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

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:

GMAT assassins aren't born, they're made,
Rich

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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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)

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)

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.

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

The way I approached this is actually through ratios:

We know that 4X = 5Y = 10Z

So what are the values of X, Y and Z that satisfy the equation?

First iteration is 4X = 5Y = 10Z.. these 3 equations "meet" at 20

4(5) = 5(4) = 10(2)

So x = 5 , y = 4 , z = 2

And since you already got the ratio, it's all about multiplying them to get the pattern

First iteration (multiplying everything by 1): x = 5 | y = 4 | z = 2
Second iteration (multiplying everything by 2): x = 10 | y = 8 | z = 2
Third iteration (multiplying everything by 3): x = 15 | y = 12 | z = 6

Now, if check out the pattern. As we proceed to more iterations.. the difference between X and Y increases:

First iteration: x = 5, y = 4 so difference is 5 - 4 = 1
Second iteration: x = 10 y = 8 so difference is 10 - 8 = 2
Third iteration: x = 15 y = 12 so difference is 15 - 12 = 3

That's the relationship between X and Y

How about the relationship between Y and Z?

First iteration: y = 4, z = 2 so sum of the two is 6
Second iteration: y = 8, z = 4 so sum is 12
Third iteration: y = 12, z = 8 so sum is 24

So we could already do a pre-thinking (like CR question) that we just need X = Y to determine what is Z

Statement (1) gives it to us:

X - Y = 6

This happens in the next iterations.. specifically the Sixth Iteration: x = 30 y = 24 (difference is 30 - 24 = 6) so z = 12

Statement (2) also gives it to us:

Y + Z = 36

This happens in the next iterations.. specifically the Sixth iteration: x = 30 y = 24 z = 12 (sum is 24 + 12 = 36) so z must be 12


If this makes sense please kudos!

Thanks for this explanation!
I know there are quite a few like me who, as soon as they get a DS question involving trivial topics like this one (simultaneous equations), ignore the data sufficiency part of DS and start scratching the pad to 'get an answer'. This is what i did in this question.

I certainly need to understand that the GMAC wouldn't like to test my equation solving/substitution skills if they throw this kinda question up to me in DS. They have the PS section to do that. BUT going with the flow and with the impulse, a started 'solving' the equations as soon as i saw them. Though i got the answer, i never knew what was actually being tested.

A nice take away- Passing a test is one thing. Knowing what was being tested is completely different

Your post was certainly useful

Hi arhumsid,

Every question that you face on the GMAT will be based on at least one pattern, so during your studies it helps to think about (and review) the patterns that you see. At the higher scoring levels, the numbers/calculations in a question really won't be a factor in whether you get the correct answer - it will be because you were able to spot (and use) whatever patterns were involved.

GMAT assassins aren't born, they're made,
Rich

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

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

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

There is only one variable, so we only need one equation; two are given from the 2 conditions, so there is high chance (D) will be our answer.

From condition 1, x-y=6 and 4x=5y. This is a sufficient condition
From condition 2, y+z=36 and 5y=10z, so this is also a sufficient condition,
making the answer D

For cases where we need 1 more equation, such as original conditions with “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 59 % chance that D is the answer, while A or B has 38% chance and C or E has 3% chance. Since D is most likely to be the answer using 1) and 2) separately according to DS definition. Obviously there may be cases where the answer is A, B, C or E.

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