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If x is a positive number less than 10, is z greater than th
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20 Feb 2014, 00:16

7

7

If x is a positive number less than 10, is z greater than the average (arithmetic mean) of x and 10?

Given: \(0<x<10\). Question: is z greater than the average of x and 10? Or: is \(z>\frac{10+x}{2}\)? --> \(2z>10+x\)?

(1) On the number line, z is closer to 10 than it is to x --> \(|10-z|<|z-x|\) --> as z is closer to 10 than it is to x, then z>x, so \(|z-x|=z-x\) --> two cases for 10-z:

A. \(z\leq{10}\) --> \(|10-z|=10-z\) --> \(|10-z|<|z-x|\) becomes: \(10-z<z-x\) --> \(2z>10+x\). Answer to the question YES.

B. \(z>{10}\) --> in this case \(2z>20\) and as \(x<10\), then \(x+10<20\), hence \(2z>10+x\). Answer to the question YES.

OR another approach: Given: x-----average-----10----- (average of x and 10 halfway between x and 10).

Now, as z is closer to 10 than it (z) is to x, then z is either in the blue area, so more than average OR in the green area, so also more than average. Answer to the question YES.

Sufficient.

(2) z = 5x --> is \(2z>10+x\)? --> is \(10x>10+x\)? --> is \(x>\frac{10}{9}\). We don't now that. Not sufficient. (we've gotten that if \(x>\frac{10}{9}\), then the answer to the question is YES, but if \(0<x\leq{\frac{10}{9}}\), then the answer to the question is NO.)

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20 Feb 2014, 02:18

3

If x is a positive number less than 10, is z greater than the average (arithmetic mean) of x and 10?

(1) On the number line, z is closer to 10 than it is to x. (2) z = 5x

On st1 i'll draw a number line. 0----------------------x-----------------z---10 We can see that z will always be greater than the average. Sufficient

St2 say that z=5x. we'll choose 2 cases: Case 1: x=2, so z=10. avarge is 6 and z is greater than the avarage. Case 2: x=1, so z=5 avarge is 5.5 and z is smaller than the avarage. Insufficient

Re: If x is a positive number less than 10, is z greater than th
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20 Feb 2014, 23:06

Option A. From S1:Is z>(x+10)/2 (1) (x+10)2 is basically the middle value between x and 10. So,if z is closer to 10 than x it would imply z>the mid value of x and 10 which is the mean. Sufficient.

From S2:If z=5x Replacing in (1) Is 5x>x+10/2 Or Is 9x>10 If x=1 ans is true If x=2. Ans is false.Not suff.

Re: If x is a positive number less than 10, is z greater than th
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21 Feb 2014, 02:59

2

Q Is z>(x+10)/2 ; provided that 0<x<10 (1) On the number line, z is closer to 10 than it is to x.

this means distance between z and x is greater than distance between z and 10 on the number line. hence we have z-x>10-z z>(x+10)/2 in the above case z is less than 10 and lies between x and 10. If z is greater 10, then it will definitely be greater than average of x+10. hence z is greater than average of x+10. therefore sufficient.

(2) z = 5x

z=5x assume x=1 average of 10 and 1 = 5.5 and z=5(1) when x=3, average of 10 and 3= 6.5 and z=15

since two different answers are possible hence 2 alone is not sufficient hence correct answer is A

Re: If x is a positive number less than 10, is z greater than th
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23 Aug 2015, 01:37

Statement 1: 0---X----Z--10 The average is the middle point between X (which is 0<x<10) and 10, and we are told that Z is closer to 10 that means Z>Average

Statement 2: 10x>x+10 (plugin 5x=z here z>(x+10)/2 --> 9x>10 ? If x=1 9<10 No, and if x=2 18>10 Yes NOT Sufficient
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Re: If x is a positive number less than 10, is z greater than th
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12 Dec 2015, 23:13

Hi,

i request your help to find out weather statement A is sufficient, when x=8 and Z =9 . Here z is closer to 10 than x, but the average of 10 and X will be equal to z.

Re: If x is a positive number less than 10, is z greater than th
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13 Sep 2016, 01:50

syamukrishnan wrote:

Hi,

i request your help to find out weather statement A is sufficient, when x=8 and Z =9 . Here z is closer to 10 than x, but the average of 10 and X will be equal to z.

Is this correct?

Thanks

No, the Statement 1 says that - Z is closer to 10 than it is to x and not that z is closer to 10 than x is. Your's is the later case.
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Re: If x is a positive number less than 10, is z greater than th
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05 Oct 2016, 10:25

2

Top Contributor

Bunuel wrote:

If x is a positive number less than 10, is z greater than the average (arithmetic mean) of x and 10?

(1) On the number line, z is closer to 10 than it is to x. (2) z = 5x

I received a PM about this question. Here's my solution:

Target question: Is z greater than the mean of x and 10?

Statement 1: On the number line, z is closer to 10 than it is to x. IMPORTANT: On the number line, the mean of two numbers will lie at the midpoint between those two numbers. So, the mean of x and 10 will lie halfway between x and 10. So, if z is closer to 10 than it is to x, then z must lie to the right of the midpoint between x and 10. This means that z must be greater than the mean of x and 10 Since we can answer the target question with certainty, statement 1 is SUFFICIENT

Statement 2: z = 5x There are several pairs of numbers that meet this condition. Here are two: Case a: x=1, z=5, in which case z is less than the mean of x and 10 (mean = 5.5) Case b: x=4, z=20, in which case z is greater than the mean of x and 10 (mean = 7) Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Re: If x is a positive number less than 10, is z greater than th
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05 Oct 2016, 12:23

If x is a positive number less than 10, is z greater than the average (arithmetic mean) of x and 10?

(1) On the number line, z is closer to 10 than it is to x. (2) z = 5x

I first rephrased the question to z>(x+10)/2 2z>x+10 2z-10>x? (This didn't help me w/ the question)

(1) On the number line, z is closer to 10 than it is to x.

I just used guess and check for this one. x=8 then (10+8)/2=9 if z is closer to 10 than to 8 it has to be more than 9 so yes x=1 then (10+1)=11/2=5.5 5.5 is the midpoint between 1 and 10 (i.e. 5.5 is the average). In order for z to be closer to 10 it need to be more than 5.5. yes suff

(2) z = 5x again I plugged in numbers x=1, z=5 then no z is not greater than the average x=2, z=10 z is greater than the average

If x is a positive number less than 10, is z greater than the average (arithmetic mean) of x and 10?

(1) On the number line, z is closer to 10 than it is to x. (2) z = 5x

0 < x < 10 Question : is z greater than the average (arithmetic mean) of x and 10?

Question : is z > (x+10)/2? OR Question : is z > Midpoint of x and 10 ?

Statement 1: On the number line, z is closer to 10 than it is to x.

i.e. z is to the right of "midpoint of 10 and x"

i.e. z is GREATER than x

SUFFICIENT

Statement 2: z = 5x

@x = 1, Midpoint = (1+10)/2 = 5.5 and z = 5 i.e. z < AVerage

@x = 2, Midpoint = (2+10)/2 = 6 and z = 10 i.e. z > AVerage

NOT SUFFICIENT

Answer: Option A
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