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# In the figure above, how many of the points on line segment PQ have

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In the figure above, how many of the points on line segment PQ have  [#permalink]

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Updated on: 10 Feb 2018, 16:09
3
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Difficulty:

75% (hard)

Question Stats:

58% (01:24) correct 42% (01:38) wrong based on 331 sessions

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In the figure above, how many of the points on line segment PQ have coordinates that are both integers?

(A) 5
(B) 8
(C) 10
(D) 11
(E) 20

Attachment:

Untitled.png [ 1.28 KiB | Viewed 4414 times ]

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Originally posted by GMATD11 on 03 Feb 2011, 12:18.
Last edited by Bunuel on 10 Feb 2018, 16:09, edited 2 times in total.
Renamed the topic, edited the question and added the OA.
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Re: In the figure above, how many of the points on line segment PQ have  [#permalink]

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09 Sep 2013, 02:42
7
6

In the figure above, how many of the points on line segment PQ have coordinates that are both integers?

(A) 5
(B) 8
(C) 10
(D) 11
(E) 20

The equation of a straight line passing through points $$P(x_1, y_1)$$ and $$Q(x_2, y_2)$$ is: $$\frac{y-y_1}{x-x_1}=\frac{y_1-y_2}{x_1-x_2}$$ (check here: math-coordinate-geometry-87652.html).

For P(0, 30) and Q(50, 0): $$\frac{y-30}{x-0}=\frac{30-0}{0-50}$$ --> $$3x + 5y = 150$$.

If x is a multiple of 5, then y will be an integer. x ranges from 0 to 50, inclusive. There are total of 11 multiples of 5 in this range: 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, and 50.

Hope this helps.
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Re: In the figure above, how many of the points on line segment PQ have  [#permalink]

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03 Feb 2011, 20:54
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1
m = (30 - 0)/(0-50)
m = -30/50 = -3/5

y = mx + c

c = 30

y = (-3/5)x + 30

For y to be an integer, the term (-3/5)x must be an integer. Thus, x must be divisible by 5.

The value of x ranges from 0 to 50

So we need to find the number of multiples of 5 between 0 and 50 (Exclusive)

5, 10, 15, 20, 25, 30, 35, 40 & 45

Ans: 9 points between A and B will have integer values for both the co-ordinates!
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Re: In the figure above, how many of the points on line segment PQ have  [#permalink]

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26 Jun 2011, 16:05
1
I think the answer should be 11. Why are 0 & 11 excluded?. Please explain.
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Re: In the figure above, how many of the points on line segment PQ have  [#permalink]

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27 Jun 2011, 04:22
maheshsrini wrote:
I think the answer should be 11. Why are 0 & 11 excluded?. Please explain.

You are right. thanks.
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Re: In the figure above, how many of the points on line segment PQ have  [#permalink]

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09 Sep 2013, 00:43
Ok Including 0 we will get an Int but how will 11 help?
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Re: In the figure above, how many of the points on line segment PQ have  [#permalink]

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14 Oct 2013, 10:17
1
First develope the equation of line with coordinates given:-
Y = MX + C

When X =0, Y=30, putting the values in above equation of line gives the value of C
30 = C

When Y=0, X=30, putting the values in above equation of line gives the value of M
M= - 3/5

The equation of Line is Y= (-3/5)X +30, Now from this equation it is clear that if we want to have both X and Y to be integers, then all values of X has to be multiple of 5, so starting from X=0 to X=50 (Coordinate limits of line PQ), we note that there are 11 integer values of X for which 11 integer values of Y exists in line PQ
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Re: In the figure above, how many of the points on line segment PQ have  [#permalink]

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10 Feb 2018, 17:31
GMATD11 wrote:

In the figure above, how many of the points on line segment PQ have coordinates that are both integers?

(A) 5
(B) 8
(C) 10
(D) 11
(E) 20

Attachment:
Untitled.png

We have -

$$3x + 5y = 150$$
$$5y = 150 - 3x ---> y = 30 - (3x/5)$$
As, $$x & y$$ has to be integers x has to be multiple of y. Also, $$x & y$$ has to be postie as it lies in the first quadrant.
$$x = {0,5,10,15,20,25,30,35,40,45,50}$$
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Re: In the figure above, how many of the points on line segment PQ have &nbs [#permalink] 10 Feb 2018, 17:31
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