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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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Updated on: 14 Nov 2014, 08:20
2
9
00:00

Difficulty:

65% (hard)

Question Stats:

64% (02:36) correct 36% (02:56) wrong based on 227 sessions

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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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I'm the Dumbest of All !!

Originally posted by shrive555 on 28 Oct 2010, 16:17.
Last edited by Bunuel on 14 Nov 2014, 08:20, 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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28 Oct 2010, 18:03
3
3
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 [ 8.16 KiB | Viewed 8665 times ]

The last point in the diagram above is 13, not 12.
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Karishma
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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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28 Oct 2010, 17:48
2
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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30 Oct 2010, 02:58
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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30 Oct 2010, 07:36
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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14 Nov 2014, 08:21
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

Similar questions to practice:
how-many-integer-points-lie-between-points-a-and-on-the-line-segment-109623.html
in-the-figure-above-how-many-of-the-points-on-line-segment-108673.html
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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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10 Sep 2017, 21:38
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
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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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02 Feb 2019, 06:32
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Re: How many points (x, y) lie on the line segment between (22, 12 2/3) an   [#permalink] 02 Feb 2019, 06:32
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