Multiplying Fractions by Fractions: Fifth Grade Mathematics

Multiplying two fractions involves multiplying the numerators to find the new numerator and multiplying the denominators to find the new denominator.

The Multiplication Algorithm

Unlike addition and subtraction, multiplying fractions does NOT require a common denominator. You multiply straight across:

+--------------------------------------------------------------------------+
|                  THE FRACTION MULTIPLICATION ALGORITHM                   |
|                                                                          |
|                     a       c       a * c                                |
|                    ---  *  ---  =  -------                               |
|                     b       d       b * d                                |
|                                                                          |
|       Example: (3/5) * (4/7) = (3 * 4) / (5 * 7) = 12/35                 |
+--------------------------------------------------------------------------+

Step-by-Step Worked Example

Multiply 5/8 by 2/3:
1. Multiply numerators: 5 * 2 = 10.

2. Multiply denominators: 8 * 3 = 24.

3. Write product: 10/24.

4. Simplify to lowest terms by dividing by GCF (2): (10 / 2) / (24 / 2) = 5/12.

Step Breakdown:
     5       2       5 * 2       10       5
    ---  *  ---  =  -------  =  ----  =  ----
     8       3       8 * 3       24       12

No Common Denominator Needed

Students often ask why common denominators are not needed for multiplication. In addition, you are combining pieces of identical size. In multiplication, the denominator itself is being partitioned: eighths are being partitioned into thirds, creating twenty-fourths. The operation naturally generates the new piece size.

Exercises and Solutions

Exercise 1: Standard Computation. Multiply 4/9 by 3/5. Compute the product by multiplying straight across, and simplify the answer to lowest terms.

Solution 1: Multiply the numerators: 4 * 3 = 12. Multiply the denominators: 9 * 5 = 45. The unsimplified product is 12/45. Find the GCF of 12 and 45: both numbers are divisible by 3 (12 / 3 = 4; 45 / 3 = 15). Divide numerator and denominator by 3: 12/45 simplifies to 4/15. The final product is 4/15.

Exercise 2: Science Laboratory Scenario. An experiment calls for 3/4 of a beaker of chemical solution. A researcher accidentally spills 1/3 of that solution. What fraction of a full beaker was spilled?

Solution 2: Find 1/3 of 3/4 by multiplying: (1/3) * (3/4). Multiply numerators: 1 * 3 = 3. Multiply denominators: 3 * 4 = 12. The product is 3/12. Simplify by dividing numerator and denominator by 3: 3/12 = 1/4. Exactly 1/4 of a full beaker was spilled.