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A set of data consists of {3, 5, 7, 9, a}. If the range of the set is

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A set of data consists of {3, 5, 7, 9, a}. If the range of the set is [#permalink]

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New post 22 Aug 2017, 22:48
2
00:00
A
B
C
D
E

Difficulty:

  25% (medium)

Question Stats:

73% (01:04) correct 27% (00:28) wrong based on 129 sessions

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A set of data consists of {3, 5, 7, 9, a}. If the range of the set is [#permalink]

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New post 22 Aug 2017, 22:57
1
Range = a-3 if a is largest
a-3=2a, a=-3 In this case a is not the largest element as assumed.
so range =3-a=2a, a=1
Answer =a=1
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Re: A set of data consists of {3, 5, 7, 9, a}. If the range of the set is [#permalink]

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New post 22 Aug 2017, 23:36
Bunuel wrote:
A set of data consists of {3, 5, 7, 9, a}. If the range of the set is equal to 2a, what is the value of a?

A. -3
B. 1
C. 3
D. 5
E. 6


range is 2a
Lets substitute value of a
a=-3 then range will be -6...not possible (range cannot be negative)
a=1 then range will be 2....not possible (range will be 9-1=8)
a=3 then range will be 6...correct (range will be 9-3=6)
a=5 then range will be 10...not possible (range will be 9-3=6)
a=6 then range will be 12..not possible (range will be 9-3=6)

Hence C
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A set of data consists of {3, 5, 7, 9, a}. If the range of the set is [#permalink]

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New post 23 Aug 2017, 00:46
If a > 9

then in that case a-3 = 2a => a = -3 hence, this is not possible.

ii ) a < 3

then in this case 9-a = 2a > a = 3

if a > 3 then the minimum value of the set will be = 3
since we know "a" cannot be greater than 9

hence range we get

9-3 = 6 => a = 3

answer = C
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Re: A set of data consists of {3, 5, 7, 9, a}. If the range of the set is [#permalink]

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New post 23 Aug 2017, 01:48
Bunuel wrote:
A set of data consists of {3, 5, 7, 9, a}. If the range of the set is equal to 2a, what is the value of a?

A. -3
B. 1
C. 3
D. 5
E. 6



I found it easier to solve this by evaluating the options! Only C satisfies.
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A set of data consists of {3, 5, 7, 9, a}. If the range of the set is [#permalink]

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New post 23 Aug 2017, 05:46
Bunuel wrote:
A set of data consists of {3, 5, 7, 9, a}. If the range of the set is equal to 2a, what is the value of a?

A. -3
B. 1
C. 3
D. 5
E. 6

I think the answer is C, but I have a question.

I did this problem in two ways and got the same answer both times. I am uneasy about the second method, however, because I can't explain, with precision, why it works. I think my brain is frozen. :-(

First I just used answer choices, then algebra.

For answer choices, only 3, Answer C, works. If a = 3, (9 - 3) = 6, and 6 = 2a.

Then I wrote:
9 - a = 2a
9 = 3a
a = 3

I think that equation is legitimate here because the range is a multiple of the difference of two numbers in the set. That's as far as I get on the explanation front.

I think I grasp for something I already know about number properties where x is greater than y. Of course x - y CAN be a multiple of y. But there's a difference between possibility and necessity, and I worry that my equation inaccurately suggests the latter.

(Waffling yet again, I think: "But these conditions mandate that necessity.")

IS the equation legitimate? If so, would someone please explain that which I can intuit but can't quite explain? Why does the equation work? Why does "a" have to be the lower limit (9 - a) of the range?

Sorry if this question is a bad case of missing the obvious. I would appreciate greatly any help!
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A set of data consists of {3, 5, 7, 9, a}. If the range of the set is [#permalink]

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New post 23 Aug 2017, 09:04
1
Hi genxer123,

Consider the set {1, 5, 7, 9, a}
In this case, the value for 'a' would be 4 as the range of numbers '2a' is 8

In this case(the problem in hand), it just so happens that a is also the lowest number,
so your equation 9 - a = 2a works.
Else, it could lead you to a wrong answer.

IMO, the only way we can do this problem is go from the answer choices.
Hope that answers your question!
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A set of data consists of {3, 5, 7, 9, a}. If the range of the set is [#permalink]

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New post 23 Aug 2017, 11:31
pushpitkc wrote:
Hi genxer123,

Consider the set {1, 5, 7, 9, a}
In this case, the value for 'a' would be 4 as the range of numbers '2a' is 8

In this case(the problem in hand), it just so happens that a is also the lowest number,
so your equation 9 - a = 2a works.
Else, it could lead you to a wrong answer.

IMO, the only way we can do this problem is go from the answer choices.
Hope that answers your question!

pushpitkc , your answer is very clear and easy to understand.

Best of all, it confirms my suspicion that something was off about that equation, especially in terms of generalization. My brain is now thawed. Thanks and kudos.
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that within me there lay an invincible summer.

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Re: A set of data consists of {3, 5, 7, 9, a}. If the range of the set is [#permalink]

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New post 25 Aug 2017, 11:05
1
Bunuel wrote:
A set of data consists of {3, 5, 7, 9, a}. If the range of the set is equal to 2a, what is the value of a?

A. -3
B. 1
C. 3
D. 5
E. 6


We have to consider the following cases: 1) a is the largest number, 2) a is the smallest number, 3) a is neither the largest number nor the smallest number. Let’s analyze each case.

Case 1: a is the largest number

If a is the largest number (i.e., at least 9), then 3 is the smallest number and we have:

a - 3 = 2a

-a = 3

a = -3

However, since a must be at least 9, this case is not viable.

Case 2: a is the smallest number

If a is the smallest number (i.e., at most 3), then 9 is the largest number and we have:

9 - a = 2a

9 = 3a

a = 3

Since a is indeed at most 3, a can be 3. Although we have found a possible value for a, let’s analyze case 3 also.

Case 3: a is neither the largest number nor the smallest number

If a is neither the largest number nor the smallest number (i.e., 3 < a < 9), then 9 is the largest number and 3 is the smallest number, and hence we have:

9 - 3 = 2a

6 = 2a

a = 3

We have validated that a = 3 works, as was found in Case 2.

Answer: C
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Re: A set of data consists of {3, 5, 7, 9, a}. If the range of the set is   [#permalink] 25 Aug 2017, 11:05
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