Class Notes: Adding Three Fractions with Different Denominators

Objective

Learn how to add three fractions with different denominators by finding a common denominator.

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Key Steps

1. Find a Common Denominator

• A simple way: Multiply all three denominators together.
  • Example 1 denominators: 3, 4, 5 → 3 × 4 × 5 = 60
  • Note: This gives a common denominator (not always the least).

2. Adjust Each Fraction

• Convert each fraction to an equivalent fraction with the common denominator.
• For each fraction, multiply both numerator and denominator by the product of the other two denominators.

Example 1:

• Given: add 1/3, 1/4, and 2/5.

| Fraction | Multiply by | Resulting Numerator | Resulting Denominator | |-------------|----------------------|---------------------|----------------------| | 1/3 | 4 × 5 = 20 / 20 | 1 × 20 = 20 | 3 × 20 = 60 | | 1/4 | 3 × 5 = 15 / 15 | 1 × 15 = 15 | 4 × 15 = 60 | | 2/5 | 3 × 4 = 12 / 12 | 2 × 12 = 24 | 5 × 12 = 60 |

3. Add the Numerators

• 20 + 15 + 24 = 59
• Write over common denominator: 59/60

4. Simplify (if possible)

• In this case, 59/60 cannot be reduced.

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Practice Example

Given:

Add 1/6 + 3/8 + 2/5

Step by Step

a. Find the Common Denominator

• 6 × 5 × 8 = 240

b. Rewrite Each Fraction

| Fraction | Multiply by | Resulting Numerator | Resulting Denominator | |-------------|----------------------|---------------------|----------------------| | 1/6 | 5 × 8 = 40 / 40 | 1 × 40 = 40 | 6 × 40 = 240 | | 3/8 | 6 × 5 = 30 / 30 | 3 × 30 = 90 | 8 × 30 = 240 | | 2/5 | 6 × 8 = 48 / 48 | 2 × 48 = 96 | 5 × 48 = 240 |

c. Add the Numerators

• 40 + 90 + 96 = 226

d. Put Over Common Denominator

• 226/240

e. Simplify

• Both numbers are even. Divide by 2:
  • 226 ÷ 2 = 113
  • 240 ÷ 2 = 120
• Final Answer: 113/120 (cannot be reduced further)

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Tips

• Multiplying all denominators gives a valid common denominator, though not always the smallest.
• Always try to simplify your final answer.
• You can multiply the numerator and denominator by the same number to create equivalent fractions.
• For faster calculation, practice mental math tricks for multiplying and dividing.

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Summary

1. Multiply all denominators for a common denominator.
2. Adjust each fraction (numerator & denominator) to that denominator.
3. Add numerators.
4. Simplify the result if possible.

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Practice: To reinforce these steps, try solving other sets of three fractions with different denominators on your own!