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If 4x = 5y = 10z, what is the value of x + y + z ?

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If 4x = 5y = 10z, what is the value of x + y + z ? [#permalink] New post 04 Jun 2006, 02:34
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If 4x = 5y = 10z, what is the value of x + y + z ?

(1) x - y = 6
(2) y + z = 36
[Reveal] Spoiler: OA

Last edited by Bunuel on 05 Apr 2015, 03:44, edited 1 time in total.
Renamed the topic, edited the question and added the OA.
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Re: If 4x = 5y = 10z, what is the value of x + y + z ? [#permalink] New post 04 Jun 2006, 10:32
SimaQ wrote:
How did you solve it?


4x=5y=10z => if we take 4x=5y than x=5y/4 thus by subst. (1) we have 5y/4-y=6

similarly for (2)
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Re: If 4x = 5y = 10z, what is the value of x + y + z ? [#permalink] New post 04 Apr 2015, 06:04
Statement (1) Sufficient:

From 4x=5y=10z ->x=5y/4 and z=5y/10=y/2
From St. (1) x-y=6 -> x=6+y, thus 6+y=5y/4 -> y=24
x=5y/4 -> x=30
z=y/2 -> z=12



Statement (2) Sufficient:

From 4x=5y=10z -> y=2z and x=5y/4
From St. (2) y+z=36 -> y=36-z, thus 2z=36-z -> z=12
y=2z -> y=24
x=5y/4 -> x= 30

Answer: D (either)
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Re: If 4x = 5y = 10z, what is the value of x + y + z ? [#permalink] New post 05 Apr 2015, 14:50
Expert's post
Hi All,

As 'layered' as this DS question looks, it's actually based on a standard Algebra rule that you WILL be tested on at least once when you take the Official GMAT: "system math." Knowing this rule will provide you with a shortcut so that you can avoid most of the work involved in this question....

We're told that 4X = 5Y = 10Z. We're asked for the value of X + Y + Z.

Fact 1: X - Y = 6

From the prompt, we also know that 4X = 5Y. We now have a "system" of equations (2 variables and 2 unique equations) involving X and Y; since there are no special situations to consider (re: absolute values, squared terms), we CAN solve for the value of X and Y and there will be just one value for each. With either of those values, we can then solve for the value of Z (since 4X = 10Z and 5Y = 10Z). With the individual values of X, Y and Z, we CAN answer the question.
Fact 1 is SUFFICIENT

Fact 2: Y + Z = 36

Here we have a similar situation to the one we saw in Fact 1. From the prompt, we know that 5Y = 10Z, so we again have a "system" of equations. The exact same shortcut applies here, so we CAN answer the question.
Fact 2 is SUFFICIENT

Final Answer:
[Reveal] Spoiler:
D


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Re: If 4x = 5y = 10z, what is the value of x + y + z ? [#permalink] New post 05 Apr 2015, 17:00
I'd only add to the expert's post that with the prompt and statement 1 you can get:

x - y = 6 and 4x = 5y (ignoring z). From 4x - 5y, you can get 4x - 5y = 0. Then, lining them up for the elimination method, you get:

x - y = 6
4x - 5y = 0

If this method is familiar to you, you'll instantly know to multiply the top equation by 4:

4x - 4y = 24
4x - 5y = 0

Subtract down and y = 24. Suppress any urge to go further, statement 1 is sufficient!

Do the same with statement 2.
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Re: If 4x = 5y = 10z, what is the value of x + y + z ? [#permalink] New post 05 Apr 2015, 20:26
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SimaQ wrote:
If 4x = 5y = 10z, what is the value of x + y + z ?

(1) x - y = 6
(2) y + z = 36


A = B = C should immediately prompt you to put them equal to k - reduce the number of variables. Bring them all to a single variable.

4x = 5y = 10z
Divide the entire equation by 20 so that your calculations with k are simple.
x/5 = y/4 = z/2 = k
x = 5k, y = 4k and z = 2k

So x+y+z = 11k

Now, whichever statement gives you the value of k, will give you the value of x+y+z

(1) x - y = 6
5k - 4k = 6 = k
You get k so you will get x+y+z. Sufficient.

(2) y + z = 36
4k + 2k = 36 = 6k
You get k so you will get x+y+z. Sufficient.

Answer (D)
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Re: If 4x = 5y = 10z, what is the value of x + y + z ? [#permalink] New post 27 Apr 2015, 14:46
From the question we know that 4x=5y=10z. Hence x=2.5z and y=2z
Statement 1. From the question we can plug in the values of x and y to find z. If we find z then we can find x and y and thus x+y+z. Sufficient
Statement 2. The same Plug in y=2z to find z. After we can find x and y and x+y+z. Sufficient
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Re: If 4x = 5y = 10z, what is the value of x + y + z ?   [#permalink] 27 Apr 2015, 14:46
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