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For the list of numbers above there is exactly one mode. Is

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by BTGmoderatorDC » Fri Aug 17, 2018 9:39 pm

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Difficulty—

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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Source: — Data Sufficiency |

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by Jay@ManhattanReview » Mon Aug 20, 2018 1:40 am
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
The question is not complete. The list of the numbers is missing. Here is the complete question and the solution.
{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.
[color=]The mode is the number that has the highest frequency of its occurrence.
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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by fskilnik@GMATH » Mon Aug 20, 2018 11:11 am
<p>
</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.
\[{\text{WLOG}}\left( * \right)\,\,\,\,a \leqslant b \leqslant c \leqslant d \leqslant e\,\,\,\,\,\,\left( {{\text{single}}\,\,{\text{Mo}}} \right)\]
\[{\text{?}}\,\,\,{\text{:}}\,\,\,{\text{Mo}}\,\,\mathop < \limits^? \,\,e - a\]
\[\left( 1 \right)\,\,\,\,a < 0\,\,\,\, \Rightarrow \,\,\,\,e - a = e + \left( { - a} \right) > e\]
\[?\,\,\,:\,\,\,{\text{Mo}} \leqslant e < e - a\,\,\,\,\, \Rightarrow \,\,\,\,{\text{Mo}} < e - a\,\,\,\,\,\,\left( {{\text{Suf}}{\text{.}}} \right)\]
\[\left( 2 \right)\,\,\,{\text{Insuf}}{\text{.}}\,\,\,\left\{ \begin{gathered}
\,\,\,\left\{ {0,0,0,0,1} \right\}\,\,\,\, \Rightarrow \,\,\,\,{\text{Yes}}\,\,\,\left( {{\text{Mo}} = 0\,\,;\,\,1 = e - a} \right) \hfill \\
\,\,\,\left\{ {0,1,1,1,1} \right\}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{No}}\,\,\,\,\,\left( {{\text{Mo}} = 1\,\,;\,\,1 = e - a} \right) \hfill \\
\end{gathered} \right.\]

(*) 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
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by fskilnik@GMATH » Mon Aug 20, 2018 11:12 am
<p>
</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.
\[{\text{WLOG}}\left( * \right)\,\,\,\,a \leqslant b \leqslant c \leqslant d \leqslant e\,\,\,\,\,\,\left( {{\text{single}}\,\,{\text{Mo}}} \right)\]
\[{\text{?}}\,\,\,{\text{:}}\,\,\,{\text{Mo}}\,\,\mathop < \limits^? \,\,e - a\]
\[\left( 1 \right)\,\,\,\,a < 0\,\,\,\, \Rightarrow \,\,\,\,e - a = e + \left( { - a} \right) > e\]
\[?\,\,\,:\,\,\,{\text{Mo}} \leqslant e < e - a\,\,\,\,\, \Rightarrow \,\,\,\,{\text{Mo}} < e - a\,\,\,\,\,\,\left( {{\text{Suf}}{\text{.}}} \right)\]
\[\left( 2 \right)\,\,\,{\text{Insuf}}{\text{.}}\,\,\,\left\{ \begin{gathered}
\,\,\,\left\{ {0,0,0,0,1} \right\}\,\,\,\, \Rightarrow \,\,\,\,{\text{Yes}}\,\,\,\left( {{\text{Mo}} = 0\,\,;\,\,1 = e - a} \right) \hfill \\
\,\,\,\left\{ {0,1,1,1,1} \right\}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{No}}\,\,\,\,\,\left( {{\text{Mo}} = 1\,\,;\,\,1 = e - a} \right) \hfill \\
\end{gathered} \right.\]

(*) 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
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by fskilnik@GMATH » Mon Aug 20, 2018 11:27 am
<p>
</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.
Image

(*) 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
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