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2010 AMC 10B Problem 9

Problem 9 of 25EasierAlgebraArithmetic

Lucky Larry’s teacher asked him to substitute numbers for a,a, b,b, c,c, d,d, and ee in the expression a−(b−(c−(d+e)))a-(b-(c-(d+e))) and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The numbers Larry substituted for a,a, b,b, c,c, and dd were 1,1, 2,2, 3,3, and 4,4, respectively. What number did Larry substitute for e?e?

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Solution

Ignoring the parentheses, Larry would get 1−2−3−4+e=e−8. 1 - 2 - 3 - 4 + e = e - 8. Evaluating with the parentheses, one would get 1−(2−(3−(4+e)))=1−(2−(−1−e))=1−(3+e)=−2−e.\begin{aligned} &1 - (2 -(3 - (4 + e)))\\ & = 1 - (2 - (-1 - e))\\ &= 1 - (3 + e) \\ &=-2 - e.\end{aligned} Both of these values are the same, so e−8=−2−e e - 8 = -2 - e e=3. e = 3. Thus, D is the correct answer.
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Tagged: order of operations · linear equation

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