If a line which has the equation 7x + 17y = 1000 is plotted on X-Y pla

Question

If a line which has the equation 7x + 17y = 1000 is plotted on X-Y plane then how many points will fall on the line in 1st quadrant which will have both x and y co-ordinates Integers?

A) 6
B) 7
C) 8
D) 9
E) 10

chetan2u · Accepted Answer

Hi,

First quadrant is the quadrant where both x and y are POSITIVE...

Let's see the equation..
7x+17y=1000
You have to find the first set of x and y that satisfies the equation and thereafter the remaining can be found..
Here subtract multiples of 7 from 1000 and check if remaining amount is div by 17..
When x is 2, y is 58... This is the first set..
Remaining values of y will be 58-7t, where t is integer..
When t is 9, 7t becomes 7*9=63 and 58-7t will become negative..

Hence t<9..
So t will take all values from 0 to 8 so 9 values..
D

GMATinsight  pl relook at the OA provided by you. Ans will be 9 and not 10..
So D is the answer and Not E

sananoor · Answer

I guess it must be D
the sets will have values as
58, 2
51,19 
44, 36
37, 53
30, 70
23, 87
16, 104
9, 121
2, 138

Chetan how can u include 0 in your answer as x and y both cant be equal to zero, or even if one is zero then other wont be an integer...if u even try drawing the line it wont pass from origin.

chetan2u · Answer

Hi,
OA is correct now because  Bunuel  has edited it from E to D.

sananoor  I have mentioned exactly that in the post that there will be 9 and not 10.
The OA originally was E which now has been changed to D by Bunuel

And I have taken t as 0 and not x and y..

sananoor · Answer

Chetan last question...can u explain he above highlighted and underlined part? how come 0?

chetan2u · Answer

Hi,

I found the max value of y as 58 and the next values will be 58- multiples of 7...
Now instead of finding values one by one, I have taken y=58-7t...
Here y would become<0, when 7t>=58..
So 7t<58... Max value of t is 8 as 7*8=56...
t can take value of 0 because y=58-7t=58-7*0=58... So y is not 0 but t is..

mdacosta · Answer

I did the problem slightly differently, can someone help where i went wrong? I got the answer as 8:

7x + 17y = 1000
rearrange to get: y = (-7/17)x + (1000/17)
solve for x intercept: (7/17)x = 1000/17
x intercept = (1000/7) or 142.XXX

then 17*(number of points on line) must be <142
17*8 = 136 so that's the number of points, 8

GMATinsight · Answer

x min = 2
x max = 138

You are missing counting the smallest solution...

Word of caution : Find the first solution for real before you take a leap to the last solution like you did.

chesstitans · Answer

this question is not so hard, but it does take time, and it may confuse test takers.
If ones look at the options, you can easily get a wrong choice by accident. Stay awake!

hellosanthosh2k2 · Answer

I solved the problem like this:

Like chetan mentioned, we have to find first multiple of 7, for which 1000 - first multiple of 7 is divisible by 17, which is 14 and remaining is 986

Now we can find the number of next multiples of 7 for which remainining is divisible by 17, by taking quotient of 1000 / LCM(7,17)
1000/119 = 8, so there are 8 multiples of 7 after 14 for which remaining is divisible by 17.
so multiple of 7 after 14 will be  14+119, 14 + 2*119, 14 + 3 * 119, ..... 14 + 8*119

So total is 9, Answer (C)

Madhavi1990 · Answer

I solved it the hard way

. Took like 3 minutes, blame the lack of coordinate grey cells. if we simplify the 17y = 1000 - 7x --> y = 1000/17 + 7/17x so if y = 0, x= 142 so if x = 0, y is 142 out of these, we will see there is a downward sloping line. So, between the two coordinates we need to find integer coordinates on the hypotenuse. So for 2 points we get 1 coordinate --> 142*2 = 284 I kept dividing by 2 till I got a remainder too small to be divided by 2. 284/2, 142/2, 71/2, 35/2, 17/2, 8/2,4/2, 2/2; and one more as it is inclusive of the points (it anyway cannot exceed beyond 10 points) --> around 9 times. Please press kudos, if this solution made sense for you

foryearss · Answer

sir , can you please explain why " Remaining values of y will be 58-7t"
I'm really confused , it took so long for me to answer this question , here's my way : 
7x + 17y = 1000 
y = (-7x + 1000 )/17
 now , + 1000-7x must be divisable by 17 , I'm stuck here , we know that x=<142 but how we can know how many valuse of (-7x + 1000 ) that are divisable by 17 where x=<142 ?
i tried back and forth ,(-7x + 1000 ) can be written as ( 1000+10x -17x ) 
now , -17x is no problem , let's take care only of 1000+10x which can be written as 10(100+x) 
now , since 0=< x =< 142 
 100=< 100+x =< 242 
let's find how many numbers divisable by 17 between 100, 242 in the classic way , the answer is 9

I appreciate any detailed explanation of other methods 
Thanks

GMATinsight · Answer

For any linear equation such as 7x+17y = 1000 The solutions always differ by co-efficient of other variables
i.e. values of x will differ by co-efficient of y i.e 17 in this case and 
i.e. values of y will differ by co-efficient of x i.e 7 in this case

Likewise if the equation is 3x+4y = 120
then values of x will differ by 4 and values of y will differ by 3

foryearss

foryearss · Answer

Thanks a lot , that really makes sense

GMATinsight · Answer

The detailed video solution of the problem is here.

Answer: Option D

https://www.youtube.com/watch?v=fYm_3OnxIoI

Kinshook · Answer

Asked: If a line which has the equation 7x + 17y = 1000 is plotted on X-Y plane then how many points will fall on the line in 1st quadrant which will have both x and y co-ordinates Integers?

x = (1000 - 17y)/7
(x,y) = {(138,2),(121,9),(104,16),(87,22),(70,29),(53,36),(36,43),(19,50),(2,57)} : 9 solutions

IMO D

bethebest96 · Answer

chetan2u  i didnt understand 58-7t .. plz explain the logic

bumpbot · Answer

Hello from the GMAT Club BumpBot!

Thanks to another GMAT Club member, I have just discovered this valuable topic, yet it had no discussion for over a year. I am now bumping it up - doing my job. I think you may find it valuable (esp those replies with Kudos).

Want to see all other topics I dig out? Follow me (click follow button on profile). You will receive a summary of all topics I bump in your profile area as well as via email.

If a line which has the equation 7x + 17y = 1000 is plotted on X-Y pla

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

If a line which has the equation 7x + 17y = 1000 is plotted on X-Y pla

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards