Problem Solving

if m and r are two numbers on the number line,what is the value of r?
(1) the distance between r and 0 is 3 times the distance between 0 and m
(2)12 is halfway between m and r


I choose answer C



OA E

Please post the question in proper section. This is a DS question you posted in PS section.

Now come to question.

Given m, r two numbers in number line. r = ?

(1) We cannot say the value of r as both m and r can take multiple values satisfying the condition.
Not Sufficient

(2) 12 is the mid point.
So m = 13, r = 11 => 12 is mid
m = 10, r = 14 => 12 is mid
Not sufficient.

Combining both,
r = 3m (if both are on same side of zero i.e both are +ve or both are -ve) ... from (1)
(m+r)/2 = 12 ... from (2)

Thus m = 6 and r = 18.

But what will happen if m becomes -ve and r is +ve. Along with this if option 2 holds true then combining both will be invalid.
r = -3m (m is -ve and r is +ve) ... from (1)
(m+r)/2 = 12 ... from(2)

Thus, m = -12 and r = 36.

Hence, r can be 36 or 18. Thus not sufficient.

IMO
E

Prompt : r=?

1) Simplify 1 .

distance from 0 to r = 3 times distance from 0 to m. In other words, we are measuring distance (a positive entity),so sign does not matter.

In equation form |r|=3|m|.

r m

6 2
6 -2
-6 2
-6 -2

All these values and many infinite other value satisfy this condition. r=2,-2. Therefore, Insufficient.

2) (r+m)/2 =12
r+m =24

r could be 2,4,5 ( anything) and still satisfy this condition.

Therefore, insufficient


Combining 1 and 2

From 2 -> r+m=24

From 1 -> |r|= 3|m|


This equation is equivalent to 2 different equations -> r=3m ; r=-3m; -r=3m ( r=-3m); -r=-3m (r=3m) [ in inequalities it would be 4 different equations because the sign flips on multiplying by -ve]

Now First substitute r=3m in 2 to get 4m=24 -> m=6 and r= 3(6)=18

Then substitute r=-3m in 2 to get -2m=24 -> m=-12 and r =-3(-12)=36

Therefore r = 18 or 36. Two different values.

hence C also is insufficient.

Answer = E