This page provides detailed solutions and explanations for the GMAT Data Sufficiency questions in Sample 1. Each solution explains why statements are sufficient or insufficient, following official GMAT logic.
Let \(L\) be the length and \(W\) the width of the rectangle. The perimeter \(P\) is given by:
\[ P = 2L + 2W = 120 \]This equation has two unknowns and cannot be solved without additional information. Statement (2) gives \(W = 20\), allowing us to find \(L\).
Both statements are required. Answer: C
To determine \(x\), the values of \(t\), \(z\), and \(w\) are needed.
From statement (2):
\[ z = 2w + 1 \Rightarrow -5 = 2w + 1 \Rightarrow w = -3 \]From statement (1):
\[ y = 2t - 1 \Rightarrow 3 = 2t - 1 \Rightarrow t = 2 \]Both statements are necessary to determine \(x\).
Answer: C
The area of a right triangle is:
\[ \text{Area} = \frac{1}{2}(\text{leg}_1)(\text{leg}_2) \]Since the triangle is isosceles, both legs are equal. Statement (1) alone gives the leg length, so the area can be found.
Using statement (2), let each leg be \(x\). By the Pythagorean theorem:
\[ x^2 + x^2 = (10\sqrt{2})^2 \]This allows computation of the area as well.
Answer: D
Statement (1) implies \(x = y\) or \(x = -y\), which is insufficient. Statement (2) implies \(x\) and \(y\) have the same sign.
Together, the statements imply \(x = y\).
Answer: C
Let \(x\), \(y\), and \(z\) represent the numbers of Mathematics, Biology, and Geography books. We seek:
\[ x + y + z \]Given:
\[ z = x + 150 \]From statement (1):
\[ \frac{x}{y} = \frac{5}{7} \]From statement (2):
\[ \frac{y}{z} = \frac{7}{8} \]All three equations together allow determination of the total.
Answer: C
From statement (1):
\[ (x + y)^2 = 16 \Rightarrow x^2 + y^2 + 2xy = 16 \]Statement (2) gives:
\[ xy = 2 \]Substituting allows computation of \(x^2 + y^2\).
Answer: C
There are two unknowns and two linear equations. Both statements are required to solve for \(x\).
Answer: C
The type of quadrilateral is not specified. Area and one side length are insufficient to determine the perimeter.
Answer: E
Since \(m\) and \(n\) are positive:
\[ m + n = 14 \]And since \(m < n\):
\[ n - m = 2 \]Multiplying:
\[ (m + n)(n - m) = n^2 - m^2 = 28 \]Thus, \(m^2 - n^2\) is determined only when both statements are used.
Answer: C
Statement (1):
\[ S = 4\pi r^2 \Rightarrow r \text{ can be found} \]Statement (2) directly gives \(r = 10\). In both cases, the volume:
\[ V = \frac{4}{3}\pi r^3 \]can be calculated.
Answer: D