Subtracting Mixed Numbers with Renaming: Fifth Grade Mathematics
Subtracting mixed numbers with renaming (borrowing) is necessary when the fraction in the minuend is smaller than the fraction in the subtrahend.
The Concept of Renaming a Whole as a Fraction
Just as you borrow 1 ten to make 10 ones in whole number subtraction, you borrow 1 whole from the whole number of a mixed number and convert it into fractional parts (d/d) to add to the existing numerator.
+--------------------------------------------------------------------------+
| THE RENAMING MECHANISM VISUALIZED |
| |
| To subtract: 6 1/5 - 2 3/5 |
| |
| Notice: You cannot subtract 3/5 from 1/5! |
| Step 1: Borrow 1 whole from 6: |
| 6 becomes 5. |
| Step 2: Convert that 1 whole into fifths: |
| 1 whole = 5/5. |
| Step 3: Add the 5/5 to the existing 1/5: |
| 5/5 + 1/5 = 6/5. |
| Step 4: Rewrite the mixed number: |
| 6 1/5 is renamed as 5 6/5. |
| Step 5: Now subtract: |
| 5 6/5 - 2 3/5 = 3 3/5. |
+--------------------------------------------------------------------------+
The Quick Renaming Shortcut:
New Numerator = Original Numerator + Denominator
For 6 1/5: borrow 1 from 6 (making 5), then add 1 + 5 = 6 in numerator -> 5 6/5.
Renaming with Unlike Denominators
When subtracting mixed numbers with unlike denominators that require renaming:
1. Find the LCD and rename fractions first.
2. Compare the numerators: if the top numerator is smaller, rename 1 whole.
3. Subtract fractions, then subtract whole numbers.
4. Simplify the final difference.
Worked Example: 5 1/6 - 2 3/4
Step 1: LCD of 6 and 4 is 12.
5 1/6 = 5 2/12
2 3/4 = 2 9/12
Step 2: 2/12 < 9/12. We must rename 5 2/12:
Borrow 1 from 5 (becomes 4).
1 whole = 12/12.
Add: 12/12 + 2/12 = 14/12.
5 2/12 becomes 4 14/12.
Step 3: Subtract:
4 14/12 - 2 9/12 = 2 5/12.
Subtracting a Mixed Number from a Whole Number
When subtracting from a plain whole number (e.g., 8 - 3 2/5), you must rename 1 whole into the required denominator:
- 8 = 7 + 1 = 7 5/5.
- 7 5/5 - 3 2/5 = 4 3/5.
Exercises and Solutions
Exercise 1: Renaming with Like Denominators. Solve 7 2/9 - 4 7/9. Show the renaming of the top mixed number step by step.
Solution 1: Compare the fractions: 2/9 is smaller than 7/9, so you cannot subtract without renaming. Borrow 1 whole from 7: the 7 becomes 6. Convert the borrowed 1 whole into ninths: 1 = 9/9. Add 9/9 to the existing 2/9: 9/9 + 2/9 = 11/9. Rewrite 7 2/9 as 6 11/9. Now subtract: 6 11/9 - 4 7/9. Subtract fractions: 11/9 - 7/9 = 4/9. Subtract whole numbers: 6 - 4 = 2. Combine parts: 2 4/9. The difference is 2 4/9.
Exercise 2: Unlike Denominators with Renaming. Solve 6 1/3 - 2 5/8. Show finding the LCD, renaming, and the final simplified answer.
Solution 2: Find the LCD of 3 and 8: 3 and 8 share no common factors, so LCD = 24. Rename fractions to twenty-fourths: 1/3 = 8/24, so 6 1/3 = 6 8/24; 5/8 = 15/24, so 2 5/8 = 2 15/24. Compare numerators: 8 < 15, requiring renaming. Borrow 1 from 6 (becomes 5). Add 24/24 to 8/24: 24 + 8 = 32/24. Rewrite 6 8/24 as 5 32/24. Subtract fractions: 32/24 - 15/24 = 17/24. Subtract whole numbers: 5 - 2 = 3. The final answer is 3 17/24.