For the list of numbers above there is exactly one mode. Is the range of the list greater than the mode of the list?
(1) At least one of the numbers in the list is negative.
(2) At least one of the numbers in the list is zero.
OA A
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For the list of numbers above there is exactly one mode. Is
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BTGmoderatorDC
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Jay@ManhattanReview
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The question is not complete. The list of the numbers is missing. Here is the complete question and the solution.BTGmoderatorDC wrote:For the list of numbers above there is exactly one mode. Is the range of the list greater than the mode of the list?
(1) At least one of the numbers in the list is negative.
(2) At least one of the numbers in the list is zero.
OA A
Source: Veritas Prep
[color=]The mode is the number that has the highest frequency of its occurrence.{a, b, c, d, e}
For the list of numbers above, there is exactly one mode. Is the range of the list greater than the mode of the list?
(1) At least one of the numbers in the list is negative.
(2) At least one of the numbers in the list is zero.
The range is the difference between the largest and the smallest number.
[/color]
We have to determine whether Range > Mode.
Let's take each statement one by one.
(1) At least one of the numbers in the list is negative.
Since we have to prove that Range > Mode, we must try to maximize the value of Mode so that we can validate the result. If at the maximum value of the mode, Range > Mode, we can conclude that Range > Mode.
Say Mode is the largest number in the set.
We know that Range = Largest - Smallest
Thus, Range = Mode - (Negative number); it is given that there is at least one negative number.
Range = Mode + |Smallest|
=> Range > Mode since |Smallest| > 0. Sufficient
(2) At least one of the numbers in the list is zero.
Say Mode is the largest number in the set.
Since Range = Largest - Smallest = Mode - 0; assuming that no number is negative in the set.
Range = Mode. Range in not less than Mode.
We have already seen in statement 1 that Range > Mode if there is at least one negative number. Thus, either Range = Mode or Range > Mode. Insufficient
The correct answer: A
Hope this helps!
-Jay
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fskilnik@GMATH
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<p>
\[{\text{?}}\,\,\,{\text{:}}\,\,\,{\text{Mo}}\,\,\mathop < \limits^? \,\,e - a\]
\[\left( 1 \right)\,\,\,\,a < 0\,\,\,\, \Rightarrow \,\,\,\,e - a = e + \left( { - a} \right) > e\]
\[?\,\,\,:\,\,\,{\text{Mo}} \leqslant e < e - a\,\,\,\,\, \Rightarrow \,\,\,\,{\text{Mo}} < e - a\,\,\,\,\,\,\left( {{\text{Suf}}{\text{.}}} \right)\]
\[\left( 2 \right)\,\,\,{\text{Insuf}}{\text{.}}\,\,\,\left\{ \begin{gathered}
\,\,\,\left\{ {0,0,0,0,1} \right\}\,\,\,\, \Rightarrow \,\,\,\,{\text{Yes}}\,\,\,\left( {{\text{Mo}} = 0\,\,;\,\,1 = e - a} \right) \hfill \\
\,\,\,\left\{ {0,1,1,1,1} \right\}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{No}}\,\,\,\,\,\left( {{\text{Mo}} = 1\,\,;\,\,1 = e - a} \right) \hfill \\
\end{gathered} \right.\]
(*) WLOG = Without loss of generality
POST-MORTEM: {0,0,1,1,2} is an example of a list that does not have a single mode (0 and 1 are two distinct modes of this list).
The solution above follows the notations and rationale taught in the <b>GMATH</b> method.
\[{\text{WLOG}}\left( * \right)\,\,\,\,a \leqslant b \leqslant c \leqslant d \leqslant e\,\,\,\,\,\,\left( {{\text{single}}\,\,{\text{Mo}}} \right)\]</p>
{a, b, c, d, e}
For the list of numbers above, there is exactly one mode. Is the range of the list greater than the mode of the list?
(1) At least one of the numbers in the list is negative.
(2) At least one of the numbers in the list is zero.
\[{\text{?}}\,\,\,{\text{:}}\,\,\,{\text{Mo}}\,\,\mathop < \limits^? \,\,e - a\]
\[\left( 1 \right)\,\,\,\,a < 0\,\,\,\, \Rightarrow \,\,\,\,e - a = e + \left( { - a} \right) > e\]
\[?\,\,\,:\,\,\,{\text{Mo}} \leqslant e < e - a\,\,\,\,\, \Rightarrow \,\,\,\,{\text{Mo}} < e - a\,\,\,\,\,\,\left( {{\text{Suf}}{\text{.}}} \right)\]
\[\left( 2 \right)\,\,\,{\text{Insuf}}{\text{.}}\,\,\,\left\{ \begin{gathered}
\,\,\,\left\{ {0,0,0,0,1} \right\}\,\,\,\, \Rightarrow \,\,\,\,{\text{Yes}}\,\,\,\left( {{\text{Mo}} = 0\,\,;\,\,1 = e - a} \right) \hfill \\
\,\,\,\left\{ {0,1,1,1,1} \right\}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{No}}\,\,\,\,\,\left( {{\text{Mo}} = 1\,\,;\,\,1 = e - a} \right) \hfill \\
\end{gathered} \right.\]
(*) WLOG = Without loss of generality
POST-MORTEM: {0,0,1,1,2} is an example of a list that does not have a single mode (0 and 1 are two distinct modes of this list).
The solution above follows the notations and rationale taught in the <b>GMATH</b> method.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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<p>
\[{\text{?}}\,\,\,{\text{:}}\,\,\,{\text{Mo}}\,\,\mathop < \limits^? \,\,e - a\]
\[\left( 1 \right)\,\,\,\,a < 0\,\,\,\, \Rightarrow \,\,\,\,e - a = e + \left( { - a} \right) > e\]
\[?\,\,\,:\,\,\,{\text{Mo}} \leqslant e < e - a\,\,\,\,\, \Rightarrow \,\,\,\,{\text{Mo}} < e - a\,\,\,\,\,\,\left( {{\text{Suf}}{\text{.}}} \right)\]
\[\left( 2 \right)\,\,\,{\text{Insuf}}{\text{.}}\,\,\,\left\{ \begin{gathered}
\,\,\,\left\{ {0,0,0,0,1} \right\}\,\,\,\, \Rightarrow \,\,\,\,{\text{Yes}}\,\,\,\left( {{\text{Mo}} = 0\,\,;\,\,1 = e - a} \right) \hfill \\
\,\,\,\left\{ {0,1,1,1,1} \right\}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{No}}\,\,\,\,\,\left( {{\text{Mo}} = 1\,\,;\,\,1 = e - a} \right) \hfill \\
\end{gathered} \right.\]
(*) WLOG = Without loss of generality
POST-MORTEM: {0,0,1,1,2} is an example of a list that does not have a single mode (0 and 1 are two distinct modes of this list).
The solution above follows the notations and rationale taught in the <b>GMATH</b> method.
\[{\text{WLOG}}\left( * \right)\,\,\,\,a \leqslant b \leqslant c \leqslant d \leqslant e\,\,\,\,\,\,\left( {{\text{single}}\,\,{\text{Mo}}} \right)\]</p>
{a, b, c, d, e}
For the list of numbers above, there is exactly one mode. Is the range of the list greater than the mode of the list?
(1) At least one of the numbers in the list is negative.
(2) At least one of the numbers in the list is zero.
\[{\text{?}}\,\,\,{\text{:}}\,\,\,{\text{Mo}}\,\,\mathop < \limits^? \,\,e - a\]
\[\left( 1 \right)\,\,\,\,a < 0\,\,\,\, \Rightarrow \,\,\,\,e - a = e + \left( { - a} \right) > e\]
\[?\,\,\,:\,\,\,{\text{Mo}} \leqslant e < e - a\,\,\,\,\, \Rightarrow \,\,\,\,{\text{Mo}} < e - a\,\,\,\,\,\,\left( {{\text{Suf}}{\text{.}}} \right)\]
\[\left( 2 \right)\,\,\,{\text{Insuf}}{\text{.}}\,\,\,\left\{ \begin{gathered}
\,\,\,\left\{ {0,0,0,0,1} \right\}\,\,\,\, \Rightarrow \,\,\,\,{\text{Yes}}\,\,\,\left( {{\text{Mo}} = 0\,\,;\,\,1 = e - a} \right) \hfill \\
\,\,\,\left\{ {0,1,1,1,1} \right\}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{No}}\,\,\,\,\,\left( {{\text{Mo}} = 1\,\,;\,\,1 = e - a} \right) \hfill \\
\end{gathered} \right.\]
(*) WLOG = Without loss of generality
POST-MORTEM: {0,0,1,1,2} is an example of a list that does not have a single mode (0 and 1 are two distinct modes of this list).
The solution above follows the notations and rationale taught in the <b>GMATH</b> method.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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fskilnik@GMATH
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<p>
(*) WLOG = Without loss of generality
POST-MORTEM: {0,0,1,1,2} is an example of a list that does not have a single mode (0 and 1 are two distinct modes of this list).
The solution above follows the notations and rationale taught in the <b>GMATH</b> method.
</p>
{a, b, c, d, e}
For the list of numbers above, there is exactly one mode. Is the range of the list greater than the mode of the list?
(1) At least one of the numbers in the list is negative.
(2) At least one of the numbers in the list is zero.
(*) WLOG = Without loss of generality
POST-MORTEM: {0,0,1,1,2} is an example of a list that does not have a single mode (0 and 1 are two distinct modes of this list).
The solution above follows the notations and rationale taught in the <b>GMATH</b> method.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
