Class Notes: Subtracting Fractions Using the Butterfly Method

Objective

Learn how to subtract two fractions using the butterfly method.

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Steps of the Butterfly Method

1. Multiply Diagonally (Positive Direction)

   • Multiply the numerator of the first fraction by the denominator of the second fraction.
   • Example: For 4/5 - 2/7, compute 4 × 7 = 28.

2. Multiply Diagonally (Negative Direction)

   • Multiply the denominator of the first fraction by the numerator of the second fraction.
   • Attach a negative sign to this product.
   • Example: 5 × 2 = 10, so it's -10.

3. Multiply the Denominators

   • Multiply the denominators of both fractions for the common denominator.
   • Example: 5 × 7 = 35.

4. Calculate the Numerator

   • Add the two products from Steps 1 and 2.
   • Example: 28 - 10 = 18.

5. Final Fraction

   • Place the result over the denominator product.
   • Example: The final answer: 18/35

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Practice Example

Subtract: 5/8 - 2/9

1. Diagonals

   • 5 × 9 = 45 (positive)
   • 8 × 2 = 16 (negative, so -16)

2. Denominator

   • 8 × 9 = 72

3. Numerator

   • 45 - 16 = 29

4. Final Answer

   • 29/72 (which cannot be reduced further)

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Tips

• Always start with the diagonal that is positive (the numerator of the first fraction × denominator of the second).
• The second diagonal (denominator of the first × numerator of the second) is subtracted.
• The common denominator is always the product of the original denominators.

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Summary

Butterfly method for subtraction:

• Multiply diagonally (left to right = +, right to left = -)
• Subtract the products
• Multiply denominators for the new denominator
• Place the result as a single fraction in simplest form

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