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polygon help

by quantskillsgmat » Wed Feb 29, 2012 11:49 pm

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Q the measure of the interior angles in a polygon are consecutive integers.The smallest angle measures 136 degrees.how many sides does this polygon hve.
a)8 b)9 c)10 d)11 e)13
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by Anurag@Gurome » Wed Feb 29, 2012 11:53 pm
quantskillsgmat wrote:Q the measure of the interior angles in a polygon are consecutive integers.The smallest angle measures 136 degrees.how many sides does this polygon hve.
a)8 b)9 c)10 d)11 e)13
Say, the number of sides of the polygon is n.
Hence, sum of the interior angles of the polygon = (n - 2)*180 degrees

Smallest angle is 136 degrees and all the interior angles are consecutive integers. Hence, the measures of the interior angles of the polygon in degrees are 136, (136 + 1), (136 + 2), ..., and (136 + n - 1).

Hence, sum of the angles = [136 + (136 + 1) + (136 + 2) + ... + (136 + n - 1)] = 136n + n(n - 1)/2

Thus, 136n + n(n - 1)/2 = (n - 2)*180
272n + n² - n = 360n - 720
n² -89n + 720 = 0
(n - 9)(n - 80) = 0

Now, if n is equal to 80, then some of the interior angles of the polygon will be greater than 180. Hence, only possible value of n is 9.

The correct answer is B.
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by amit28it » Thu Mar 01, 2012 5:01 am
My answer for your question is B Baecause the number of sides of the polygon is n.
Hence, sum of the interior angles of the polygon = (n - 2)*180 degrees

Thus, 136n + n(n - 1)/2 = (n - 2)*180
272n + n² - n = 360n - 720
n² -89n + 720 = 0
(n - 9)(n - 80) = 0



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by GMATGuruNY » Thu Mar 01, 2012 5:18 am
quantskillsgmat wrote:Q the measure of the interior angles in a polygon are consecutive integers.The smallest angle measures 136 degrees.how many sides does this polygon hve.
a)8 b)9 c)10 d)11 e)13
The sum of the interior angles must be a multiple of 180.
The sum of consecutive integers = (number of integers)(average of the biggest and smallest).
Thus, the sum of the smallest angle and the biggest angle is likely to be a multiple of 10.
Since the units digit of the smallest angle is 6 (136), the units digit of the biggest angle is likely to be 4 (144).
Between 136 and 144, inclusive, the number of integers = biggest-smallest+1 = 144-136+1 = 9.
If there are 9 sides:
Average of the biggest and smallest angles = (144+136)/2 = 140.
Sum of the angles = number*average = 9*140.
Since 9 is a multiple of 9 and 140 is a multiple of 20, the sum of the angles is a multiple of 180.
Success!

The correct answer is B.
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by krusta80 » Thu Mar 01, 2012 9:03 am
quantskillsgmat wrote:Q the measure of the interior angles in a polygon are consecutive integers.The smallest angle measures 136 degrees.how many sides does this polygon hve.
a)8 b)9 c)10 d)11 e)13
Total degrees of the interior angles of an n-sided (or n-angled) polygon equals 180*n-360.

Rather than go through some messy calculations, let's note that the sum must end in a 0...so we can just look at the last digit of our total after adding each new angle. The first digit in each row below represents to last digit of the sum up to that point (ie. first total for one side is 136, so we see a 6). The second digit represents the last digit of the next value in sequence (ie. 137 in the first row). etc..

Cycle approach:

6+7
3+8
1+9
0+0
0+1
1+2
3+3
6+4
0+5 ---> THIS WORKS for n=9 (ninth row), BUT LET'S KEEP CHECKING TO RULE OUT 10,11, and 13
5+6
1+7
8+8
6+9
5+0 DONE

Answer is B


Keep in mind that this worked out nicely, since none of the other choices yielded a total ending in 0. I just thought this would be an interesting alternative to an algebraic approach.
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by krusta80 » Thu Mar 01, 2012 9:17 am
Anurag@Gurome wrote:
quantskillsgmat wrote:Q the measure of the interior angles in a polygon are consecutive integers.The smallest angle measures 136 degrees.how many sides does this polygon hve.
a)8 b)9 c)10 d)11 e)13
Say, the number of sides of the polygon is n.
Hence, sum of the interior angles of the polygon = (n - 2)*180 degrees

Smallest angle is 136 degrees and all the interior angles are consecutive integers. Hence, the measures of the interior angles of the polygon in degrees are 136, (136 + 1), (136 + 2), ..., and (136 + n - 1).

Hence, sum of the angles = [136 + (136 + 1) + (136 + 2) + ... + (136 + n - 1)] = 136n + n(n - 1)/2

Thus, 136n + n(n - 1)/2 = (n - 2)*180
272n + n² - n = 360n - 720
n² -89n + 720 = 0
(n - 9)(n - 80) = 0

Now, if n is equal to 80, then some of the interior angles of the polygon will be greater than 180. Hence, only possible value of n is 9.

The correct answer is B.
A good formula to know was derived here:

sum(x through x+n-1) = x*n + nC2

Let's try it out for x = 1 and n = 5:

sum(1 through 5) = 1*5 + 5C2 = 5 + 10 = 15
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by alanforde800Maximus » Mon Jul 30, 2018 10:12 pm
GMATGuruNY wrote:
quantskillsgmat wrote:Q the measure of the interior angles in a polygon are consecutive integers.The smallest angle measures 136 degrees.how many sides does this polygon hve.
a)8 b)9 c)10 d)11 e)13
The sum of the interior angles must be a multiple of 180.
The sum of consecutive integers = (number of integers)(average of the biggest and smallest).
Thus, the sum of the smallest angle and the biggest angle is likely to be a multiple of 10.
Since the units digit of the smallest angle is 6 (136), the units digit of the biggest angle is likely to be 4 (144).
Between 136 and 144, inclusive, the number of integers = biggest-smallest+1 = 144-136+1 = 9.
If there are 9 sides:
Average of the biggest and smallest angles = (144+136)/2 = 140.
Sum of the angles = number*average = 9*140.
Since 9 is a multiple of 9 and 140 is a multiple of 20, the sum of the angles is a multiple of 180.
Success!

The correct answer is B.
Hello Mitch,

Might be a stupid question but can you help me understand how come 144 came into picture? Is it coz 136 + (n-1) will be the largest angle. Therefore, putting n =9 this will result to 144, is that so?
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by Scott@TargetTestPrep » Fri Aug 10, 2018 5:56 pm
quantskillsgmat wrote:Q the measure of the interior angles in a polygon are consecutive integers.The smallest angle measures 136 degrees.how many sides does this polygon hve.
a)8 b)9 c)10 d)11 e)13
Let's assume that the polygon has n sides. Recall that the sum of the angle measures of an n-sided polygon is 180(n - 2). We are given the smallest angle of the polygon measures 136 degrees, so ones after that are 136 + 1, 136 + 2, and so on, ending with the largest angle measure of 136 + n - 1. Therefore, we can create the equation:

136 + (136 + 1) + (136 + 2) + ... + (136 + n - 1) = 180(n - 2)

(135 + 1) + (135 + 2) + (135 + 3) + ... + (135 + n) = 180n - 360

On the left hand side of the equation, we can rearrange the terms as:

(135 + 135 + 135 + ... + 135) + (1 + 2 + 3 + ... + n) = 180n - 360

There are a total of n repetitions of 135, so the sum of the 135's is 135n; thus, we have:

135n + (1 + 2 + 3 + ... + n) = 180n - 360

1 + 2 + 3 + ... + n = 45n - 360

We use the shortcut formula for the sum of n consecutive integers to re-express the left side of the equation:

n(n + 1)/2 = 45n - 360

n^2 + n = 90n - 720

n^2 - 89n + 720 = 0

(n - 80)(n - 9) = 0

n = 80 or n = 9

Since only 9 is given, the correct choice is B.

Answer: B

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