Intuition behind a method of adding two fractions

[Juwane]

One way to add two fractions is to multiply the numerators of both fractions with each other's denominator, then adding the two products, gives us the numerator of the final result. Then we multiply together the denominators of each other--this gives us the denominator of the final result.

Consider this example:

$$\frac{2}{3} + \frac{4}{5} = \frac{(5\times2)+(3\times4)}{(3\times5)} = \frac{22}{15}$$

Why are we able to do that? I'm not looking for a rigorous proof, just the intuition behind it.

 

[Bohrok]

To add fractions, they need the same denominator. We "change" the fractions to have a common denominator by the identity property x·1 = x and then simplify the result.
$$\frac{2}{3} + \frac{4}{5} = \frac{2}{3}\times\frac{5}{5} + \frac{4}{5}\times\frac{3}{3}$$

$$= \frac{2\times 5}{3\times 5} + \frac{4\times 3}{5\times 3} = \frac{(5\times 2) + (3\times 4)}{3\times 5}$$

I suppose the "intuition" is to make the denominators the same and then combine the fractions. What you have could be used as a formula for adding fractions, bypassing all the steps that lead up to that result.

 

[Tac-Tics]

This reminds me of something I told one of my friends a long time ago while driving down the highway. We saw a sign saying 3/4 mile to our exit. I mentioned to her, "there's only three quarter-miles to go." She thought about it for a second and was amused that she never thought of it as three "quarter miles." In her mind, it was always "three quarters" of a whole mile.

The nice thing is the two are interchangeable. Any hungry child will tell you that half a pie isn't the same as a third of a pie. But which is bigger? How do you compare halves and thirds? You convert each to an equal amount of sixths-pies. Then it's clear. If you cut a pie into 6 even pieces, you need to take 3 to have taken half the pie, so 1/2 = 3/6. And in the same way, you need 2/6 of a pie to equal that third-pie.

It's obvious with pie and pictures. But once you distill it to numbers, it does appear mysterious.

 

[HallsofIvy]

One way of thinking about it is that the denominators of fractions are really "units". That is, if we talk about "4 feet" or "3 yards", the "4" and "3" alone would be meaningless without specifying what units we are using. If we were to add those, we would NOT get "7" anything! We must have the same units. Here, it is simplest to change "3 yards" to 9 feet, since there are 3 yards to a foot. "4 feet plus 3 yards" is the same as "4 feet plus 9 feet= (4 + 9) feet= 13 feet.

Similarly, two add 2/3 plus 4/5, I need to get the same "units". One "third" is not any integer multiple of one "fifth" or vice versa but one "third" is 5 "fifteenths" so 2 "thirds" is 10 "fifteenths" and one "fifth" is 3 "fifteenths" so 4 "fifths" is 12 "fifteenths".

2 thirds plus 4 fifths= 10 fifteenths plust 12 fifteenths= 22 fifteenths.
2/3+ 4/5= 10/15+ 12/15= 22/15.