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How many points (x, y) lie on the line segment between (22, 12 2/3) an

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How many points (x, y) lie on the line segment between (22, 12 2/3) an [#permalink]

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New post Updated on: 14 Nov 2014, 09:20
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How many points (x, y) lie on the line segment between (22, 12 2/3) and (7, 17 2/3) such that x and y are both integers?

A. 4
B. 5
C. 7
D. 8
E. 9

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Originally posted by shrive555 on 28 Oct 2010, 17:17.
Last edited by Bunuel on 14 Nov 2014, 09:20, edited 1 time in total.
Renamed the topic and edited the question.
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Re: How many points (x, y) lie on the line segment between (22, 12 2/3) an [#permalink]

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New post 28 Oct 2010, 19:03
3
2
The line joins the points (7, 17 2/3) and (22, 12 2/3).
The slope of the line will be (17 2/3 - 12 2/3)/(7 - 22) = -1/3

What does a slope of -1/3 imply? It implies that for every decrease of 1 unit in the value of y, x will increase by 3 units or in other words, for every decrease of 1/3 in value of y, x will increase by 1 unit.

Start from the left end point. When value of y becomes 17 i.e. a reduction of 2/3 , value of x will be increase by 2 and will become 9. When value of y reduces by another 1 unit, value of x will increase by 3 units and will become 12 and so on... You will have 5 integral values of y (17, 16, 15, 14 and 13). For each of these values, x will be integral.
Attachment:
Ques.jpg
Ques.jpg [ 8.16 KiB | Viewed 6262 times ]


The last point in the diagram above is 13, not 12.
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Re: How many points (x, y) lie on the line segment between (22, 12 2/3) an [#permalink]

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New post 28 Oct 2010, 18:48
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1
slope = (17 2/3 - 12 2/3) / (7 - 22) = -1/3
y = mx + b => 12 2/3 = -22/3 + b => b = 20
y = -x/3 + 20

Only integer values work, and the only multiples of 3 between 7 and 22 for x values are 9, 12, 15, 18 and 21, thus 5 points.
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Re: How many points (x, y) lie on the line segment between (22, 12 2/3) an [#permalink]

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New post 30 Oct 2010, 03:58
VeritasPrepKarishma wrote:
The line joins the points (7, 17 2/3) and (22, 12 2/3).
The slope of the line will be (17 2/3 - 12 2/3)/(7 - 22) = -1/3

What does a slope of -1/3 imply? It implies that for every decrease of 1 unit in the value of y, x will increase by 3 units or in other words, for every decrease of 1/3 in value of y, x will increase by 1 unit.

Start from the left end point. When value of y becomes 17 i.e. a reduction of 2/3 , value of x will be increase by 2 and will become 9. When value of y reduces by another 1 unit, value of x will increase by 3 units and will become 12 and so on... You will have 5 integral values of y (17, 16, 15, 14 and 13). For each of these values, x will be integral.
Attachment:
Ques.jpg


The last point in the diagram above is 13, not 12.


I felt this as very time consuming. I got the answer the same way you have suggested. Is there any other way to solve this quickly?
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Re: How many points (x, y) lie on the line segment between (22, 12 2/3) an [#permalink]

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New post 30 Oct 2010, 08:36
It takes under a minute to get to the solution since you know that x will be integral and you only have to count integral values from 12 2/3 to 17 2/3.

Nevertheless, let me think. If I come up with a solution that takes even less time, I will let you know...
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Re: How many points (x, y) lie on the line segment between (22, 12 2/3) an [#permalink]

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New post 14 Nov 2014, 09:21
shrive555 wrote:
How many points (x, y) lie on the line segment between (22, 12 2/3) and (7, 17 2/3) such that x and y are both integers?

A. 4
B. 5
C. 7
D. 8
E. 9


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How many points (x, y) lie on the line segment between (22, 12 2/3) an [#permalink]

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New post 10 Sep 2017, 22:38
The fastest way I know:
you find the slope (17 2/3 - 12 2/3) / (7 - 22) = -1/3
you find the y intercept by taking x y for any point y = mx + b = 12 2/3 = -22/3 + b b = 20
then create a function for the line y = -1/3*x + 20

domain for the x (7 to 22)
Between 17 2/3 and 12 2/3 there are 5 integer values of y, 13, 14, 15, 16, 17
so y = -1/3*x + 20
13 = -1/3*x + 20 (-1/3*x = 7)
14 = -1/3*x + 20 (-1/3*x = 6)
when you resolve first 2 the rest is going much faster

however takes some time a bit more than >2mins
for such kind of questions main to pay attention of the points are on the line or between 2 suggested points
How many points (x, y) lie on the line segment between (22, 12 2/3) an   [#permalink] 10 Sep 2017, 22:38
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