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If m and r are two numbers on a number line, what is the

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Source: — Data Sufficiency |

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by Jay@ManhattanReview » Thu Aug 23, 2018 9:15 pm
BTGmoderatorDC wrote:If m and r are two numbers on a number line, what is the value of r?

(1) The distance between r and 0 is 3 times the distance between m and 0.
(2) 12 is halfway between m and r.

OA E

Source: GMAT Prep
Let's take each statement one by one.

(1) The distance between r and 0 is 3 times the distance between m and 0.

There are two situations:

1. m and r on the same side of 0:

=> Say the distance between 0 and m is x, then the distance between 0 and r is 3x

=> Distance between m and r is 2x.

=> |r - m| = 2x

2. m and r on the opposite sides of 0:

=> Say the distance between 0 and m is x, then the distance between 0 and r is 3x

=> Distance between m and r is 4x.

=> |r + m| = 4x

Insufficient.

(2) 12 is halfway between m and r.

=> |m| + |r| = 24

This does not help to get the value of r. Insufficient.

(1) and (2) together

We have

|m| + |r| = 24 ---(1)
|r - m| = 2x ---(2)
|r + m| = 4x ---(3)

Even after combining the two statements, we can't get the unique value of r.

The correct answer: E

Hope this helps!

-Jay
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by fskilnik@GMATH » Mon Aug 27, 2018 1:11 pm
BTGmoderatorDC wrote:If m and r are two numbers on a number line, what is the value of r?

(1) The distance between r and 0 is 3 times the distance between m and 0.
(2) 12 is halfway between m and r.
? = r

(1) |r| = 3 |m|
Insufficient

Take m = 0, then ? = 0
Take m = 1, then |r| = 3, hence ? = 3 or -3

(2) (m+r)/2 = 12 or, equivalently, m+r = 24
Insufficient

Take m = 24, then ? = 0
Take m = 0, then ? = 24

(1+2) Consider (as particular cases)
(A) r > 0 and m > 0 , hence from (1) we have r = 3m and in (2) we have m = 6 (and r = 18). Check that ? = 18 is viable.
(B) r > 0 and m < 0 , hence from (1) we have r = -3m and in (2) we have m = -12 (and r = 36). Check that ? = 36 is viable.

Hence (1+2) is Insufficient

The above follows the notations and rationale taught in the GMATH method.
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by GMATGuruNY » Mon Aug 27, 2018 1:20 pm
BTGmoderatorDC wrote:If m and r are two numbers on a number line, what is the value of r?

(1) The distance between r and 0 is 3 times the distance between m and 0.
(2) 12 is halfway between m and r.
Translate the words into math.
The DISTANCE between x and 0 is |x|.
HALFWAY between two numbers is the AVERAGE of the two numbers.

Statement 1: The distance between r and 0 is 3 times the distance between m and 0.
|r| = 3|m|
r = 3m or r=-3m.
No way to solve for r.
INSUFFICIENT.

Statement 2: 12 is halfway between m and r.
(m+r)/2 = 12
m+r = 24.
No way to solve for r.
INSUFFICIENT.

Statements 1 and 2 combined:
Substituting r=3m into m+r=24, we get:
m+3m=24
4m=24
m=6.
r=3m=18.

Substituting r=-3m into m+r=24, we get:
m+(-3m)=24
-2m=24
m=-12.
r=-3m=-3(-12)=36.

Since it's possible that r=18 or that r=36, INSUFFICIENT.

The correct answer is E.
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by masoom j negi » Fri Dec 21, 2018 8:42 pm
Statement 1. The distance between r and 0 is 3 times the distance between m and 0.
r = 3m or -3m.
Since we don't know the distance of 'm' from 0, we can't find the value of 'r'.
Hence, Insufficient.
Statement 2. 12 is halfway between m and r. This gives, m + r = 2 x 12 = 24
(m, r) could be ( 11,13), (10,14),(9,15)..........
Hence, Insufficient.
Statement 1 & 2 toghether. Combining the results of statement 1 & 2, we get
m + 3m = 24 i.e. m = 6 and r = 18
m - 3m = 24 i.e. m = -12 and r = 36. Hence, Insufficient.
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