Adding & Subtracting Mixed Numbers - Sixth Grade Mathematics
Adding and subtracting mixed numbers combines whole number arithmetic with fraction skills, requiring careful handling of regrouping, decomposition, and common denominators.
Two Distinct Computational Approaches
There are two primary methods for adding and subtracting mixed numbers. Both are mathematically sound, and choosing the right method depends on the numbers involved.
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| MIXED NUMBER ARITHMETIC STRATEGIES |
+-------------------+-----------------------+-----------------------------+
| Method | Procedure | Best Applied To |
+-------------------+-----------------------+-----------------------------+
| Whole-and-Fraction| Operate on whole parts| Addition problems, and |
| Decomposition | and fractional parts | subtraction where no complex|
| | separately; regroup | borrowing across wholes |
+-------------------+-----------------------+-----------------------------+
| Improper Fraction | Convert both mixed | Subtraction with borrowing, |
| Conversion | numbers to improper | expressions involving |
| | fractions first | small denominators |
+-------------------+-----------------------+-----------------------------+
Adding Mixed Numbers: Step-by-Step
When adding mixed numbers using the whole-and-fraction decomposition method: First, find a common denominator for the fraction components and scale both fractions. Second, add the whole numbers together. Third, add the fraction numerators together over the common denominator. Fourth, if the fractional sum is an improper fraction, convert it into a mixed number and add its whole number part to the existing whole number total.
Evaluate: 3 and 3/4 + 2 and 5/6
Step 1: Find LCD of 4 and 6. LCM(4, 6) = 12.
3/4 = 9/12
5/6 = 10/12
Step 2: Add whole numbers: 3 + 2 = 5.
Step 3: Add fractions: 9/12 + 10/12 = 19/12.
Step 4: Regroup improper fraction:
19/12 = 1 and 7/12.
Step 5: Combine whole numbers:
5 + 1 and 7/12 = 6 and 7/12.
Subtracting Mixed Numbers and Regrouping (Borrowing)
Subtraction presents a unique challenge when the fractional part of the first number (minuend) is smaller than the fractional part of the second number (subtrahend). In such cases, you must borrow 1 whole from the minuend's whole number part and decompose it into fractional units.
Evaluate: 5 and 1/3 - 2 and 3/4
Step 1: Common Denominator: LCD(3, 4) = 12.
5 and 1/3 = 5 and 4/12
2 and 3/4 = 2 and 9/12
Problem becomes: 5 and 4/12 - 2 and 9/12.
Step 2: Notice that you cannot subtract 9/12 from 4/12 directly!
Decompose 1 from whole number 5:
5 = 4 + 1 = 4 + 12/12.
Add the 12/12 to 4/12:
4 + (12/12 + 4/12) = 4 and 16/12.
Step 3: Now subtract:
Whole numbers: 4 - 2 = 2.
Fractions: 16/12 - 9/12 = 7/12.
Final Result: 2 and 7/12.
The Decomposition Principle
Remember that borrowing 1 from the whole number means adding d/d to the fraction, where d is the common denominator. If the denominator is 12, borrowing 1 gives you 12 twelfths. If the denominator is 8, borrowing 1 gives you 8 eighths. Simply add the denominator to the numerator: 4 + 12 = 16 twelfths.
Subtraction via Improper Fractions (Alternative)
You can also solve 5 and 1/3 - 2 and 3/4 by converting both to improper fractions: 5 and 1/3 = 16/3 = 64/12. 2 and 3/4 = 11/4 = 33/12. Subtract numerators: 64/12 - 33/12 = 31/12. Convert back to mixed number: 31 / 12 = 2 with a remainder of 7, giving 2 and 7/12. This method avoids borrowing entirely, though it involves larger numerators.
Exercises
Exercise 1. Calculate the sum: 4 and 2/5 + 3 and 7/10. Express your final answer as a mixed number in simplest form.
Exercise 2. Calculate the difference: 6 and 1/6 - 2 and 3/4 using the regrouping (borrowing) method. Show the regrouped fraction clearly.
Exercise 3. A carpenter had a wooden board measuring 8 and 1/4 feet long. He cut off a piece measuring 3 and 5/8 feet. What is the length of the remaining board?
Solutions
Solution to Exercise 1. For 4 and 2/5 + 3 and 7/10: find LCD of 5 and 10, which is 10. Scale 2/5 to tenths: 2/5 = 4/10. The problem is 4 and 4/10 + 3 and 7/10. Add the whole numbers: 4 + 3 = 7. Add the fractions: 4/10 + 7/10 = 11/10. Convert 11/10 to a mixed number: 11/10 = 1 and 1/10. Combine with the whole numbers: 7 + 1 and 1/10 = 8 and 1/10.
Solution to Exercise 2. For 6 and 1/6 - 2 and 3/4: find LCD of 6 and 4, which is 12. Convert fractions to twelfths: 6 and 2/12 - 2 and 9/12. Since 2/12 is smaller than 9/12, borrow 1 from 6: 6 becomes 5, and 1 whole equals 12/12. Add 12/12 to 2/12 to get 14/12. The problem is now 5 and 14/12 - 2 and 9/12. Subtract whole numbers: 5 - 2 = 3. Subtract fractions: 14/12 - 9/12 = 5/12. The final result is 3 and 5/12.
Solution to Exercise 3. Subtract 3 and 5/8 from 8 and 1/4: LCD of 4 and 8 is 8. Convert 1/4 to eighths: 1/4 = 2/8, giving 8 and 2/8 - 3 and 5/8. Borrow 1 from 8: 8 becomes 7, and add 8/8 to 2/8 to obtain 10/8. The expression becomes 7 and 10/8 - 3 and 5/8. Subtract whole numbers: 7 - 3 = 4. Subtract fractions: 10/8 - 5/8 = 5/8. The remaining board is 4 and 5/8 feet long.