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

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

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

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

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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).

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

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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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New post 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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New post 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).

Answer: A.



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. :roll:

Is there any other method to solve this?
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Re: How many integer points lie between points A and В on the line segment [#permalink]

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New post 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).

Answer: A.



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. :roll:

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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PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

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

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

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New post 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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Collection of Questions:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: How many integer points lie between points A and В on the line segment   [#permalink] 31 May 2017, 10:54
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