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What is range of the six positive integers: 1, p, 12, q, 15, r?  [#permalink]

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What is range of the six positive integers: 1, p, 12, q, 15, r?

(1) $$1 < p < q < 5 < r < 18$$

(2) $$p + q + r < 18$$ This question was provided by e-GMAT for the Game of Timers Competition _________________
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Re: What is range of the six positive integers: 1, p, 12, q, 15, r?  [#permalink]

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3
What is range of the six positive integers: 1, p, 12, q, 15, r?

(1) 1<p<q<5<r<18
r can take values larger than 15, i.e. 16 or 17 which may increase the range. NOT SUFFICIENT.

(2) p+q+r<18
p, q & r are positive integers.
Let us take p & q minimum i.e. p=1 & q=1 to make r as large as possible.
1+1+r<18
=>r<16
r can be maximum 15 and cannot be less than 1.
Therefore, range of 6 integers = 15-1 = 14
SUFFICIENT

Statement 2 alone is SUFFICIENT.

IMO B
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Re: What is range of the six positive integers: 1, p, 12, q, 15, r?  [#permalink]

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1
(1) 1<p<q<5<r<18

Range of the integers equals the difference between the biggest number and the smallest number
1st statement allows us to find the range: 18-1=17
Sufficient

(2) p+q+r<18
p,q can equal 1 and r can be 16, then the range will be 16-1=15
Also, p,q,r - can be smaller than 15, then the range will be 15-1=14 (for example)
Not enough to answer the question

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Re: What is range of the six positive integers: 1, p, 12, q, 15, r?  [#permalink]

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3
1
What is range of the six positive integers: 1, p, 12, q, 15, r?

(1) 1<p<q<5<r<18: if r = 15, the range= 15-1=14, but if r = 16, range = 16-1=15, range can't be determined exactly

(2) p+q+r<18 --> correct: case-1: if r = 15, p+q < 3 => p=q=1, case-2: if r=16, p+q < 2 not possible because p & q are positive integer so p+q should be >= 2. So r should be <=15, range = 15-1=14

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What is range of the six positive integers: 1, p, 12, q, 15, r?  [#permalink]

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1
1
Option 1:
1<p<q<5<r<181<p<q<5<r<18

r can be greater than 15 or less than 15. if 5<r<15 then the range is 14. If 15<r<18 then range is more than 15.
Insufficient

Option 2: p+q+r<18
If p=q=1 then r<16. since r is a positive integer r can be 15 or less. Hence range will be 14.
If one of p or q is not equal to 1. r being a positive integer will be less than 15. Here too the range will be 14
Sufficient

Ans : B

Originally posted by ruchik on 08 Jul 2019, 08:10.
Last edited by ruchik on 10 Jul 2019, 20:30, edited 1 time in total.
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What is range of the six positive integers: 1, p, 12, q, 15, r?  [#permalink]

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IMO B

range of set = highest- lowest value
#1
1<p<q<5<r<18
p=2or3
q=3or4 if p=2
r=6 to 17
insufficient as we get many possible values of r >15 as well range value will vary
#2
p+q+r<18
p=q=1 and r=15
so range ; 15-1 ; 14
IMO B

What is range of the six positive integers: 1, p, 12, q, 15, r?

(1) 1<p<q<5<r<181<p<q<5<r<18

(2) p+q+r<18

Originally posted by Archit3110 on 08 Jul 2019, 08:12.
Last edited by Archit3110 on 09 Jul 2019, 08:08, edited 1 time in total.
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Re: What is range of the six positive integers: 1, p, 12, q, 15, r?  [#permalink]

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1
Statement 1-
r can be any integer from 6 to 17(inclusive)
p can be 2 or 3
q can be 3 or 4

Hence
i) if r is less than or equal to 15, range of 6 number is 15-1=14
ii) if r is 16, range of 6 numbers is 16-1=15
iii) if r is 17, range of 6 numbers is 17-1=16

Insufficient

Statement 2
p+q+r<18
maximum value p, q or r can take is 17-1-1=15
Hence the greatest possible integer among 6 number is 15
Range of 6 numbers= 15-1=14
Sufficient.

IMO B
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Re: What is range of the six positive integers: 1, p, 12, q, 15, r?  [#permalink]

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1
We are asked for the range of the following set of positive numbers: 1, p, 12, q, 15, r
The range will be [largest number] - [smallest number]
We note that we know the smallest number is 1 as all numbers are positive.
We do not know that the numbers are in order, so the largest number could be 15, or a larger number.
We also do not know if the numbers are all different.

Statement 1: This tells us that all numbers are different, and that all numbers <18 therefore the range will be <17-1=16.
We also know that the largest number will be either r or 15. However, from the given information, r could be anything between 6 and 16. Not sufficient

Statement 2: If p+q+r < 18, p,q,r could be 1, 1, 1 (sum of 3) or 15, 1, 1 (sum of 17). From this information, we can see that none of the variables can be larger than 15. Therefore, the largest number is 15, and the range is 15-1=14. Sufficient
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Re: What is range of the six positive integers: 1, p, 12, q, 15, r?  [#permalink]

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1
Statement 1: Range = 18-1=17. So, 1 alone can give the range.
Statement 2: From p+q+r<18, and given, all are positive integers, Either p, q or r cannot be greater than 15. So, Range =15-1=14. So 2 alone can give the range.
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Re: What is range of the six positive integers: 1, p, 12, q, 15, r?  [#permalink]

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1
We have 6 numbers (1, p, 12, q, 15, r) and all of them are positive. We do not know if the order provided is the correct one or the numbers are given in the random order. Our need is to find the range of numbers (difference between smallest and highest number). Let us analyze statements one by one.

Statement 1:
It is given that 1 < p < q < 5 < r < 18. The lowest number in the succession is 1, and the highest number is r. Thus, the number r is in the range 5 < r < 18, i.e. it can be any number in this range. The highest number in the range was 15, and we do not know whether r = 6 or r = 16. Thus, the information provided is not sufficient.
Statement 1 is insufficient.

Statement 2:
It is given that p + q + r < 18.
We are also given that all numbers are positive i.e. p > 0, q > 0 and r > 0. It means that the minimum value for all these numbers should be at least 1.
Let us find the maximum value we can get from this equation. If we assume that p = q = 1, then 1 + 1 + r < 18 i.e. r < 16. As we already have number 15 in our number set, we can conclude that the range is 15 - 1 = 14. The information given is sufficient.
Statement 2 is sufficient.

P.S. I actually answered the question incorrectly, since in the Statement 1 I was deceived by the equation and incorrectly identified the range. Thus, my answer was D - each statement alone is sufficient, which is not the case. I was able to realize it when was writing down an explanation for my answer.
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Re: What is range of the six positive integers: 1, p, 12, q, 15, r?  [#permalink]

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C.

Range is: maximum - minimum.

From st 1: p and q are greater than 1 and r is greater than 5 and less than 18. So r can be less than 12 or greater than 15. So we dont know what is the maximum number. Not Sufficient
From st 2: This statement itself is clearly not sufficient since p,q and r can have various integer solutions.

Combine both: we can certainly now say that r is less than 15 since p and q are both greater than or equal to 2. So range has to be 15 -1 = 14.
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What is range of the six positive integers: 1, p, 12, q, 15, r?  [#permalink]

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1
answer is B in my opinion.
In st 1, if r is 17, range is 16, if r is 16, range is 15. Different values, thus not suff.
St 2, we know that p, q, r combined are less than 18 or at most 17. Largest number R can be is 15 and min number p and q are 1 each (as all integers are positive) so in any case range is 14 (15-1). Note: of course R can be less than 15, but we don't care then because we already have 15 in out set. Thus, st 2 is suff., So B

Originally posted by mira93 on 08 Jul 2019, 08:22.
Last edited by mira93 on 08 Jul 2019, 22:32, edited 1 time in total.
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Re: What is range of the six positive integers: 1, p, 12, q, 15, r?  [#permalink]

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1) Not sufficient. The statement does not lead to the value of P, Q or R.
2) not sufficient. same as statement 1.
1&2: sufficient.

P+Q=2+3 = 5
18-5=13=r
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Re: What is range of the six positive integers: 1, p, 12, q, 15, r?  [#permalink]

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Quote:
What is range of the six positive integers: 1, p, 12, q, 15, r?
(1) 1<p<q<5<r<18
(2) p+q+r<18

given: all 6 are positive integers
rule: range=largest-smallest

(1) 1<p<q<5<r<18: if r=17, then rng=17-1=16; if r=16, then rng=16-1=15; two different ranges, insufficient.
(2) p+q+r<18: if p,q,r≤17, and p,q=1, then r≤17-2≤15, so largest=r or 15 and smallest is 1; range=15-1=14; sufficient.

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Re: What is range of the six positive integers: 1, p, 12, q, 15, r?  [#permalink]

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Range=Highest- Lowest
Q:
1, p, 12, q, 15, r

We need to figure out values of
p,q,r.

1. clear shows that least value is 1 and highest is 18.
A is sufficient.

2. p+q+r <18.
and nos are positive.
so in no scenario p,q,r can change the highest value. but can be equal to the lowest value of 1.
B is sufficient.

So, IMO d

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Re: What is range of the six positive integers: 1, p, 12, q, 15, r?  [#permalink]

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1
From (1)

If r=16, range is 16-1 = 15,
If r=13, range is 15-1 = 14

Insufficient

From (2)

The maximum value while one of the numbers can take is 15 in which case the other two numbers are 1

Range is 15-1 = 14

For any other combination of p,q,r, we get 1<p,q,r<15

Range in this case is again, 15-1=14

Sufficient

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What is range of the six positive integers: 1, p, 12, q, 15, r?  [#permalink]

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1
1, p, 12, q, 15, r

(1) 1<p<q<5<r<18
r can take any value between 6 and 17.
If r=17, range = 17-1=16
If r=10, range = 15-1 = 14

(2) p+q+r<18
Given p,q,r are +ve integers
The least value they can take is 1.
To maximize the value of one variable, minimize the other two
Lets say p=q=1, this implies r = 15. so the maximum value can be 15.
Which implies that the highest value in the given list will be 15.
Therefore range = 15-1 = 14

Hence statement 2 alone is sufficient.

Option B

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Originally posted by prashanths on 08 Jul 2019, 08:27.
Last edited by prashanths on 09 Jul 2019, 00:39, edited 1 time in total.
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Re: What is range of the six positive integers: 1, p, 12, q, 15, r?  [#permalink]

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B alone is sufficient as 15 will be the highest number as per B

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Re: What is range of the six positive integers: 1, p, 12, q, 15, r?  [#permalink]

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IMO, (B)

Given that we have six positive integers: 1, p, 12, q, 15, r

Statement (1) has 1<p<q<5<r<18

Now, we know, p and q are greater than 1 and less than 5, so could be 2, 3 or 4 - which are all > 1, so min. value of this set remains 1. Now, coming to the maximum value, we're only told that 5<r<18, hence, if r<15, then the maximum value of the set would be 15 and the corresponding range would be 15-1 = 14, BUT if r>15, then the maximum value of the set would be r and the range would then be r-1

Since we're not sure about the value of r, Statement (1) is INSUFFICIENT

Looking at Statement (2) now

p+q+r<18

Since, p, q and r are all positive integers, the maximum value of their sums can be 17 (since their sum also has to have an integral value), now although we don't know here which of p, q, r is largest and which is the smallest, but that's inconsequential here, since the maximum value any number can take is 15 (minimising the other two numbers at their lowest possible values, i.e. 1), and hence, regardless of what the values and inequalities between p, q and r are, the smallest value cannot be less than 1 and the largest value cannot exceed 15, which coincides with the smallest and the largest values of the known numbers of the set, and hence, the range will ALWAYS be 15 - 1 = 14

Thus, Statement (2) is SUFFICIENT

And hence, (B) is the correct answer choice
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Re: What is range of the six positive integers: 1, p, 12, q, 15, r?  [#permalink]

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What is range of the six positive integers: 1, p, 12, q, 15, r?

(1) 1<p<q<5<r<18

(2) p+q+r<18

statement 1 : insufficient
r can be 16, 15
range can differ
15-1 = 14
or 16 - 1= 15

2) Insufficent
as p, q and r can all there be either 6,6,6
or 1,2,15 or 1,1, 16

Combine both
we get
given 1< p<q
thus
take min value of p and q equal to 2 and 2
r = 14
hence 15 is greatest
range = 15-1 =14
thus C Re: What is range of the six positive integers: 1, p, 12, q, 15, r?   [#permalink] 08 Jul 2019, 08:30

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