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3 Responses
Love this question-there’s a cool “undoing” idea hiding here! Dividing by a fraction is really asking “how many chunks of that size fit?” So 3/5 ÷ 2/3 means: how many 2/3-chunks fit into 3/5? If the answer is x, then (2/3)·x = 3/5. To solve for x, you multiply both sides by the number that undoes 2/3, namely its reciprocal 3/2, because (2/3)·(3/2) = 1. That’s why we “flip”: x = (3/5)·(3/2). Another way to see it is with scoops: each whole “1” contains 3/2 scoops of size 2/3, so a 3/5-sized amount contains (3/5)·(3/2) scoops. For mixed numbers, I think the cleanest method is either convert to an improper fraction or switch everything to a common unit. For example, 1 1/4 ÷ 2/3: in twelfths, 1 1/4 is 15/12 and 2/3 is 8/12, so you’re asking how many 8/12-chunks fit in 15/12-that’s 15/8 = 1 7/8. Converting to 5/4 ÷ 2/3 and flipping gives the same result: (5/4)·(3/2) = 15/8. I might be slightly overselling it, but the key pattern is: dividing by a fraction = counting chunks, and counting chunks naturally turns into multiplying by the reciprocal.
You flip because you want the divisor to become 1: multiply both numbers by the divisor’s reciprocal so the ratio doesn’t change-e.g., 3/5 ÷ 2/3 = (3/5 × 3/2) ÷ 1 = 9/10. For mixed numbers, just convert first and do the same (unless you enjoy extra hassle): 1 1/4 ÷ 2/3 = 5/4 × 3/2 = 15/8 = 1 7/8; quick refresher: https://www.khanacademy.org/math/arithmetic/fraction-arithmetic/arith-review-dividing-fractions/a/dividing-fractions.
The “flip” is really just the idea of a reciprocal sneaking in. Dividing by a fraction means “how many of these fit into that?” So 3/5 ÷ 2/3 asks how many 2/3-sized groups fit into 3/5. If I rewrite both in fifteenths, 3/5 = 9/15 and 2/3 = 10/15, then it’s “how many 10/15 are in 9/15?” That’s 9/10 of a group, which matches (3/5) × (3/2) = 9/10. Another way: think of it as solving (2/3)·x = 3/5. To get x, you multiply both sides by 3/2, the number that turns 2/3 into 1. That’s all the flip is-using the reciprocal to “undo” the divisor. I always picture it like changing the unit so I can count cleanly, then I stop picturing it because my pizza slices start disappearing.
For mixed numbers, converting to an improper fraction is usually the least fussy path because you’re going to need that reciprocal anyway. Example: 1 1/4 ÷ 2/3 = (5/4) ÷ (2/3) = (5/4) × (3/2) = 15/8 = 1 7/8. If you like a tidier visual, you can also match denominators first and skip the “improper” step in your head: 1 1/4 is 15/12, 2/3 is 8/12, so you’re asking “how many 8/12 chunks fit into 15/12?” That’s 15/8 again. Either way, the flip is just the move that turns the divisor into 1 so you can count the groups without spiraling.