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3 Responses
I used to chant “keep, change, flip” too and worry it was more spell than math. The real idea is: dividing by something means “how many of those fit into this?” So 2/3 ÷ 5/6 asks, “How many 5/6-cup scoops fit into 2/3 of a cup?” Since a 5/6 scoop is bigger than what you have, the answer should be less than 1. To compute it, use the fact that dividing by a number is the same as multiplying by its reciprocal (the number that makes 1 when multiplied by it). So 2/3 ÷ 5/6 = 2/3 × 6/5, because 5/6 × 6/5 = 1. That gives 12/15 = 4/5, which matches the story: a bit less than a whole scoop fits. If you like the “clear the denominator” view, think of the complex fraction (2/3)/(5/6): multiply top and bottom by 6 to wipe out the denominator in 5/6, and you get (2·6/3)/(5) = 4/5-same result, same reason.
Another way to see it is as an “undoing” move: division by y is answering “what q makes q·y = x?” For x = 2/3 and y = 5/6, we solve q·(5/6) = 2/3 by multiplying both sides by 6/5, giving q = (2/3)·(6/5). That’s the flip-and-multiply rule in action-no magic, just using the multiplicative inverse. If the pizza picture felt fuzzy, you’re not wrong; a closer analogy is measuring scoops: flipping the fraction is like swapping to the “undo” measuring cup that turns a scoop-size into 1. A nice walkthrough with visuals is here: https://www.khanacademy.org/math/arithmetic/fraction-arithmetic/arith-review-dividing-fractions/a/dividing-fractions-article
Flip-and-multiply works because dividing by a number means multiplying by its reciprocal; for fractions, 1/(c/d) = d/c since (c/d)(d/c) = 1, so 2/3 ÷ 5/6 = 2/3 × 6/5-aka “how many 5/6-sized chunks fit into 2/3?”
Does that picture click, or want a quick common-denominator version too; here’s a tidy walkthrough: https://www.khanacademy.org/math/arithmetic/fraction-arithmetic/arith-review-dividing-fractions/a/dividing-fractions
Dividing by a fraction means “find x so that (5/6)·x = 2/3,” and multiplying both sides by 6/5 (the reciprocal) undoes the 5/6, giving x = (2/3)·(6/5)-so flip-and-multiply is just solving that one-step equation. I still remember my lightbulb moment in 7th grade when I scribbled “reciprocals undo!” in the margin and suddenly every fraction division felt like a tiny algebra victory.