How is the butterfly method different from making equivalent fractions?

Answer 1

The butterfly method is making equivalent fractions, just not obviously.

However, the different is that in making equivalent fractions you have to choose the new denominator (ideally this should be the lowest common multiple (LCM) of the denominators to avoid big numbers, but it doesn't have to be), whereas with the butterfly method the new denominator is always the product of the original two denominators.

For example: what is 3/4 + 1/8?

Using LCM:

lcm(4, 8) = 8

→ 3/4 = (3x2)/(4x2) = 6/8 (as 4 goes into 8 two times, multiply top and bottom by 2)

→ 1/8 stays the same as it already has 8 in the denominator

→ 3/4 + 1/8 = 6/8 + 1/8 = (6+1)/8 = 7/8

Using product of the denominators:

4x8 = 32

→ 3/4 = (3x8)/(4x8) = 24/32 (as 4 goes into 32 eight times, multiply top and bottom by 8)

→ 1/8 = (1x4)/(8x4) = 4/32 (as 8 goes into 32 four times, multiply top and bottom by 4)

→ 3/4 + 1/8 = 24/32 + 4/32 = (24+4)/32 = 28/32 = (7x4)/(8x4) = 7/8

Using the butterfly method:

Left antenna: 3x8 = 24

Right antenna: 4x1 = 4

Body: 4x8 = 32

Calculation: (24+4)/8 = 28/32 = (7x4)/(8x4) = 7/8

Which you should notice is the same as the last part of the equivalent fractions method when the product of the denominators is used.