whole number

✏️ Example: Estimate 8 3/5 + 3 1/8

Step 1: Round each mixed number

• 8 3/5: Is 3/5 ≥ 1/2? Yes (3/5 = 0.6) → Round to 9

• 3 1/8: Is 1/8 ≥ 1/2? No (1/8 = 0.125) → Round to 3

Step 2: Add: 9 + 3 = 12

Estimate: About 12

2. Add and Subtract Mixed Numbers: Without Regrouping

Definition: When adding or subtracting mixed numbers without regrouping, the fraction parts can be directly added or subtracted without borrowing.

📝 Steps (Without Regrouping):

Find LCD of the fractional parts
Convert fractions to equivalent fractions with LCD
Add or subtract the fraction parts
Add or subtract the whole number parts
Simplify if needed

✏️ Example: 5 1/4 + 2 1/6

Step 1: Find LCD of 4 and 6 = 12

Step 2: Convert fractions:

• 5 1/4 = 5 3/12

• 2 1/6 = 2 2/12

Step 3: Add fractions: 3/12 + 2/12 = 5/12

Step 4: Add whole numbers: 5 + 2 = 7

Answer: 7 5/12

3. Add Mixed Numbers with Unlike Denominators (With Regrouping)

Definition: When the sum of the fraction parts equals or exceeds 1, regroup by converting the improper fraction to a mixed number and adding it to the whole number part.

📝 Steps (With Regrouping):

Find LCD and convert fractions
Add the fractional parts
If fraction ≥ 1, convert to mixed number
Add whole numbers (including any from regrouping)
Simplify final answer

✏️ Example: 4 5/6 + 3 3/4

Step 1: LCD of 6 and 4 = 12

• 4 5/6 = 4 10/12

• 3 3/4 = 3 9/12

Step 2: Add fractions: 10/12 + 9/12 = 19/12

Step 3: 19/12 = 1 7/12 (regroup!)

Step 4: Add whole numbers: 4 + 3 + 1 = 8

Answer: 8 7/12

4. Subtract Mixed Numbers with Unlike Denominators (Borrowing/Regrouping)

Definition: When the fraction being subtracted is larger than the fraction you're subtracting from, you must borrow 1 from the whole number part.

📝 Steps (With Borrowing):

Find LCD and convert fractions
Compare fraction parts
If first fraction < second fraction, borrow 1 from whole number
Convert borrowed 1 to fraction with same denominator
Add borrowed fraction to original fraction
Subtract fractions, then whole numbers

Borrowing Formula:
If subtracting and a/b < c/d, borrow 1 = d/d from whole number

✏️ Example: 7 1/4 - 3 2/3

Step 1: LCD of 4 and 3 = 12

• 7 1/4 = 7 3/12

• 3 2/3 = 3 8/12

Step 2: Compare: 3/12 < 8/12 (Need to borrow!)

Step 3: Borrow 1 from 7: 7 becomes 6

Step 4: 1 = 12/12, so 3/12 + 12/12 = 15/12

Step 5: Rewrite: 6 15/12 - 3 8/12

Step 6: Subtract: 15/12 - 8/12 = 7/12

Step 7: Whole numbers: 6 - 3 = 3

Answer: 3 7/12

5. Add and Subtract Mixed Numbers (Combined Operations)

Definition: Problems that combine both addition and subtraction of mixed numbers in the same expression.

✏️ Example: 5 1/2 + 3 1/4 - 2 1/8

Step 1: LCD = 8

• 5 1/2 = 5 4/8

• 3 1/4 = 3 2/8

• 2 1/8 = 2 1/8

Step 2: Work left to right: First add

5 4/8 + 3 2/8 = 8 6/8

Step 3: Then subtract

8 6/8 - 2 1/8 = 6 5/8

Answer: 6 5/8

6. Add and Subtract Mixed Numbers: Word Problems

Definition: Apply mixed number operations to solve real-world problems.

✏️ Example 1: Measurement Problem

John ran 2 3/4 miles on Monday and 1 1/2 miles on Tuesday. How far did he run in total?

Solution:

Add: 2 3/4 + 1 1/2

LCD = 4

2 3/4 + 1 2/4 = 3 5/4 = 4 1/4

Answer: 4 1/4 miles

✏️ Example 2: Comparison Problem

Sarah had 5 1/3 cups of flour. She used 2 3/4 cups for a recipe. How much flour is left?

Solution:

Subtract: 5 1/3 - 2 3/4

LCD = 12

5 4/12 - 2 9/12

Borrow: 4 16/12 - 2 9/12 = 2 7/12

Answer: 2 7/12 cups left

7. Add and Subtract Fractions and Mixed Numbers in Recipes

Definition: Practical application in cooking measurements, often involving doubling recipes or adjusting ingredient amounts.

✏️ Example: Doubling a Recipe

A recipe calls for 2 1/4 cups of sugar and 1 3/4 cups of flour. If you double the recipe, how much of these two ingredients do you need in total?

Solution:

Sugar: 2 1/4 × 2 = 4 2/4 = 4 1/2 cups

Flour: 1 3/4 × 2 = 3 6/4 = 4 1/2 cups

Total: 4 1/2 + 4 1/2 = 9 cups

Answer: 9 cups total

8. Add and Subtract Fractions and Mixed Numbers: Multi-Step Word Problems

Definition: Problems requiring multiple operations to solve, combining addition and subtraction of mixed numbers.

✏️ Example: Multi-Step Problem

Maria had 10 1/2 yards of ribbon. She used 3 1/4 yards for a project, bought 2 3/8 more yards, then used another 1 1/2 yards. How much ribbon does she have now?

Solution:

Step 1: Start with 10 1/2 yards

Step 2: Subtract first use: 10 1/2 - 3 1/4 = 10 2/4 - 3 1/4 = 7 1/4

Step 3: Add new ribbon: 7 1/4 + 2 3/8 = 7 2/8 + 2 3/8 = 9 5/8

Step 4: Subtract second use: 9 5/8 - 1 1/2 = 9 5/8 - 1 4/8 = 8 1/8

Answer: 8 1/8 yards remaining

9. Complete Addition and Subtraction Sentences with Mixed Numbers

Definition: Find the missing mixed number in an equation by working backwards.

✏️ Examples:

Example 1: 3 1/2 + ___ = 7 1/4

Solution: Subtract to find the missing addend

7 1/4 - 3 1/2 = 7 1/4 - 3 2/4 = 3 3/4

Answer: 3 3/4

Example 2: ___ - 2 2/3 = 4 1/6

Solution: Add to find the starting number

4 1/6 + 2 2/3 = 4 1/6 + 2 4/6 = 6 5/6

Answer: 6 5/6

10. Compare Sums and Differences of Mixed Numbers

Definition: Evaluate and compare the results of two different mixed number operations.

✏️ Example: Compare (5 1/2 + 2 1/4) ___ (10 - 1 3/8)

Left side: 5 1/2 + 2 1/4

= 5 2/4 + 2 1/4 = 7 3/4

Right side: 10 - 1 3/8

= 9 8/8 - 1 3/8 = 8 5/8

Compare: 7 3/4 vs 8 5/8

7 < 8, so 7 3/4 < 8 5/8

Answer: (5 1/2 + 2 1/4) < (10 - 1 3/8)

Quick Reference Chart

Operation	When to Use	Key Steps
Addition (No Regroup)	Fraction sum < 1	LCD → Add fractions → Add wholes
Addition (Regroup)	Fraction sum ≥ 1	Convert improper → Add to whole
Subtraction (No Borrow)	First fraction > Second	LCD → Subtract directly
Subtraction (Borrow)	First fraction < Second	Borrow 1 → Convert → Subtract