A student draws 35 lines in the XY plane, none of which are vertical

Question

A student draws 35 lines in the XY plane, none of which are vertical and 15 of which have a positive slope. Of those with positive slope, 1/3 also have a positive Y-intercept. If 23 lines have a Y-intercept less than or equal to zero, how many lines contain no point in Quadrant I?
A: 0
B: 7
C: 13
D: 20
E: 35

chetan2u · Accepted Answer

Hi...
35 lines.. 
15 positive slope that if x increases y increases and vice versa..
So 1/3 have a positive y intercept, so remaining 2/3 or 10 have negative y intercept or y is 0
But total 23 have negative or zero y intercept, so 23-10=13 of these have negative slope..
So these 13 will never cross the quad I

Ans 13

Izzyjolly · Answer

Here is my approach.
Line contains no point in Quadrant I means Line has Y-intercept ≤ 0 AND negative slope 
Of the 15 lines with  positive  slope, 1/3 also have a positive Y-intercept --> remaining 2/3 * 15 = 10 lines have Y-intercept ≤ 0
Of the 23 lines have Y-intercept ≤ 0, 10 lines have Y-intercept ≤ 0 AND positive slope --> remaining 23-10 = 13 lines Y-intercept ≤ 0 and negative slope
Answer C.

Nikhil · Answer

chetan2u

Shouldn't the remaining 10 lines have zero or negative y-intercept ? The final answer doesn't change anyway.

chetan2u · Answer

Yes , it will be 0 or negative...
You are correct. Doesn't effect the answer so didn't mention it.

GyMrAT · Answer

There are 8 categories of lines

(i) positive slope, positive y-intercept - x

(ii) positive slope, passing through origin - y

(iii) positive slope, negative y-intercept - z

(iv) negative slope & positive y-intercept - a

(v) negative slope & passing through origin - b

(vi) negative slope & negative y-intercept - c

(vii) parallel to x - axis & with positive y-intercept - p

(viii) parallel to x - axis & with negative y-intercept - q

Lines x, y, z, a, p pass through Quadrant I, we are asked to find (b + c + q)

Given x + y + z + a + b + c + p + q = 35

x + y + z = 15

x = 15/3 = 5, hence y + z = 10

y + z + b + c + q = 23

hence b + c + q = 23 - 10 = 13

Answer C.

Thanks,
GyM

stne · Answer

Total 35 lines 
23 of those lines have y intercept  \leq  0
5 of those lines have y intercept  >  0 
what about the remaining seven lines ? 
if they don't have a zero y intercept, or negative y intercept or positive y intercept , what kind of intercept do they have ?

Is it possible to have a line with NO y intercept? Isn't it the same as having a zero Y intercept? 
if those 7 lines pass through the origin , then y intercept should be 0 and we should have 23 +7 lines having y intercept  \leq 0

Please account for those 7 lines .
Thanks.

stne · Answer

That can be easily understood , can you help me understand , about the y intercept of the remaining 7 lines , positive 5 y intercept ,zero and less = 23 Y intercept , that makes 28, what about the y intercept of the remaining 7 lines ?

Izzyjolly · Answer

5 of the  15 lines which have a positive slope  have y intercept  >  0, not 5 of the  total 35 lines  have y intercept  >  0

In general, it is possible to have lines with NO y-intercept. Those lines never intersect the y-axis, or in other words, those lines are vertical (parallel to the y-axis). It is not the same as having a zero y-intercept. If a line is vertical and has a zero y-intercept, it  is  y-axis.
In this question, the stem clearly states that "A student draws 35 lines in the XY plane,  none of which are vertical. .." so it's not possible to have lines with NO y-intercept in this case.

Hope it's clear.

stne · Answer

I think you did not understand my question correctly.

May be I am missing something 
total 35 lines , 15 have positive slope and 20 have 0 or negative slope ( 20 + 15 = 35)
out of 15 having positive slope, 5 have positive Y intercept 
So 10 having positive slope have 0 or negative Y intercept 
So total 15 lines have positive slope = 5 lines have positive y intercept and 10 lines have , zero or negative y intercept.
also we know total 23 lines have y intercept  \leq  0

so 23 + 5 = 28 , what kind of y intercept do the remaining 7 lines have ?
hope now my question is clear.

Edit: 
Thank you , understood it , the answer I was looking for was this :

Those 7 lines have positive y intercept but negative slope .

So here is the break up

Total 12 have positive y intercept out of which 5 have positive slope and 7 have zero or negative slope 
total 23 have zero or negative Y intercept out of which 10 have positive slope and 13 have 0 or negative slope .

total 35 lines are now accounted for .

Izzyjolly · Answer

Drawing a 2x2 table may help you see it easier

KanishkM · Answer

Matrix method was easier here

---------------------(+ive Slope)---------(-ive slope)-----Total
+ive Y Intercept----------5----------------------7------------12
-ive Y Intercept----------10--------------------13-----------23 
----------------------------15--------------------20-----------35

-ive slope and -ive Y Intercept will be responsible for answering how many lines contain no point in Quadrant I
13

C

martinfzb · Answer

out of 15 lines with posivitive slop, 5 lines have positive intercept on Y, 10 lines has negative or 0.

so out of the 23 lines with a Y-intercept less than or equal to zero, 
there are must contains 10 positive slop which mentioned in positive slop. 
and then 23-10=13 left with negative slop. 
those 13 line with negative slop and Y intercept <=0 must not cross the quadrant I.

bumpbot · Answer

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A student draws 35 lines in the XY plane, none of which are vertical

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A student draws 35 lines in the XY plane, none of which are vertical

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