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U05 • Lesson 23 of 105

Fraction Operations with Mixed Numbers

Extends addition, subtraction, multiplication, and division of fractions to include mixed numbers. Students learn to add and subtract mixed numbers with regrouping, multiply a whole number by a mixed number, and solve real-world problems involving fractional quantities greater than one.

Upper Elementary • 3-5

Prerequisites: U04, M14, M15

Key Concepts

  • adding mixed numbers with regrouping
  • subtracting mixed numbers with borrowing
  • multiplying whole numbers by mixed numbers
  • solving word problems with mixed number operations

Fraction Operations with Mixed Numbers

Now that you understand mixed numbers and improper fractions, it is time to add, subtract, multiply, and divide with them. The key skill is knowing when to convert and how to regroup.

Adding Mixed Numbers

  1. Add the whole numbers.
  2. Add the fractions (find a common denominator if needed).
  3. If the fraction sum is improper, regroup it into the whole number.

Worked Example 1: Adding Mixed Numbers

Find 2 3/4 + 1 2/4.

  1. Whole numbers: 2 + 1 = 3
  2. Fractions: 3/4 + 2/4 = 5/4
  3. 5/4 is improper! 5/4 = 1 1/4. Add the extra 1 to the whole number: 3 + 1 = 4.
2 3/4 + 1 2/4 = 4 1/4

Subtracting Mixed Numbers

Sometimes the fraction you are subtracting is larger than the one you are subtracting from. Then you need to borrow 1 from the whole number.

Worked Example 2: Subtracting with Borrowing

Find 5 1/3 - 2 2/3.

  1. Look at the fractions: 1/3 - 2/3? We cannot do this! Borrow 1 from 5.
  2. Rewrite 5 1/3 as 4 4/3 (borrow 1 whole = 3/3, so 1/3 + 3/3 = 4/3).
  3. Now subtract fractions: 4/3 - 2/3 = 2/3.
  4. Subtract whole numbers: 4 - 2 = 2.
5 1/3 - 2 2/3 = 2 2/3

Multiplying a Whole Number by a Mixed Number

Convert the mixed number to an improper fraction first, then multiply.

Worked Example 3: Multiplying

Find 3 x 2 1/5.

  1. Convert 2 1/5 to improper: 2 x 5 + 1 = 11/5.
  2. Multiply: 3 x 11/5 = 33/5.
  3. Convert back: 33/5 = 6 3/5.
3 x 2 1/5 = 6 3/5

Word Problems with Mixed Numbers

Problem-Solving Strategy

Read carefully to decide which operation to use: "How much altogether?" means add. "How much more?" means subtract. "Groups of" or "times" means multiply.

Common Mistake

When adding fractions with different denominators, students sometimes add the denominators too. That is wrong! Find a common denominator, convert the fractions, then add only the numerators. For example: 1/3 + 1/4 = 4/12 + 3/12 = 7/12, NOT 2/7.

Real-World Connection

A recipe calls for 2 1/2 cups of flour. If you want to make a triple batch, you need 3 x 2 1/2 = 7 1/2 cups of flour. Mixed number operations are essential in cooking!

Practice Problems

1. Add: 3 2/5 + 4 4/5

Show Answer

Whole numbers: 3+4=7. Fractions: 2/5+4/5=6/5=1 1/5. Regroup: 7+1=8. Answer: 8 1/5.

2. Add: 1 1/3 + 2 1/2

Show Answer

Common denominator is 6. 1/3=2/6, 1/2=3/6. Whole numbers: 1+2=3. Fractions: 2/6+3/6=5/6. Answer: 3 5/6.

3. Subtract: 6 1/4 - 3 3/4

Show Answer

1/4 < 3/4, so borrow: 6 1/4 = 5 5/4. Now: 5/4 - 3/4 = 2/4 = 1/2. Whole: 5-3=2. Answer: 2 1/2.

4. Multiply: 4 x 1 3/8

Show Answer

1 3/8 = 11/8. 4 x 11/8 = 44/8 = 5 4/8 = 5 1/2.

5. Maria ran 3 1/2 miles on Monday and 2 3/4 miles on Tuesday. How far did she run in total?

Show Answer

Common denominator is 4. 3 2/4 + 2 3/4. Whole: 3+2=5. Fractions: 2/4+3/4=5/4=1 1/4. Regroup: 5+1=6. Answer: 6 1/4 miles.

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