What is the value of x^3 + y^3? GMAT data sufficiency

Question: What is the value of x³ + y³?

Statement (1) x + y = 12
Statement (2) x − y = 8

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.

Answer: C

Solution and Explanation:

Approach Solution 1:
First Proposition: x + y = 12
This seems like too broad of a statement, so we’ll try out some different parameters.
It is possible to satisfy assertion 1 with a wide range of x and y values. Some examples:
If x = 12 and y = 0, then (a) is correct. As an example, x3 + y3 = 123 + 03 = 123.
Situation B: x = 10, y = 2. If so, then x³ + y³ = 103 + 23 = 1008.
Statement 1 is insufficient, as it does not allow us to definitively answer the research question.
Statement Two: x + y = 8
Similar to the first statement, this data warrants more testing, so let's do that now.
Many combinations of x and y numbers are consistent with the second statement. Two examples:
It's Case A if x = 8 and y = 0. Here, x³ + y³ = 0³ + 8³= 8³.
Situation B: x = 10, y = 2. If so, then x³ + y³ = 10³ + 2³ = 1008.
Statement 2 is insufficient because it does not enable us to definitively answer the research issue.
Taken together, points 1 and 2
That x + y = 12 is the result of Statement 1.
Based on Statement 2, we know that xy = 8.
In this case, we have two separate 2-variable linear equations.
Since we could find values for x and y in this system, we would know the solution to the research question.
As a result, the merged claims are the adequate answer.
Correct option: C

Approach Solution 2:
We can solve this problem by using algebraic manipulation.
First, we note that (x + y)³ = x³ + y³ + 3xy(x + y). Therefore, x³ + y³ can be expressed as (x + y)³ - 3xy(x + y).
Statement 1 tells us that x + y = 12, but it does not provide any information about xy. Therefore, we cannot determine the value of x³ + y³ from statement 1 alone.
Statement 2 tells us that x - y = 8. We can manipulate this equation to solve for xy as follows:
x - y = 8
(x - y)² = 64
x² - 2xy + y² = 64
xy = (x² + y² - 64) / 2
Now we can substitute x + y = 12 and xy = (x² + y² - 64) / 2 into our expression for x³ + y³:
x³ + y³ = (x + y)³ - 3xy(x + y)
x³ + y³ = 12³ - 3 ((x² + y² - 64) / 2) (12)
x³ + y³ = 1728 - 18(x² + y² - 64)
x³ + y³ = 18 (64 - x² - y²) + 1728
We still need information about x² + y² to determine the value of x³ + y³
To solve for the values of x and y, we can add the two given equations:
(x + y) + (x - y) = 12 + 8
2x = 20
x = 10
Substituting x = 10 into either equation, we can solve for y:
x + y = 12
10 + y = 12
y = 2
Now that we know the values of x and y, we can find the value of x³ + y³:
x³ + y³ = 10³ + 2³
x³ + y³ = 1000 + 8
x³ + y³ = 1008
Therefore, the value of x³ + y³ is 1008.
Correct option: C

Approach Solution 3:
By manipulating the algebra, we can find a solution to this issue.
To begin with, we observe that (x + y)³ = x³ + y³ + 3xy (x + y). As a result, (x + y)³ - 3xy (x + y) can be used to express x³ + y³.
Statement 1 states that x + y equals 12, yet it says nothing about xy. Because of this, we are unable to infer the value of x³ + y³ from assertion 1 alone.
According to Statement 2, x - y equals 8. In order to solve for xy, we can alter this equation as follows:
(x - y)² = 64 x² - 2xy + y² = 64 x - y = 8
xy = (x² + y² - 64) / 2
In our expression for x³ + y³, we can now insert x + y = 12 and xy = (x² + y² - 64) / 2:
x³ + y³ = (x + y)
x + y = 3 - 3xy (x + y) = 12
x³ + y³ = 1728 - 18 (x² + y² - 64) = 3 - 3 ((x² + y² - 64) / 2) (12)
x³ + y³ = 18 (64 - x² - y²) + 1728
To ascertain the value of x³ + y³, we still want knowledge of x² + y².
We may add the two equations above to find the values of x and y:
(x + y) + (x - y) = 12 + 8
2x = 20
x = 10
We can find y by substituting x = 10 into either equation:
x + y = 12
10 + y = 12
y = 2
With the values of x and y, we can calculate the value of x³ + y³ as follows:
x³ + y³ = 10³ + 2³, 1000 + 8 and 1008 respectively.
Hence, the result of x³ + y³ is 1008.
Correct option: C

“What is the value of x³ + y³? GMAT data sufficiency" - is a topic of the GMAT data sufficiency section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.

To understand GMAT data sufficiency questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude in reasoning and mathematics. The GMAT Quantitative test's problem-solving phase consists of a question and two statements. By using mathematics to answer the question, the candidate must select the appropriate response among five choices which states which statement is sufficient to answer the problem. The data sufficiency section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.

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