Adding with the Butterfly

To add 3/5 plus 2/7: cross-multiply to get the new numerators — 3 times 7 equals 21, and 5 times 2 equals 10. Multiply the denominators: 5 times 7 equals 35. Now add: 21/35 plus 10/35 equals 31/35. One method, no searching for common denominators, and it works for any two fractions.

Subtraction works identically, just subtract instead of add. 5/6 - 1/4: cross products are 5 × 4 = 20 and 6 × 1 = 6. Denominator: 6 × 4 = 24. Result: 20/24 - 6/24 = 14/24 = 7/12. The butterfly method automates the common-denominator step.