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What is the mathematical reason that letters and units cancel from multiplication?

For example, 1/2x * 4x = 2 and 1/(2 ounces) * 4 ounces = 2

The x's and the ounces cancel. What is the reason for the cancellation or why or how does this occur?

To reduce a fraction to lowest terms, you cancel out any numbers that are factors in both numerator and denominator. Any units and variables that are common factors are treated the same way as the numbers.

When reducing the fraction 6/2, you are answering the question "how many times does 2 go into 6?"—3 times. That is why 6/2 = 3.

How many times does 1 foot go into 1 foot?—it goes in once. Thus, (1 ft)/(1 ft) reduces to 1.
How many times does x go into 2x? If x≠0, it goes in twice. (2x)/x reduces to 2.

I hope this representation helps.
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#### Janet Yang

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I can answer questions in Algebra, Basic Math, Calculus, Differential Equations, Geometry, Number Theory, and Word Problems. I would not feel comfortable answering questions in Probability and Statistics or Topology because I have not studied these in depth.

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