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2020 AMC 10A Problem 3

Problem 3 of 25EasierAlgebra

Assuming a≠3,a\neq3, b≠4,b\neq4, and c≠5,c\neq5, what is the value in simplest form of the following expression? a−35−c⋅b−43−a⋅c−54−b\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}

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Solution

Rewrite the denominator factors as 5−c=−(c−5)5-c=-(c-5), 3−a=−(a−3)3-a=-(a-3), and 4−b=−(b−4)4-b=-(b-4). The expression becomes (a−3)(b−4)(c−5)−(a−3)(b−4)(c−5)=−1\dfrac{(a-3)(b-4)(c-5)}{-(a-3)(b-4)(c-5)}=-1. Thus, A is the correct answer.

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Concepts: algebraic manipulation · fraction

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.