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08 Jul 2022, 01:30
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Points a, b, c, d are all on a number line and d > a. If b is twice as far from d as it is from a and c is twice as far from a as it is from d, then (c - b)/(d - a) could equal all of the following answer choices EXCEPT?
A. -1
B. 1/3
C. 1
D. 2
E. 3
Are You Up For the Challenge: 700 Level Questions
A. -1
B. 1/3
C. 1
D. 2
E. 3
Are You Up For the Challenge: 700 Level Questions
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08 Jul 2022, 12:56
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Since it is given that \(d>a\), the denominator of the fraction \(\frac{c-b}{d-a}\) is always positive.
Now, consider the number line with \(a\) and \(d\) on it, and let the mid point exactly between them be denoted by \(m\).
Since \(b\) is twice as far from \(d\) as it is from \(a\), clearly \(b\) will lie to the left of \(m\). We can apply a similar logic to say that \(c\) will lie to the right of \(m\).
Hence, \(c\) will always lie to the right of \(b\), which tells us that \(c-b >0\).
\(\Rightarrow\) The numerator is also positive, hence the fraction has to be positive, which makes Option A the odd one out.
Now, consider the number line with \(a\) and \(d\) on it, and let the mid point exactly between them be denoted by \(m\).
Since \(b\) is twice as far from \(d\) as it is from \(a\), clearly \(b\) will lie to the left of \(m\). We can apply a similar logic to say that \(c\) will lie to the right of \(m\).
Hence, \(c\) will always lie to the right of \(b\), which tells us that \(c-b >0\).
\(\Rightarrow\) The numerator is also positive, hence the fraction has to be positive, which makes Option A the odd one out.
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11 Jul 2022, 13:48
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Though by logic I can arrive at option (A) because c will always be greater than b, I think mathematically there are only 4 cases and 3 unique values of the fraction (c-b)/(d-a) possible - that is - 5/3, 1/3, and 3. Maybe I am doing sleep math because I am too worn out, or maybe this actually contains an error. Can anyone please check?
Case 1: A--x--B--x--C--x--D
Required ratio = x/3x = 1/3
Case 2: A--x--B--2x--D--3x--C
Required ratio = 5x/3x = 5/3
Case 3: B--3x--A--2x--C--x--D
Required ratio = 5x/3x = 5/3
Case 4: B--x--A--x--D--x--C
Required ratio = 3x/x = 3
Case 1: A--x--B--x--C--x--D
Required ratio = x/3x = 1/3
Case 2: A--x--B--2x--D--3x--C
Required ratio = 5x/3x = 5/3
Case 3: B--3x--A--2x--C--x--D
Required ratio = 5x/3x = 5/3
Case 4: B--x--A--x--D--x--C
Required ratio = 3x/x = 3
