Multiplying Fractions and Whole Numbers: Fifth Grade Mathematics

Multiplying a fraction by a whole number combines repeated addition concepts with fractional scaling.

Visualizing as Repeated Addition

When multiplying a whole number by a fraction, such as 4 * (2/3), the operation represents repeated addition of that fractional amount: 4 * (2/3) = 2/3 + 2/3 + 2/3 + 2/3 = 8/3 = 2 2/3.

+--------------------------------------------------------------------------+
|                  REPEATED ADDITION ON A NUMBER LINE                      |
|                                                                          |
|       Problem: 3 * (3/4)                                                 |
|                                                                          |
|   <---+---+---+---+---+---+---+---+---+---+---+---+                      |
|       0      1/4     2/4     3/4     4/4     5/4     6/4     7/4     8/4 |
|                                                                          |
|       Jump 1: 0 to 3/4                                                   |
|       Jump 2: 3/4 to 6/4                                                 |
|       Jump 3: 6/4 to 9/4 = 2 1/4                                         |
|                                                                          |
|       3 * (3/4) = 9/4 = 2 1/4                                            |
+--------------------------------------------------------------------------+

The Computation Algorithm

To multiply a fraction and a whole number:
1. Write the whole number as a fraction over 1: w = w/1.

2. Multiply the numerators together: w * a.

3. Multiply the denominators together: 1 * b = b.

4. Simplify the resulting fraction and convert to a mixed number if improper.

Algorithmic Setup:
      2       5     2       5 * 2      10
  5 * -   =  --- * ---  =  -------  =  ---  =  3 1/3
      3       1     3       1 * 3       3

Why Denominators Stay the Same When Multiplying Whole Numbers

Notice that multiplying a whole number by a fraction multiplies only the numerator. The denominator indicates the size of each slice; multiplying by 5 means you have 5 times as many slices, not that the slices changed in size.

Exercises and Solutions

Exercise 1: Standard Computation with Simplification. Multiply 6 * (3/8). Show the whole number written as a fraction over 1, the numerator multiplication, and simplify the product to a mixed number in lowest terms.

Solution 1: Write the whole number 6 as 6/1: (6/1) * (3/8). Multiply numerators: 6 * 3 = 18. Multiply denominators: 1 * 8 = 8. The resulting fraction is 18/8. Simplify 18/8 by dividing numerator and denominator by 2 to get 9/4. Convert 9/4 to a mixed number: divide 9 by 4, which gives 2 with a remainder of 1. The final product is 2 1/4.

Exercise 2: Real-World Portion Problem. A pet owner feeds their dog 2/3 of a cup of dry food twice a day for 7 days. How many total cups of food does the dog consume in one week?

Solution 2: First, find daily consumption: 2 * (2/3) = 4/3 cups per day. Next, multiply daily consumption by 7 days: 7 * (4/3) = (7/1) * (4/3) = 28/3 cups. Convert 28/3 to a mixed number: divide 28 by 3, which gives 9 with a remainder of 1. The dog consumes 9 1/3 cups of food in one week.