On the number line, if w x, y is the midpoint between w and z, and x

Question

On the number line, if  w > x ,  y  is the midpoint between  w  and  z , and  x  is twice as far from  z  as it is from  w , then  \frac{(w – y)}{(z – x)}  could equal which of the following?

A) -2

B)  -\frac{1}{4}

C) 0

D)  \frac{1}{4}

E)  \frac{3}{2}

davedekoos · Accepted Answer

With number line problems it always helps tremendously to draw it out.

Start with the second statement, y is the midpoint between w and z. Then add in x to satisfy the first condition. You shoud get two scenarios that look like this:

x------w---y---z (1)
z------y--x----w (2)

Looking at scenario (1) first, say the distance between  x  and  w  is  d . Then the distance between  w  and  y  is  d/2 , and the distance between  w  and  z  is  d .

Now, what we are being asked to find is:
 \frac{w-y}{z-x}=\frac{-\frac{1}{2}d}{2d} = -\frac{1}{4}

This is one of our answer choices, so we can stop there, and the answer is B.

BUT, if you had started with number line scenario (2), then you could say the distance between  z  and  w  is  d . Then the distance between  y  and  w  is  d/2  and the distance between  z  and  x  is  2d/3 . Now what we're being asked to find is:
 \frac{w-y}{z-x}=\frac{\frac{1}{2}d}{-\frac{2}{3}d}=-\frac{3}{4}

This is not one of our answer choices, so if we had started with scenario (2), we would have to then go and work out scenario (1).

The tricky part of this question could be recognizing that there are 2 different possible scenarios for the arrangement of w, x, y, and z on the number line. One of the biggest clues that there could be multiple arrangements is the use of the word  could  in the question

This is essentially saying that the expression could also equal something else, implying that there is another scenario and other possible answers not listed in the answer choices. Always be on the lookout for telltale words like  could  in questions.

chetan2u · Answer

lets imagine a number line... 
 x is to the left of w...
y is between w and z ...
and x is twice the distance from z as from w...

two scenarios... 
 
1)z is to the left of w.. 
sequence would be z,y,x,w

2)z is to the right of w.. 
sequence would be x,w,y,z..

we can check for the values in eq  \frac{(w−y)}{(z−x)}  in both the scenarios..
I tried2nd scenario and got the answer, so would try that ..

the seq is x,w,y,z..
let w-x=a, z-x=2a..
so w-y=-1/2 *a..
z-x=2a

so  \frac{(w−y)}{(z−x)}= -(\frac{1}{2} *a)/2a=-(\frac{1}{4}) ..
ans B

ENGRTOMBA2018 · Answer

Based on the information given, you need to visualize a couple of possible scenarios to rule out obvious outliers from the given options. You are given w>x and that |x-z| = 2 |x-w| ---> a possible scenario on the numberline is -----0---z----y----x---w. This scenario shows that the value of z-y < 0 while w-y > 0 ---> (w-y)/(z-x) < 0 ---> eliminate D and E. Now to C), you can clearly see that you can get (w-y)/(z-x) = 0 by having w=y and this is not possible as it is given that y is midway between w and z and hence y eq w Thus you now have 2 options remaining, A and B. Take A) now. Lets assume (w-y)/(z-x) =2 ---> w-y = 2(z-x) and do substitute y = (w+z)/2 to get, w/2 - z/2 = 2z-2x...(1) Additionally, from the above scenario, you know that x-z = 2/3 * (w-z) ....(2) Thus from substituting value of x from (2) to (1), you end up getting on rearranging the terms, w=z. This is not possible as this will make 'y' = w= z. Thus A ) is not possible, leaving B as the only possible scenario. Hence B is the correct answer. Hope this helps.

BrainLab · Answer

Just insert these numbers in the expression  \frac{(w – y)}{(z – x)}  and you'll get the answer (B)

ragsSF · Answer

I get that the answer is B

Using  BrainLab  diagram from above (it seems I can't post URLs unless I have 5 posts),

Aren't w, x, y, z points on the line segment, so (w–y) would be the distance between w and y (non-negative)?

Thus  \frac{(w–y)}{z-x}  should be :

2/8 = 1/4 ?

Kinshook · Answer

x-------------------------------w

x-------------------------------w---------------y-----------------z
or 
z------------------------------------------------y-----------------x---------------------------------w

(w-y)/(z-x) = -1/4

(w-y)/(z-x) = 3/(-4)=-3/4

IMO B

bumpbot · Answer

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On the number line, if w > x, y is the midpoint between w and z, and x

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