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The value of x is one quarter of z. The sum of x, y, and z

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The value of x is one quarter of z. The sum of x, y, and z  [#permalink]

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New post 01 Mar 2014, 07:44
2
3
00:00
A
B
C
D
E

Difficulty:

  25% (medium)

Question Stats:

78% (01:47) correct 22% (01:51) wrong based on 301 sessions

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The value of x is one quarter of z. The sum of x, y, and z is equal to 26. If the value of y is twice the value of z, what is the largest factor of the sum of y and z?

A. 2
B. 3
C. 8
D. 12
E. 24

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Jerry Rice
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Re: The value of x is one quarter of z. The sum of x, y, and z  [#permalink]

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New post 01 Mar 2014, 07:45
1
In this problem, we are given the information to set up the following three equations with three unknowns:

x = z/4
x + y + z = 26
y = 2z

Using the method of substitution, we can now solve for each of the three unknowns.

Since the first equation, x = z/4, is an equation for x in terms of z, and the third equation, y = 2z, is an equation for y in terms of z, we can replace x and y in the second equation by their equivalent expressions as follows:

x + y + z = (z/4) + (2z) + z = 26

We are left with an equation with just one unknown z. We can now solve for z:

(z/4) + (2z) + z = 26
z/4 + 8z/4 + 4z/4 = 13z/4 = 26
z/4 = 2
z = 8

Now that we know z, we can easily solve for y, then compute y + z (note: we can also solve for x, but since we are not interested in the value of x there is no need to do so):

y = 2z = 2(8) = 16
y + z = 16 + 8 = 24

Since the largest factor of any number is the number itself, the largest factor of the sum of y and z is 24.

The correct answer is E.
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Re: The value of x is one quarter of z. The sum of x, y, and z  [#permalink]

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New post 24 Aug 2015, 19:51
1
Hi All,

While most Test Takers would treat this as a 'system' algebra question and solve it that way, there are some interesting Number Property patterns in this prompt that will actually allow you to use the answer choices (and some pattern-matching) to get to the solution.

To start, I'm going to deal with the last piece of information first:

Y is twice Z

The prompt asks for the BIGGEST factor of the SUM of Y and Z. Since Y = 2Z, the sum of Y and Z is (2Z) + Z = 3Z. This means that the BIGGEST factor is a MULTIPLE OF 3 (and likely one of the larger multiples of 3 in the answer choices).

The second piece of information tells us...

X+Y+Z = 26

By extension...

X+(Y+Z) = 26
X+(larger multiple of 3) = 26

From the answer choices, there are really only 2 options that make any sense: Answers D and E....

With Answer D:
IF....Y+Z = 12, then X= 14, but....

The first piece of information tells us that X = 1/4 of Z, so X has to be relatively small....In this example, it's LARGER than Z, so D is NOT the answer. At this point, the answer would have to be E, but here's the proof...

IF...Y+Z = 24 and Y= 2Z....
Y = 16
Z = 8

X is 1/4 of Z, so X = (1/4)(8) = 2

X+Y+Z=26
2+16+8=26

All of the numbers 'fit' what we were told, so this MUST be the answer.

Final Answer:

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Rich
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Re: The value of x is one quarter of z. The sum of x, y, and z  [#permalink]

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New post 14 Sep 2016, 07:15
i luckily picked some numbers...
z=8, x=2, y=16.
now, z+y=24. E
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Re: The value of x is one quarter of z. The sum of x, y, and z  [#permalink]

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Re: The value of x is one quarter of z. The sum of x, y, and z   [#permalink] 18 Dec 2018, 09:15
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