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A line passes through point p(1,1) and q(100,1000). How many other poi

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A line passes through point p(1,1) and q(100,1000). How many other points with integer co-ordinates are on the line and strictly between points p and q ?

a) 0
b) 2
c) 3
d) 8
e) 9
[Reveal] Spoiler: OA
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Re: A line passes through point p(1,1) and q(100,1000). How many other poi [#permalink]

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New post 28 Jan 2015, 07:23
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ans D.....
p(1,1) and q(100,1000)... so the ratio in which the increase is ,increase of x by 10 results in increase of 100 in y coord..
so for x, there are 8 such points for which y is also int...
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Re: A line passes through point p(1,1) and q(100,1000). How many other poi [#permalink]

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New post 28 Jan 2015, 08:09
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chetan2u wrote:
ans D.....
p(1,1) and q(100,1000)... so the ratio in which the increase is ,increase of x by 10 results in increase of 100 in y coord..
so for x, there are 8 such points for which y is also int...



Can you please elaborate?
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Re: A line passes through point p(1,1) and q(100,1000). How many other poi [#permalink]

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DesiGmat wrote:
chetan2u wrote:
ans D.....
p(1,1) and q(100,1000)... so the ratio in which the increase is ,increase of x by 10 results in increase of 100 in y coord..
so for x, there are 8 such points for which y is also int...



Can you please elaborate?


hi...
the Y coord is increasing by(1000-1) or999 as x increases by (100-1) or 99..
so the ratio of increase is 99:999 or 11:111...
it means with every increase of x by 11 , y increases by 111..
so the value of 12(1+11) will have y as 112(1+111).. 23(12+11) will have 223(112+111) and so on till 89 will have 889..
value of x as 1 and 100 is not taken so 8 values 12,23,34,45,56,67,78,89 with corresponding int values for Y
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Re: A line passes through point p(1,1) and q(100,1000). How many other poi [#permalink]

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Let's say point p is at the origin and q is at the (99,999) (this does not change the problem).
The line pq has the equation y =\(\frac{999}{99}x\)= \(\frac{111}{11}x\) .

Now the co-ordinates of y will be integer, when x takes the value of 11,22,33,44,55,66,77,88.

hence answer is D.
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Re: A line passes through point p(1,1) and q(100,1000). How many other poi [#permalink]

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New post 29 Jan 2015, 06:11
Line equation y = mx + c....Using the inform...m = 111/11....so y = 111/11x + c......put y=x= 1....c = -100/11
Line eqn :-11y = 111x - 100......x = (11y+100)/111...limit x till 100....only 8 values permissible
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Re: A line passes through point p(1,1) and q(100,1000). How many other poi [#permalink]

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New post 07 Mar 2015, 23:50
Hi,

I didnt get the solution.Why can't 99 also be the value ??
When x is 99 ,y can still be an integer..??
Can some1 please explain the problem
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Re: A line passes through point p(1,1) and q(100,1000). How many other poi [#permalink]

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New post 30 Jun 2015, 06:03
the slope we will get as 11/111.
Then equation of the line y=11/111x+c
if we take for every mulitple of 111, the value of y is integer, then we will have 9 values.

Can someone pls tell me where i am going wrong?
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Re: A line passes through point p(1,1) and q(100,1000). How many other poi [#permalink]

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New post 10 Aug 2015, 20:54
jayanthjanardhan wrote:
the slope we will get as 11/111.
Then equation of the line y=11/111x+c
if we take for every mulitple of 111, the value of y is integer, then we will have 9 values.

Can someone pls tell me where i am going wrong?


Hi,

The error is that c is not an integer. Am I right in suggesting that you have assumed c to be an integer?
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Re: A line passes through point p(1,1) and q(100,1000). How many other poi [#permalink]

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New post 17 Aug 2015, 11:14
The slope is y2-y1/x2-x1. So, it is 1000-1/100-1 = 999/99 = 111/11

Also, the value of C needs to be calculated (to check whether it is integer). In this case, the value of C is not an integer = -100/11.

Hope this helps.
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Re: A line passes through point p(1,1) and q(100,1000). How many other poi [#permalink]

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New post 18 Sep 2015, 03:58
AryamaDuttaSaikia wrote:
The slope is y2-y1/x2-x1. So, it is 1000-1/100-1 = 999/99 = 111/11

Also, the value of C needs to be calculated (to check whether it is integer). In this case, the value of C is not an integer = -100/11.

Hope this helps.

Hi Aryama,

Can you extend your explanation further
I mean now we have got the equation of the line as

y= (11/111)x -100/11

Now how to go ahead with this from here
What values one should place here to get integer value of y

Thanks
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A line passes through point p(1,1) and q(100,1000). How many other poi [#permalink]

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Hi folks, here is how i solved the problem :

From the two point coordinates, we obtain the following equation Y= (111/11)*X - 100/11; This equation can be written as follow : (111*X - 100)/11 or More better (100(X-1) + 11 X)/11 .

Now the problem here is that we should obtain an INTEGER Y, in other words (100(X-1) + 11 X) should be divisible by 11.

11X is obviously divisible by 11 so we can neglect it, now our unique friend is 100(X-1) which should be divisible by 11.

X ranges from 1 to 100 , thus the possible values for X to be divisible by 11 are : 12, 23,34,45,56,67,78,89 . So we have 8 values, thus Answer (D) is correct.

Hope it helps :)
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Re: A line passes through point p(1,1) and q(100,1000). How many other poi [#permalink]

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Slope is calculated by using the formula :

m = y2-y1/x2-x1

Slope intercept format (Equation of the line)

Y = mx+c

In this equation substitute the values of " m "and one of the given coordinates (x,y) to get the value of C.

c= -100/11

So y becomes y=(111/11)x-100/11

= (111x-100) /11

X =1 So we get Y = 1 (given co-ordinates). We want values between (1,1) and (100, 1000)

The next values of X that make Y as integer is 11 (divisor is 11) more than the first value.

12,23 (12+11),34,45,56,67,78,89 = Total 8 values.
HOPE IT HELPS!!
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