Adding Fractions with Unlike Denominators: Fifth Grade Mathematics

Adding fractions with unlike denominators requires renaming the fractions into equivalent forms with a common denominator before combining numerators.

The Step-by-Step Addition Algorithm

To add fractions with unlike denominators, execute this four-step procedure:
1. Find the Least Common Denominator (LCD).

2. Rename each fraction into an equivalent fraction with the LCD.

3. Add the numerators while keeping the common denominator.

4. Simplify the resulting sum to lowest terms or a mixed number.

+--------------------------------------------------------------------------+
|                  ADDING FRACTIONS WITH UNLIKE DENOMINATORS               |
|                                                                          |
|       Problem:  2/3 + 1/4                                                |
|                                                                          |
|       Step 1: Find LCD of 3 and 4:                                       |
|               Multiples of 3: 3, 6, 9, 12, 15...                         |
|               Multiples of 4: 4, 8, 12, 16...  --->  LCD = 12            |
|                                                                          |
|       Step 2: Rename fractions to twelfths:                              |
|               2/3 = (2 * 4) / (3 * 4) = 8/12                             |
|               1/4 = (1 * 3) / (4 * 3) = 3/12                             |
|                                                                          |
|       Step 3: Add numerators:                                            |
|               8/12 + 3/12 = 11/12                                        |
|                                                                          |
|       Step 4: Check simplest form: GCF(11, 12) = 1. Done!               |
+--------------------------------------------------------------------------+

Visualizing 1/2 + 1/3 Using Sixths:
  1/2 = 3/6:  [===|===|===|---|---|---] (3 sixths)
  1/3 = 2/6:  [===|===|---|---|---|---] (2 sixths)
  Sum:        3/6 + 2/6 = 5/6

Adding Three Fractions with Unlike Denominators

When adding three fractions (such as 1/2 + 2/5 + 3/10), find the common multiple of all three denominators simultaneously:
- Multiples of 10: 10, 20, 30...

- 10 is divisible by 2 (2 * 5 = 10) and by 5 (5 * 2 = 10).

- Rename all three to tenths: 5/10 + 4/10 + 3/10 = 12/10.

- Simplify: 12/10 = 6/5 = 1 1/5.

Exercises and Solutions

Exercise 1: Standard Unlike Fraction Addition. Compute 3/5 + 1/6. Show the steps for finding the LCD, writing equivalent fractions, adding numerators, and checking simplest form.

Solution 1: Step 1: Find LCD of 5 and 6. Since 5 and 6 share no common factors, LCD = 5 * 6 = 30. Step 2: Rename fractions: 3/5 = (3 * 6) / (5 * 6) = 18/30; 1/6 = (1 * 5) / (6 * 5) = 5/30. Step 3: Add numerators over the common denominator: 18/30 + 5/30 = (18 + 5)/30 = 23/30. Step 4: Check simplest form: 23 is a prime number and does not divide into 30. The fraction 23/30 is in simplest form.

Exercise 2: Sum Resulting in a Mixed Number. Solve 5/8 + 3/4. Express the final answer as a mixed number in simplest form.

Solution 2: Denominator 8 is a multiple of 4 (4 * 2 = 8), so LCD = 8. Rename 3/4 to eighths: (3 * 2) / (4 * 2) = 6/8. Keep 5/8 as it is. Add numerators: 5/8 + 6/8 = 11/8. Convert the improper fraction 11/8 into a mixed number: divide 11 by 8, which gives 1 with a remainder of 3. The final answer is 1 3/8.