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Sample Question 6:
Step 1: Define a new variable \(z\) to represent the amount that Xiaoli rounded up and down by.
Step 2: According to the problem, Xiaoli's calculation was \((x+z) - (y-z)\).
Step 3: Simplify that to get \(x+z-y+z = x-y+2z\).
Step 4: Observe that \(x-y+2z\) is greater than \(x-y\), as \(2z\) is a positive value (since \(z\) is the amount rounded up or down by, assumedly a positive number).
Step 5: Therefore, her estimate is larger than \(x-y\), which corresponds to answer choice (A).
In order to estimate the value of \(x-y\) where \(x\) and \(y\) are real numbers with \(x > y > 0\), Xiaoli rounded \(x\) up by a small amount, rounded \(y\) down by the same amount, and then subtracted her rounded values. Which of the following statements is necessarily correct?
\(\textbf{(A)}\) Her estimate is larger than \(x-y\) \(\textbf{(B)}\) Her estimate is smaller than \(x-y\) \(\textbf{(C)}\) Her estimate equals \(x-y\) \(\textbf{(D)}\) Her estimate equals \(y - x\) \(\textbf{(E)}\) Her estimate is \(0\)
Answer Keys
Question 6: ASolutions
Question 6Step 1: Define a new variable \(z\) to represent the amount that Xiaoli rounded up and down by.
Step 2: According to the problem, Xiaoli's calculation was \((x+z) - (y-z)\).
Step 3: Simplify that to get \(x+z-y+z = x-y+2z\).
Step 4: Observe that \(x-y+2z\) is greater than \(x-y\), as \(2z\) is a positive value (since \(z\) is the amount rounded up or down by, assumedly a positive number).
Step 5: Therefore, her estimate is larger than \(x-y\), which corresponds to answer choice (A).