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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) \(yy_1=\frac{y_2y_1}{x_2x_1}(xx_1)\) \((x_1,y_1) = (5,7)\) \((x_2,y_2) = (10,3)\) \(y7=\frac{37}{105}(x5)\) \(y7=2(x5)\) \(y+2x=17\) 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=72=5; x=6+1=7 > y=52=3; x=7+1=8 > y=32=1; x=8+1=9 > y=12=1; x=9+1=10 > y=12=3. So there are 4 points with integer coordinates: (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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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=72=5; x=6+1=7 > y=52=3; x=7+1=8 > y=32=1; x=8+1=9 > y=12=1; x=9+1=10 > y=12=3. So there are 4 points with integer coordinates: (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. 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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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=72=5; x=6+1=7 > y=52=3; x=7+1=8 > y=32=1; x=8+1=9 > y=12=1; x=9+1=10 > y=12=3. So there are 4 points with integer coordinates: (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. 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.



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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, 10:54




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