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# Let q represent the integer length of a side of a triangle. If r repre

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Math Expert
Joined: 02 Sep 2009
Posts: 55276
Let q represent the integer length of a side of a triangle. If r repre  [#permalink]

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15 Jul 2015, 03:21
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45% (medium)

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60% (01:28) correct 40% (01:36) wrong based on 146 sessions

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Let q represent the integer length of a side of a triangle. If r represents the number of distinct values for q such that we can create a triangle with lengths q, 9, and 13, what is the value of r?

A. 5
B. 17
C. 18
D. 22
E. 29

Kudos for a correct solution.

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Re: Let q represent the integer length of a side of a triangle. If r repre  [#permalink]

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15 Jul 2015, 04:36
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2
Bunuel wrote:
Let q represent the integer length of a side of a triangle. If r represents the number of distinct values for q such that we can create a triangle with lengths q, 9, and 13, what is the value of r?

A. 5
B. 17
C. 18
D. 22
E. 29

Kudos for a correct solution.

Sides of triangle are q, 9 and 13

for a triangle to exist
Sum of the two sides > third side

i.e. For Smallest value of q
q + 9 > 13
i.e. q > 4

i.e. For Largest value of q
13 + 9 > q
i.e. q < 22

i.e. q can be any Integer from 5 to 21
Total Possible values of q = r = {5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21}

i.e. r = 17

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Re: Let q represent the integer length of a side of a triangle. If r repre  [#permalink]

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15 Jul 2015, 04:51
1
Bunuel wrote:
Let q represent the integer length of a side of a triangle. If r represents the number of distinct values for q such that we can create a triangle with lengths q, 9, and 13, what is the value of r?

A. 5
B. 17
C. 18
D. 22
E. 29

Kudos for a correct solution.

The trick here is to know that for a given traingle with sides a,b,c , for any side the following relation holds:

|a-b| < c < a+b (for ALL triangles).

Thus, per the given question, 13-9< q < 13+9 ---> 4 <q < 22 ---> This gives us 17 integer values for q and thus r= 17. B is the correct answer.
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Joined: 11 Sep 2013
Posts: 7
Re: Let q represent the integer length of a side of a triangle. If r repre  [#permalink]

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15 Jul 2015, 22:32
1
As per triangle inequality, length of q will be greater than the difference of 2 sides and less than the sum of 2 sides.
So 4<q<22

The number of distinct values or r will then be 21-5/1 + 1 = 17 (coz 4 and 22 will not be included)

Option B
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Joined: 20 Jul 2011
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Re: Let q represent the integer length of a side of a triangle. If r repre  [#permalink]

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17 Jul 2015, 22:22
Bunuel wrote:
Let q represent the integer length of a side of a triangle. If r represents the number of distinct values for q such that we can create a triangle with lengths q, 9, and 13, what is the value of r?

A. 5
B. 17
C. 18
D. 22
E. 29

Kudos for a correct solution.

We know that, the sum of any two sides of a triangle should be greater than the third side (a+b > c)
and the difference between any two sides should be less than the third side (a-b<c)

9+13 < q < 13-9
22 < q < 4

q ranges from 5 to 21. No of items in this range = 21-5+1 = 17.

Option B
Math Expert
Joined: 02 Sep 2009
Posts: 55276
Re: Let q represent the integer length of a side of a triangle. If r repre  [#permalink]

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19 Jul 2015, 13:11
Bunuel wrote:
Let q represent the integer length of a side of a triangle. If r represents the number of distinct values for q such that we can create a triangle with lengths q, 9, and 13, what is the value of r?

A. 5
B. 17
C. 18
D. 22
E. 29

Kudos for a correct solution.

800score Official Solution:

From the triangle inequality rule, the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. So, the three inequalities are
q + 9 > 13,
9 + 13 > q,
and q + 13 > 9.

The third equation holds for any positive value of q. So solve the first two inequalities for q, so q > 4 and 22 > q. So the value of q is 4 &lt; q &lt; 22.

Since the lengths of the sides of the triangle are of integer value, the lowest possible integer/side can be 5 and the highest possible integer/side can be 21 (q = 5, 6, 7, 8, 9 ..., 19, 20, or 21). So, r, the number of numbers in this set, must equal 21 – (5 – 1) = 17.

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Re: Let q represent the integer length of a side of a triangle. If r repre  [#permalink]

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21 Dec 2018, 17:13
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Re: Let q represent the integer length of a side of a triangle. If r repre   [#permalink] 21 Dec 2018, 17:13
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# Let q represent the integer length of a side of a triangle. If r repre

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