If 4x = 5y = 10z, What is the Value of x + y + z ? (1) x - y = 6 (2) y + z = 36

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

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

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are not sufficient.

“If 4x = 5y = 10z, what is the value of x + y + z ? (1) x - y = 6 (2) y + z = 36” – is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Quantitative Review". GMAT Quant section consists of a total of 31 questions. GMAT Data Sufficiency questions consist of a problem statement followed by two factual statements. GMAT data sufficiency comprises 15 questions which are two-fifths of the total 31 GMAT quant questions.

Solution and Explanation:

Approach 1:

Though this Data Sufficiency question looks layered, it's actually based on a standard Algebra rule : "system math." Knowing this rule will provide candidates with a shortcut so that they 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.
Now, let us consider each statements one at a time.

Statement 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.
After we get the values, with either of those values, we can then solve for the value of Z (since 4X = 10Z and 5Y = 10Z).
Now once we get the value ofZ, we can find the individual values of X, Y and Z.

Hence, we CAN answer the question from statement 1.
Hence, statement 1 is sufficient.

Statement 2: Y + Z = 36

Here we have a similar situation to the one we saw in statement 1.
From the question, 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 and find the individual values.
Hence, statement 2 is sufficient.
Since, both the statements can be answered separately, D is the correct answer.

Approach 2:

We will check each statements one by one.

Statement 1:

From 4x=5y=10z
->x=5y/4 and z=5y/10=y/2

From the statement 1, we get:
x-y=6 -> x=6+y,
thus 6+y=5y/4
-> y=24

Now, putting the values:
x=5y/4 -> x=30
z=y/2 -> z=12

Hence, statement 1 is sufficient.

Statement 2:

From 4x=5y=10z
-> y=2z and x=5y/4
Fromthe statement 2, we get:
y+z=36
-> y=36-z,
thus 2z=36-z
-> z=12
Now, putting the value:
y=2z -> y=24
x=5y/4 -> x= 30
Hence, statement 2 is sufficient.

Since both the statements are sufficient, D is the correct answer.

Approach 3:

A = B = C should immediately prompt to put them equal to k

• We need to 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 the value of k, will give the value of x+y+z

(1) x - y = 6
5k - 4k = 6 = k
So we will get k and in turn will get x+y+z.
Hence, it is Sufficient.

(2) y + z = 36
4k + 2k = 36 = 6k
We get k so we will also get x+y+z.
This is also Sufficient.

Since both are sufficient, D is the correct answer.

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