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On the GMAT we can often see such statement: \(k\) is halfway between \(m\) and \(n\) on the number line. Remember this statement can ALWAYS be expressed as:

\(\frac{m+n}{2}=k\).

Also on the GMAT when we see the distance between x and y, this can be expressed as \(|x-y|\).

BACK TO THE QUESTION.

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

(1) The distance between r and zero is 3 times the distance between m and zero --> \(|r-0|=3|m-0|\) --> \(|r|=3|m|\) --> \(r=3m\) OR \(r=-3m\). Clearly insufficient.

(2) 12 is halfway between m and r --> \(\frac{r+m}{2}=12\) --> \(r+m=24\). Clearly insufficient.

(1)+(2) \(r=3m\) OR \(r=-3m\) and \(r+m=24\).

\(r=3m\) --> \(r+m=3m+m=24\) --> \(m=6\) and \(r=18\) OR \(r=-3m\) --> \(r+m=-3m+m=24\) --> \(m=-12\) and \(r=36\)

s1) let m-0=x, then r-0=3x ( x can be 1,2,3,4,......anything)...not suff s2) m-12=12-r or r-12=12-m....(can have any value)...not suff

s1)+s2)if m-12=1,2,3,4,5.... then 12-r=3,6,9,12,15...any thing same for r-12...hence from bth also we are not getting any particular value so E

The answer is correct, but there is some problems in solution:

(1) When you write: m=x and r=3x, it's not right: if m=x, then r=3x OR r=-3x, as |r|=3|m|.

(2) You wrote: m-12=12-r or r-12=12-m. If you look at it you'll see that these two equations are the same and derived from \(\frac{m+r}{2}=12\).

Again: Statement: distance between r and x, is three times the distance between m and x can be expressed as \(|r-x|=3|m-x|\).

Statement: \(k\) is halfway between \(m\) and \(r\) on the number line can be expressed as:

\(\frac{m+r}{2}=k\).

thanks bunuel i cud not express it in a correct manner , but my intention was same since i tried on no. line and i got it in a easier way ,however cudn't express that in my post(appologies for that), on a no. line it was clear that the position of m and r is not fix with bth the given information hence insuff....
_________________

GMAT is not a game for losers , and the moment u decide to appear for it u are no more a loser........ITS A BRAIN GAME

ST1: The distance between r and 0 is 3 times the distance between m and 0 Since the statement has the term 'distance' in it, it signifies that we are not consider the -ve or +ve possibility of the number position. Hence ST1 can be written algebrically as: |r-0| = 3|m-0| ---> |r| = 3|m| Clearly NOT SUFF as m could be anything and even if m is constant, r could be -3m or 3m

ST2: 12 is halfway between m & r is clearly NOT SUFF as the same is true for (m=11,r = 13) , (m=10,r = 14)....

Both ST1 and ST2 together would give us: m = 6 and r = 18, m=-12 & r = 36 ...etc..Hence NOT SUFF....

OA as D.... ... Not sure..
_________________

Cheers! JT........... If u like my post..... payback in Kudos!!

|Do not post questions with OA|Please underline your SC questions while posting|Try posting the explanation along with your answer choice| |For CR refer Powerscore CR Bible|For SC refer Manhattan SC Guide|

For any positive integer n the length is defined as the number of prime numbers whose product equals n. So for 75 the length is 3 since 75 = 3 * 5 * 5. How many 2 digit numbers have a length of 6.

a) None b) One c) Two d) Three e) Four

OA is provided.

Which is the easiest way to solve this?

You have to choose minimum prime factors to get the maximum length;

\(2^6=64\) has a length of 6. (Kept minimum prime factor as 2) \(2^5*3^1=96\) has a length of 6. (Substituted one prime factor of 2 with 3)

Thus the answer is 2.

I don't know any better way to approach the problem.

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.

I used the number line and tested two cases.

Case 1. Assume m is negative and r is positive, each dashed segments ----- is 1x

m ----- 0 ----- ----- ----- r

m = -x r = 3x (3 times the distance between m and 0)

If 12 is the midpoint, the graph becomes:

m ----- 0 ----- 12 ----- ----- r

That means x = 12 and m = -12 and r = 36

Case 2. Assume both m and r are positive

0 ----- m ----- ----- r

m = x r = 3x

Add 12 as the mid point:

0 ----- m ----- 12 ----- r

Therefore m = 6 and r = 18

This shows that even if you combine the 2 statements, you still can't get a unique answer. Therefore the answer should be E.

If m and n are two numbers on 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

A) The distance between r and 0 & m and 0 does not help even identify if r is to the left (negative) or right (positive) of zero.

B) 12 is the mid point between m & r, but m & r can both be on the positive side of the line, or negative side of the line or one is positive and the other negative, so cannot identify the value of r

Combining the two

Does not help here since r can be a positive value or negative value, m can be lesser than r or greater than r . This does not help

If m and r are two numbers on 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

*Edited the question. It should be r instead of n.

Stmnt 1 alone: Too many values possible. Say r = 3, m = 1 OR r= 6, m = 2 etc Stmnt 2 alone: Again too many values possible. Think 12 is in the middle. m and r are equidistant from it so m = 11, r = 13 OR m = 10, r = 14 etc

Both together: Focus on the logic behind it. You don't need to do any calculations. We are looking for two values equidistant from 12. Let's say both m and r are at 12 initially. Their distance from 0 is the same i.e. 12 at this point. As they both start moving away from 12 simultaneously, the distance of m from 0 is reducing and that of r from 0 is increasing. There will be point when the distance of m from 0 will be a third of the distance of r from 0. This will be our first pair (shown in blue). Let's say they keep moving. m will finally reach 0 when its distance from 0 is 0 while r will be at 24. Then m will move in the negative range and its distance from 0 will start increasing. Distance of r from 0 is continuing to increase. There will be a point again when distance of m from 0 is a third of the distance of r from 0 (shown in red).

Re: Another DS good example If m and n are two numbers on number [#permalink]

Show Tags

16 Nov 2011, 07:27

1

This post received KUDOS

VeritasPrepKarishma wrote:

SwapnilRanadive wrote:

Another DS good example

If m and r are two numbers on 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

*Edited the question. It should be r instead of n.

\(|a-b|=c\) ---> Distance of \(a\) from \(b\) equals \(c\)

Question: r=?

Statement 1:

\(|r-0|=3*|m-0|\) ---> \(r=3*m\) or \(r=-3*m\),

\(r\) depends on \(m\), and since we don't know \(m\), Insufficient.

Statement 2:

Number line is like a set with consecutive numbers. Since this set is an evenly spaced set we know that median=average.

Re: If m and r are two numbers on a number line, what is the [#permalink]

Show Tags

19 Jul 2013, 19:35

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. |r-0| = 3|m-0| |r|=3|m|

This gives us no basis for what r is. For example: m=3 |r|=3|m| |r|=3|3| |r|=9 r=9, r=-9 INSUFFICIENT

(2) 12 is halfway between m and r. (m+r)/2=12 As with #1, any number of values could be substituted in for m and r to get a valid solution. INSUFFICIENT

1+2) |r|=3|m| and (m+r)/2=12 m+r=24

|r|=3|m| r=3m OR r=-3m

m+(3m)=24 4m=24 m=6 OR m+(-3m)=24 2m=24 m=12

We have two different values for m. This means, when you plug into |r|=3|m| you get two different values for r. INSUFFICIENT

TIP: On the GMAT we can often see such statement: \(k\) is halfway between \(m\) and \(n\) on the number line. Remember this statement can ALWAYS be expressed as:

\(\frac{m+n}{2}=k\).

Also on the GMAT when we see the distance between x and y, this can be expressed as \(|x-y|\).

Back to the question: If m and r are two numbers on a number line, what is the value of r?

(1) The distance between r and zero is 3 times the distance between m and zero --> \(|r-0|=3|m-0|\) --> \(|r|=3|m|\) --> \(r=3m\) OR \(r=-3m\). Clearly insufficient.

(2) 12 is halfway between m and r --> \(\frac{r+m}{2}=12\) --> \(r+m=24\). Clearly insufficient.

(1)+(2) \(r=3m\) OR \(r=-3m\) and \(r+m=24\).

\(r=3m\) --> \(r+m=3m+m=24\) --> \(m=6\) and \(r=18\) OR \(r=-3m\) --> \(r+m=-3m+m=24\) --> \(m=-12\) and \(r=36\)

Two different values for \(r\). Not sufficient.

Answer: E.

Bunuel, can you explain how \(|r|=3|m|\) --> \(r=3m\) OR \(r=-3m\)?

TIP: On the GMAT we can often see such statement: \(k\) is halfway between \(m\) and \(n\) on the number line. Remember this statement can ALWAYS be expressed as:

\(\frac{m+n}{2}=k\).

Also on the GMAT when we see the distance between x and y, this can be expressed as \(|x-y|\).

Back to the question: If m and r are two numbers on a number line, what is the value of r?

(1) The distance between r and zero is 3 times the distance between m and zero --> \(|r-0|=3|m-0|\) --> \(|r|=3|m|\) --> \(r=3m\) OR \(r=-3m\). Clearly insufficient.

(2) 12 is halfway between m and r --> \(\frac{r+m}{2}=12\) --> \(r+m=24\). Clearly insufficient.

(1)+(2) \(r=3m\) OR \(r=-3m\) and \(r+m=24\).

\(r=3m\) --> \(r+m=3m+m=24\) --> \(m=6\) and \(r=18\) OR \(r=-3m\) --> \(r+m=-3m+m=24\) --> \(m=-12\) and \(r=36\)

Two different values for \(r\). Not sufficient.

Answer: E.

Bunuel, can you explain how \(|r|=3|m|\) --> \(r=3m\) OR \(r=-3m\)?

\(|r|=3|m|\) means that the distance from r to 0 is thrice the distance from m to 0:

-----0--m-----r------ r-----m--0--------------

--m--0--------r------ r--------0--m------------

If r and m have the same sign (cases A and B), then r=3m but if r and m have different signs (cases C and D), then r=-3m.