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# How many integer points lie between points A and В on the line segment

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Manager
Joined: 10 Feb 2011
Posts: 110
How many integer points lie between points A and В on the line segment  [#permalink]

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18 Feb 2011, 10:25
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Difficulty:

95% (hard)

Question Stats:

53% (01:29) correct 47% (01:39) wrong based on 362 sessions

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How many integer points lie between points A and В on the line segment AB, if A is (5, 7) and В is (10, –3)?

(A) 4
(B) 5
(C) 6
(D) 10
(E) 15
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Joined: 02 Sep 2009
Posts: 47161
Re: How many integer points lie between points A and В on the line segment  [#permalink]

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18 Feb 2011, 11:07
4
7
banksy wrote:
204. How many integer points lie between points A and В on the line segment AB, if A is (5, 7) and В is (10, –3)?
(A) 4
(B) 5
(C) 6
(D) 10
(E) 15

You can notice that while we moved 5 units to the right (from x=5 to x=10) we moved 10 units down (from y=7 to y=-3) thus every unit to the right = exactly 2 units down. So each of 4 integer coordinates of x between 5 and 10 (6, 7, 8 and 9) will have an integer pair of y (5, 3, 1, and -1).

To demonstrate:
x=5, y=7;
x=5+1=6 --> y=7-2=5;
x=6+1=7 --> y=5-2=3;
x=7+1=8 --> y=3-2=1;
x=8+1=9 --> y=1-2=-1;
x=9+1=10 --> y=-1-2=-3.

So there are 4 points with integer co-ordinates: (6, 5), (7, 3), (8, 1), and (9, -1).

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Re: How many integer points lie between points A and В on the line segment  [#permalink]

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18 Feb 2011, 10:55
4
2
equation of line passing through two points (x1,y1) and (x2,y2)

$$y-y_1=\frac{y_2-y_1}{x_2-x_1}(x-x_1)$$

$$(x_1,y_1) = (5,7)$$
$$(x_2,y_2) = (10,-3)$$

$$y-7=\frac{-3-7}{10-5}(x-5)$$

$$y-7=-2(x-5)$$

$$y+2x=17$$

Plug in values for x from 6 to 9 and see whether y will be an integer.

x=6; y=5
x=7; y=3
x=8; y=1
x=9; y=-1

Total 4 points that are integers.

Ans: "A"
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##### General Discussion
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Posts: 47161
Re: How many integer points lie between points A and В on the line segment  [#permalink]

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21 Oct 2016, 02:47
1
Bunuel wrote:
duahsolo wrote:
How many integer points lie between points A and B on the line segment AB, if A is (5, 7) and B is (10, -3)?

A) 4
B) 5
C) 6
D) 10
E) 15

Merging topics. Please refer to the discussion above.

Similar questions to practice:
in-the-figure-above-how-many-of-the-points-on-line-segment-108673.html
how-many-points-x-y-lie-on-the-line-segment-between-22-12-2-3-an-103881.html

Hope it helps.
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Re: How many integer points lie between points A and В on the line segment  [#permalink]

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10 Jan 2017, 20:20
Between the points (5,7) and (10,-3) I got the slope as -2. I then kept increasing x by 1 and accordingly reduced y by -2. Got 4 integer points until (10,-3)
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Re: How many integer points lie between points A and В on the line segment  [#permalink]

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22 May 2017, 13:29
Bunuel wrote:
banksy wrote:
204. How many integer points lie between points A and В on the line segment AB, if A is (5, 7) and В is (10, –3)?
(A) 4
(B) 5
(C) 6
(D) 10
(E) 15

You can notice that while we moved 5 units to the right (from x=5 to x=10) we moved 10 units down (from y=7 to y=-3) thus every unit to the right = exactly 2 units down. So each of 4 integer coordinates of x between 5 and 10 (6, 7, 8 and 9) will have an integer pair of y (5, 3, 1, and -1).

To demonstrate:
x=5, y=7;
x=5+1=6 --> y=7-2=5;
x=6+1=7 --> y=5-2=3;
x=7+1=8 --> y=3-2=1;
x=8+1=9 --> y=1-2=-1;
x=9+1=10 --> y=-1-2=-3.

So there are 4 points with integer co-ordinates: (6, 5), (7, 3), (8, 1), and (9, -1).

Hi

Can someone please correct my thought process here. What i thought was to find the distance between the two and see the integers points on that distance.
Distance came out to be 11.18.Considering integer coordinates we should have 11 integer points?
I know i am wrong somewhere, but need help to mend my thought process.

Is there any other method to solve this?
Math Expert
Joined: 02 Sep 2009
Posts: 47161
Re: How many integer points lie between points A and В on the line segment  [#permalink]

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23 May 2017, 06:16
nishantt7 wrote:
Bunuel wrote:
banksy wrote:
204. How many integer points lie between points A and В on the line segment AB, if A is (5, 7) and В is (10, –3)?
(A) 4
(B) 5
(C) 6
(D) 10
(E) 15

You can notice that while we moved 5 units to the right (from x=5 to x=10) we moved 10 units down (from y=7 to y=-3) thus every unit to the right = exactly 2 units down. So each of 4 integer coordinates of x between 5 and 10 (6, 7, 8 and 9) will have an integer pair of y (5, 3, 1, and -1).

To demonstrate:
x=5, y=7;
x=5+1=6 --> y=7-2=5;
x=6+1=7 --> y=5-2=3;
x=7+1=8 --> y=3-2=1;
x=8+1=9 --> y=1-2=-1;
x=9+1=10 --> y=-1-2=-3.

So there are 4 points with integer co-ordinates: (6, 5), (7, 3), (8, 1), and (9, -1).

Hi

Can someone please correct my thought process here. What i thought was to find the distance between the two and see the integers points on that distance.
Distance came out to be 11.18.Considering integer coordinates we should have 11 integer points?
I know i am wrong somewhere, but need help to mend my thought process.

Is there any other method to solve this?

Slope matters. For example, the distance between (0, 0) and (1, 10) is $$\sqrt{101} \approx 10$$ but 0 integer points lie between them. On the other hand the distance between (0, 0) and (0, $$\sqrt{101}$$) is also $$\sqrt{101} \approx 10$$ and 10 integer points lie between them.

Hope it's clear.
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Re: How many integer points lie between points A and В on the line segment  [#permalink]

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31 May 2017, 10:31
Hi Bunuel,
If the question states that "integer points lie between points A and В", it means that A and B are excluded?
I mistakenly chose 6 points given the context did not state clearly inclusive or exclusive.
Math Expert
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Posts: 47161
Re: How many integer points lie between points A and В on the line segment  [#permalink]

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31 May 2017, 10:54
sayiurway wrote:
Hi Bunuel,
If the question states that "integer points lie between points A and В", it means that A and B are excluded?
I mistakenly chose 6 points given the context did not state clearly inclusive or exclusive.

Better question would make it clearer but if nothing is mentioned, between usually means not inclusive.
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Concentration: Operations, Social Entrepreneurship
How many integer points lie between points A and В on the line segment  [#permalink]

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11 Oct 2017, 10:06
I have calculated the slope which comes out to be -2 and post this i am not able to reach or figure out the next step?

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Posts: 236
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GPA: 3.4
Re: How many integer points lie between points A and В on the line segment  [#permalink]

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05 Nov 2017, 12:03
Take the general equation of a line; $$y=mx+c$$

Now substitute the two points A and B,

A(5,7) :$$7=5m+c$$...................(1)
B(10,-3) :$$-3=10m+c$$...................(2)

Subtracting (1) and (2)

$$10=-5m$$
$$m=-2$$..................(3)

Substituting (3) in (2), we get c=17

Now the equation of the line is; $$y=-2x+17$$
Notice that the boundary of x is given by the x co-ordinates of points A & B. Hence 5<=x<=10

Since the question asks for the number of points between A & B. (keyword = between), there are 4 integer values of x.
x=6,7,8 and 9. We can safely say that for all these values of x, y will be an integer and hence the answer is 4 points or A
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Re: How many integer points lie between points A and В on the line segment  [#permalink]

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08 Nov 2017, 17:37
banksy wrote:
How many integer points lie between points A and В on the line segment AB, if A is (5, 7) and В is (10, –3)?

(A) 4
(B) 5
(C) 6
(D) 10
(E) 15

Let’s determine the equation of the line that connects the two given points. We can start with the slope.

slope = change in y/change in x

slope = (-3 - 7)/(10 - 5) = -10/5 = -2

So, we have:

y = -2x + b

Let’s substitute 5 and 7 for x and y, respectively, to calculate b.

7 = -2(5) + b

7 = -10 + b

17 = b

So, we have:

y = -2x + 17

We see that whenever x is an integer in this equation, y will also be an integer, and therefore (x, y) will be an integer point. Furthermore, when x is between 5 and 10, the corresponding points on the line will be between A and B. Thus, the points on the line whose x coordinates are 6, 7, 8, and 9 are the only integer points between A and B. There are four of them.

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Re: How many integer points lie between points A and В on the line segment &nbs [#permalink] 08 Nov 2017, 17:37
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