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The exterior angles of a quadrilateral are in the ratio 1:4:4:6. What

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The exterior angles of a quadrilateral are in the ratio 1:4:4:6. What  [#permalink]

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New post 11 Jul 2017, 22:28
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The exterior angles of a quadrilateral are in the ratio 1:4:4:6. What is the difference between the largest and the smallest interior angles of the quadrilateral?

(A) 110
(B) 60
(C) 140
(D) 120
(E) 100

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The exterior angles of a quadrilateral are in the ratio 1:4:4:6. What  [#permalink]

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New post 12 Jul 2017, 00:12
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Gnpth wrote:
The exterior angles of a quadrilateral are in the ratio 1:4:4:6. What is the difference between the largest and the smallest interior angles of the quadrilateral?

(A) 110
(B) 60
(C) 140
(D) 120
(E) 100


Sum of all exterior angles is 360 so we can just use fractions for largest and smallest proportions to find their corresponding angle in degrees.

Sum of proportions = \(1 + 4 + 4 + 6 = 15\)
Smallest exterior angle:

\(1/15 * 360 = 24\)

Largest exterior angle:

\(6/15 * 360 = 144\)

interior angle + exterior angle = 180.

interior angle = 180 - exterior angle.

Naturally largest interior angle will be corresponding to the smallest exterior angle and smallest interior angle will be angle corresponding to the largest exterior angle.

Largest interior angle = 180 - smallest exterior angle = \(180 - 24 = 156\)
Smallest interior Angle = 180 - largest exterior angle = \(180 - 144 = 36\)

Difference = \(156 - 36 = 120\)
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The exterior angles of a quadrilateral are in the ratio 1:4:4:6. What  [#permalink]

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New post 12 Jul 2017, 01:15
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We know that the sum of the exterior angles is 360 degree for any quadrilateral.
Sum of proportions = \(1+4+4+6=15\)

Hence, each part(of the angle) is \(\frac{1}{15}∗360=24\)
Hence the exterior angles are 24,96,96 and 144.

The sum of the subsequent interior angle and the exterior angle is always 180 degree
Therefore, the largest interior angle is 180 - 24 = 146 and the smallest interior angle is 180-144 = 26
The difference is 146 - 26 = 120

Hence, the difference between the interior angles(largest and smallest) is 120 degree(Option D)
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Re: The exterior angles of a quadrilateral are in the ratio 1:4:4:6. What  [#permalink]

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New post 12 Jul 2017, 05:58
Gnpth wrote:
The exterior angles of a quadrilateral are in the ratio 1:4:4:6. What is the difference between the largest and the smallest interior angles of the quadrilateral?

(A) 110
(B) 60
(C) 140
(D) 120
(E) 100




can anyone tell me if my approach is correct:)

the ratio is 1:4:4:6

it means that
1 = 360
4 = 90
4 = 90
6 = 60

so 90+90+60 = 240

360 -240 = 120 :)
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Re: The exterior angles of a quadrilateral are in the ratio 1:4:4:6. What  [#permalink]

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New post 12 Jul 2017, 07:00
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The sum of Internal angles of a Quadrilateral is 360°. The lines of the sides of the Quadrilateral can be extended to form external angles (see attachment). Hence the sum of external angles =360°
Now sides are in Ratio 1:4:4:6.
Let the common multiplier be x
therefore 1x+4x+4x+6x=15x.
15x=360°(Sum of angles)
X=24
Side with the lowest ratio=1x24°=24°
Side with the highest Ratio =24x6°=144
Diff= 120.


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Re: The exterior angles of a quadrilateral are in the ratio 1:4:4:6. What  [#permalink]

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New post 12 Jul 2017, 09:32
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dave13 wrote:
Gnpth wrote:
The exterior angles of a quadrilateral are in the ratio 1:4:4:6. What is the difference between the largest and the smallest interior angles of the quadrilateral?

(A) 110
(B) 60
(C) 140
(D) 120
(E) 100




can anyone tell me if my approach is correct:)

the ratio is 1:4:4:6

it means that
1 = 360
4 = 90
4 = 90
6 = 60

so 90+90+60 = 240

360 -240 = 120 :)


No, this doesn't seem to be the right approach. You may have got this right this time. Look at other explanations from pushpitkc, jedit and gps5441.
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Re: The exterior angles of a quadrilateral are in the ratio 1:4:4:6. What  [#permalink]

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New post 16 Jul 2017, 17:13
Gnpth wrote:
The exterior angles of a quadrilateral are in the ratio 1:4:4:6. What is the difference between the largest and the smallest interior angles of the quadrilateral?

(A) 110
(B) 60
(C) 140
(D) 120
(E) 100


Since the sum of the exterior angles of a polygon is 360 degrees, we can create the following equation:

x + 4x + 4x + 6x = 360

15x = 360

x = 24

The smallest exterior angle is 24, which corresponds to an interior angle of 180 - 24 = 156 degrees. The largest exterior angle is 6 x 24 = 144, which corresponds to an interior angle of 180 - 144 = 36 degrees.

Thus, the difference between the largest and smallest interior angles is 156 - 36 = 120 degrees.

Answer: D
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Re: The exterior angles of a quadrilateral are in the ratio 1:4:4:6. What  [#permalink]

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New post 17 Jul 2017, 11:19
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Gnpth wrote:
The exterior angles of a quadrilateral are in the ratio 1:4:4:6. What is the difference between the largest and the smallest interior angles of the quadrilateral?

(A) 110
(B) 60
(C) 140
(D) 120
(E) 100


We can avoid unnecessary calculation in the following manner:

Interior angle= 180°- exterior angle.
Let the exterior angles of the quadrilateral are x,4x,4x,6x

Hence difference between largest and the smallest interior angles = (180-x) -(180-6x) =5x.
Now sum of exterior angles of a quadrilateral is 360; hence x+4x+4x+6x=360; i.e. 15x=360; 5x=120.
Hence D is the answer.
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Re: The exterior angles of a quadrilateral are in the ratio 1:4:4:6. What  [#permalink]

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New post 26 Jul 2017, 20:40
Gnpth wrote:
The exterior angles of a quadrilateral are in the ratio 1:4:4:6. What is the difference between the largest and the smallest interior angles of the quadrilateral?

(A) 110
(B) 60
(C) 140
(D) 120
(E) 100


First, the answer is either 60 or 120, as it has to be a multiple of 5 (range of ratio i.e 6-1), 6, and 15. Second, the sum of exterior angles of n=4 is 180(2)= 360. Therefore, the answer is 360/3= 120 (D) since the range is a third of the ratio i.e 5/15.
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The exterior angles of a quadrilateral are in the ratio 1:4:4:6. What  [#permalink]

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New post 26 Jul 2017, 20:53
dave13 wrote:
Gnpth wrote:
The exterior angles of a quadrilateral are in the ratio 1:4:4:6. What is the difference between the largest and the smallest interior angles of the quadrilateral?

(A) 110
(B) 60
(C) 140
(D) 120
(E) 100




can anyone tell me if my approach is correct:)

the ratio is 1:4:4:6


it means that
1 = 360
4 = 90
4 = 90
6 = 60

so 90+90+60 = 240

360 -240 = 120 :)


IMO, this approach is wrong. Remember you are dealing with ratio and the whole is (1+4+4+6), making each ratio a fraction of 15 and each corresponding angle out of 360. Here 1 is simply 1/15, 6 is 6/15 and so on. The question wants the difference between the biggest and smallest angles, which happens to be <(6-1)/15> *360 = 120.
No need to shoehorn the answer, check other explanations on this thread.

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The exterior angles of a quadrilateral are in the ratio 1:4:4:6. What   [#permalink] 26 Jul 2017, 20:53
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