Multiplying Mixed Numbers: Fifth Grade Mathematics

Multiplying mixed numbers requires converting each mixed number into an improper fraction before applying the multiplication algorithm.

The Core Conversion Method

Never multiply the whole numbers together and the fractions together (e.g., 2 1/2 * 3 1/3 does NOT equal 6 1/6). You must convert all mixed numbers into improper fractions first.

+--------------------------------------------------------------------------+
|                  MULTIPLYING MIXED NUMBERS STEP-BY-STEP                  |
|                                                                          |
|       Problem:  2 1/3 * 1 4/5                                            |
|                                                                          |
|       Step 1: Convert each mixed number to an improper fraction:         |
|               2 1/3 = (2 * 3 + 1)/3 = 7/3                                |
|               1 4/5 = (1 * 5 + 4)/5 = 9/5                                |
|                                                                          |
|       Step 2: Cross-cancel if possible:                                  |
|               Cancel 3 in denominator with 9 in numerator:               |
|               3 becomes 1; 9 becomes 3.                                  |
|                                                                          |
|       Step 3: Multiply remaining numerators and denominators:            |
|               (7 * 3) / (1 * 5) = 21/5                                   |
|                                                                          |
|       Step 4: Convert improper fraction to mixed number:                 |
|               21 / 5 = 4 with remainder 1  --->  4 1/5                   |
+--------------------------------------------------------------------------+

Why Multiplying Wholes and Fractions Separately Fails

If you attempt 2 1/2 * 3 1/3 by multiplying wholes (2 * 3 = 6) and fractions (1/2 * 1/3 = 1/6), you are omitting the two cross terms from the Distributive Property: (2 * 1/3) and (3 * 1/2). Those omitted terms contribute 2/3 + 1 1/2 = 2 1/6, meaning your answer of 6 1/6 is short by more than 2 whole units!

The Four Partial Products of (2 + 1/2) * (3 + 1/3):
  1. Whole * Whole:       2 * 3   = 6
  2. Whole * Fraction:    2 * 1/3 = 2/3
  3. Fraction * Whole:    1/2 * 3 = 1 1/2
  4. Fraction * Fraction: 1/2 * 1/3 = 1/6
  Total Sum = 6 + 2/3 + 1 1/2 + 1/6 = 8 2/6 = 8 1/3.

Exercises and Solutions

Exercise 1: Standard Mixed Number Product. Calculate 3 3/4 * 2 2/5. Show the conversion to improper fractions, cross-canceling, and conversion back to a mixed number.

Solution 1: Convert 3 3/4 to an improper fraction: (3 * 4 + 3)/4 = 15/4. Convert 2 2/5 to an improper fraction: (2 * 5 + 2)/5 = 12/5. Write the multiplication expression: (15/4) * (12/5). Cross-cancel: 15 and 5 are divisible by 5 (15 becomes 3; 5 becomes 1). 12 and 4 are divisible by 4 (12 becomes 3; 4 becomes 1). Multiply remaining factors: (3 * 3) / (1 * 1) = 9/1 = 9. The product is the whole number 9.

Exercise 2: Recipe Scaling Word Problem. A catering chef needs to prepare 3 1/2 batches of a dessert recipe. The original recipe calls for 1 3/4 cups of brown sugar. How many cups of brown sugar must the chef use?

Solution 2: Multiply 3 1/2 by 1 3/4. Convert to improper fractions: 3 1/2 = 7/2; 1 3/4 = 7/4. Multiply numerators: 7 * 7 = 49. Multiply denominators: 2 * 4 = 8. Product as an improper fraction is 49/8. Convert 49/8 to a mixed number: divide 49 by 8, which gives 6 with a remainder of 1 (8 * 6 = 48). The chef must use 6 1/8 cups of brown sugar.