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# Line M has a y-intercept of –4, and its slope must be an integer-multi

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Math Expert
Joined: 02 Sep 2009
Posts: 52392
Line M has a y-intercept of –4, and its slope must be an integer-multi  [#permalink]

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10 Mar 2015, 05:16
1
34
00:00

Difficulty:

95% (hard)

Question Stats:

43% (03:07) correct 57% (03:03) wrong based on 264 sessions

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Line M has a y-intercept of –4, and its slope must be an integer-multiple of 1/7. Given that Line M passes below (4, –1) and above (5, –6), how many possible slopes could Line M have?

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

Kudos for a correct solution.

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Re: Line M has a y-intercept of –4, and its slope must be an integer-multi  [#permalink]

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15 Mar 2015, 21:22
Bunuel wrote:
Line M has a y-intercept of –4, and its slope must be an integer-multiple of 1/7. Given that Line M passes below (4, –1) and above (5, –6), how many possible slopes could Line M have?

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

Kudos for a correct solution.

MAGOOSH OFFICIAL SOLUTION:

Well, for starters, zero is a multiple of every number, and a line with slope zero, the horizontal line y = –4 passes below (4, –1) and above (5, –6). That’s horizontal line is our starting point.

The point (4, –1) is over 4, up 3 from the y-intercept (0, –4). A line with a slope of +3/4 would go straight from (0, –4) to (4, –1). Thus, we need a slope that is less than +3/4. Notice that 3/4 = 21/28. Notice that 5/7 = 20/28, so this would be less than 3/4. Therefore, +1/7 through +5/7 will all slope up, obviously above (5, –6), and all will pass below (4, –1). That’s five upward sloping lines.

The point (5, –6) is over 5, down 2, from the y-intercept (0, –4). A line with a slope of –2/5 would go straight from (0, –4) to (5, –6). Thus, we need a slope that is more than –2/5; another way to say that is, we need a negative slope whose absolute value is less than +2/5. Well, 2/5 = 14/35, while 2/7 = 10/35 and 3/7 = 15/35, so (2/7) < (2/5) < (3, 7). The negatively sloping lines obviously pass below (4, –1), but only two of them, –1/7 and –2/7, pass above (5, –6).

That’s one horizontal line, five upward sloping lines, and two downward sloping lines, for a total of eight.

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Re: Line M has a y-intercept of –4, and its slope must be an integer-multi  [#permalink]

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17 Apr 2017, 09:04
6
3
The slope of the line M has to be in between the slope of line A (passing through (0, -4) and (4, -1)) and line B(passing through (5, -6) and (0, -4))

Now, as we know slope of line with two given points = (y2-y1)/(x2-x1)
Therefore ,
slope of line A => 3/4
slope of line B => -2/5

Therefore slope of line M => -2/5 < m < 3/4

As given in question, m is integer multiple of 1/7 therefore m =k/7 (where k is any integer)
-2/5 < k/7 < 3/4
-14/5 < k < 21/4
-2.8 < k < 5.1

and as we know k is integer
Therefore possible values of k would be -2, -1, 0, 1, 2, 3 ,4 ,5
Total = 8 values (C Answer)
##### General Discussion
Manager
Joined: 14 Sep 2014
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Concentration: Technology, Finance
WE: Analyst (Other)
Re: Line M has a y-intercept of –4, and its slope must be an integer-multi  [#permalink]

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14 Mar 2015, 13:07
3
Let m = slope of Line M
Using the y-intercept and the given points, we can find that $$-\frac{2}{5} < m < \frac{3}{4}$$
Since we need to think in multiples of $$\frac{1}{7}$$, let's convert this to $$-\frac{14}{35} < m < \frac{21}{28}$$

Multiplying $$\frac{1}{7}$$ by the consecutive integers between -2 and 5, inclusive, will give us values that fall in this range.
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Joined: 26 Dec 2011
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Re: Line M has a y-intercept of –4, and its slope must be an integer-multi  [#permalink]

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02 Jun 2015, 10:43
sterling19 wrote:
Let m = slope of Line M
Using the y-intercept and the given points, we can find that $$-\frac{2}{5} < m < \frac{3}{4}$$
Since we need to think in multiples of $$\frac{1}{7}$$, let's convert this to $$-\frac{14}{35} < m < \frac{21}{28}$$

Multiplying $$\frac{1}{7}$$ by the consecutive integers between -2 and 5, inclusive, will give us values that fall in this range.

Please elaborate the last sentence....how can values faill in the range after multiplying 1/7 between -2 and 5.
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Re: Line M has a y-intercept of –4, and its slope must be an integer-multi  [#permalink]

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14 Jul 2016, 10:08
sterling19 wrote:
Let m = slope of Line M
Using the y-intercept and the given points, we can find that $$-\frac{2}{5} < m < \frac{3}{4}$$
Since we need to think in multiples of $$\frac{1}{7}$$, let's convert this to $$-\frac{14}{35} < m < \frac{21}{28}$$

Multiplying $$\frac{1}{7}$$ by the consecutive integers between -2 and 5, inclusive, will give us values that fall in this range.

Hi Sterling,

Regards
Manager
Joined: 17 Sep 2015
Posts: 93
Re: Line M has a y-intercept of –4, and its slope must be an integer-multi  [#permalink]

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27 Jul 2016, 22:01
1
ravi2107 wrote:
sterling19 wrote:
Let m = slope of Line M
Using the y-intercept and the given points, we can find that $$-\frac{2}{5} < m < \frac{3}{4}$$
Since we need to think in multiples of $$\frac{1}{7}$$, let's convert this to $$-\frac{14}{35} < m < \frac{21}{28}$$

Multiplying $$\frac{1}{7}$$ by the consecutive integers between -2 and 5, inclusive, will give us values that fall in this range.

Hi Sterling,

Regards

Trying to make sense of it :

-14/35 < m < 21/28

so m must be less than 21/28
20/28 = 5/7 so we can take values from 20/28 or 5/7 to 0
ie 5/7, 4/7, 3/7, 2/7, 1/7 and 0 (6 values)

now m must be > -14/35 ie -2/5
take m as -10/35 ie - 2/7
so two values -2/7 and -1/7

total values = 6 + 2 (8) (C)
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Re: Line M has a y-intercept of –4, and its slope must be an integer-multi  [#permalink]

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28 Oct 2016, 08:55
But if the slope must be a integer multiple of 1/7 then how can there be 8 slopes? this would mean that slope can only be +/- 1,2,3 and so on.. Can anyone resolve this?
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Re: Line M has a y-intercept of –4, and its slope must be an integer-multi  [#permalink]

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10 Jan 2017, 18:44
I am confused here too. I got to the point of -2/5 < m < 3/4. After that I couldn't figure how to get integer multiple of 1/7.
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Re: Line M has a y-intercept of –4, and its slope must be an integer-multi  [#permalink]

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08 Apr 2017, 11:12
what is meant by 'integer multiple' of 1/7? does that mean multiples such as 1 2 3 4, or -1, -2, -3, -4 or something else? Thanks
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Re: Line M has a y-intercept of –4, and its slope must be an integer-multi  [#permalink]

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17 Apr 2017, 23:40
Step 1.

Find range for possible slope values i.e., -0.4(-2/5) to 0.75(3/4)

Step 2.

We know from question stems that the limits of the above range are not included in our target values and that the values of slope have to be multiples of 0.14(1/7). Hence, only possible values are

-0.28, -0.14, 0, 0.14, 0.18, 0.42, 0.56 and 0.70

a total of 8 values.
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Re: Line M has a y-intercept of –4, and its slope must be an integer-multi  [#permalink]

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03 Jun 2018, 06:10
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Re: Line M has a y-intercept of –4, and its slope must be an integer-multi &nbs [#permalink] 03 Jun 2018, 06:10
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