In the figure above, how many of the points on line segment PQ have

Question

https://gmatclub.com/forum/download/file.php?id=21007 In the figure above, how many of the points on line segment PQ have coordinates that are both integers? (A) 5 (B) 8 (C) 10 (D) 11 (E) 20 Untitled.png

Bunuel · Accepted Answer

https://gmatclub.com/forum/download/file.php?id=21007 In the figure above, how many of the points on line segment PQ have coordinates that are both integers? (A) 5 (B) 8 (C) 10 (D) 11 (E) 20 The equation of a straight line passing through points P(x_1, y_1) and Q(x_2, y_2) is: \frac{y-y_1}{x-x_1}=\frac{y_1-y_2}{x_1-x_2} (check here: math-coordinate-geometry-87652.html). For P(0, 30) and Q(50, 0): \frac{y-30}{x-0}=\frac{30-0}{0-50} --> 3x + 5y = 150 . If x is a multiple of 5, then y will be an integer. x ranges from 0 to 50, inclusive. There are total of 11 multiples of 5 in this range: 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, and 50. Answer: D. Hope this helps.

AmrithS · Answer

m = (30 - 0)/(0-50)
m = -30/50 = -3/5

y = mx + c

c = 30

y = (-3/5)x + 30

For y to be an integer, the term (-3/5)x must be an integer. Thus, x must be divisible by 5.

The value of x ranges from 0 to 50

So we need to find the number of multiples of 5 between 0 and 50 (Exclusive)

5, 10, 15, 20, 25, 30, 35, 40 & 45

Ans: 9 points between A and B will have integer values for both the co-ordinates!

maheshsrini · Answer

I think the answer should be 11. Why are 0 & 11 excluded?. Please explain.

fluke · Answer

You are right. thanks.

Phoenix72 · Answer

Ok Including 0 we will get an Int but how will 11 help?

amitbharadwaj7 · Answer

First develope the equation of line with coordinates given:-
Y = MX + C

When X =0, Y=30, putting the values in above equation of line gives the value of C
30 = C

When Y=0, X=30, putting the values in above equation of line gives the value of M
M= - 3/5

The equation of Line is Y= (-3/5)X +30, Now from this equation it is clear that if we want to have both X and Y to be integers, then all values of X has to be multiple of 5, so starting from X=0 to X=50 (Coordinate limits of line PQ), we note that there are 11 integer values of X for which 11 integer values of Y exists in line PQ

rahul16singh28 · Answer

We have -

3x + 5y = 150 
 5y = 150 - 3x ---> y = 30 - (3x/5) 
As,  x & y  has to be integers x has to be multiple of y. Also,  x & y  has to be postie as it lies in the first quadrant. 
 x = {0,5,10,15,20,25,30,35,40,45,50}

Probus · Answer

The alternate method would be usiing the slope to find the points on the line

Since we know that slope is  \frac{-3}{5}  we know that y intercept is 30 and x intercept is 50.

We have negative slope 
 If we increase x then x decreases. So if we start at point (0,30) then we can say that next point on this line would be 3 units down from 30 and 5 units right of 0 we will get our point at (5,27). Now other point on this line would be 3 units down from 27 and 5 units right to 5 which would be (10,24) Similarly we see that we get

(15, 21) , (20, 18). ( 25, 15), (30,12) (35,9) (40 ,6) (45,3) (50,0)

So in all we have 11 Points

aru010 · Answer

Using INTERCEPT Form of straight line : x/xo + y/yo = 1 
(xo = x intercept & y0 = y intercept)

So here eq is : x/50 + y/30 = 1 
=> x = (1-y/30) 50 
=> x = 50-5y/3
Hence all values of y divisble by 3 will give an integer value 
now y range is  , so 11 values

kshv · Answer

Shouldn’t our approach be to also find the range of Y and then select the answer which would be the lesser of the two? ( the question over here has range 11 for both X and Y but had the ranges been different we would have to calculate ranges for both X and Y - hence a little more time had to be invested in this quest.)

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Kinshook · Answer

https://gmatclub.com/forum/download/file.php?id=21007 
Asked: In the figure above, how many of the points on line segment PQ have coordinates that are both integers?

Equation of the line PQ: -

(y - 0) = (30-0)/(0-50) (x-50)
y = -3/5 (x-50)
5y = - 3x + 150
3x + 5y = 150

(x,y) = {(0,30),(5,27),(10,24),(15,21),(20,18),(25,15),(30,12),(35,9),(40,6),(45,3),(50,0)}: 11 points

IMO D

adewale223 · Answer

When you get the equation of the line and you have determined that for y to be an integer x must be a multiple of 5, instead of listing which may cost you some seconds just divide 50 by 5. That gives 10, then add x is 0 to give you 11 as your answer.

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tryorcry · Answer

Lesson learned : how to create line equation from two coordinates , 5y + 3x = 150 
then, x = (150-5y)/3
in order to make x an integer , y must be a multiple of 3 and since y can be from 0 to 30 inclusive, so y can be 0,3,6,9,12,15,18,21,24,27,30 i.e 11 values

In the figure above, how many of the points on line segment PQ have

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In the figure above, how many of the points on line segment PQ have

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