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Back in elementary school, I learned a great magic trick to compare two positive fractions. Here’s my magic trick in action. Consider the fractions 5/19 and 6/23. I multiply the numerator of the first and the denominator of the second, and get 5 · 23 = 115. I multiply the numerator of the second and the denominator of the first, and get 6 · 19 = 114. Because 115 > 114, I know that 5/19 > 6/23 .

 

Why does this magic trick work? In other words, if a, b, c, and d are positive, then why does ad > bc mean that a/b > c/d ?

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Given ad > bc, and we also know all the variables are positive (so when dividing/multiplying you don't have to switch the inequality), you can use the division property and divide both sides by d that still makes the inequality true: a > bc/d.

Then by the same property, you can keep the inequality true by dividing both sides by b, and you obtain a/b > c/d.

Same way to think of it as the example 5/19 > 6/23, because multiplying 23 to both sides won't change the fact that 5 * 23 / 19 > 6, logically speaking, if you fairly multiply each side by the same number, then it's obvious that the bigger side will be even bigger now. After you fiddle around with the division property, you can obtain a/b > c/d.

 Dec 4, 2022

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