Below is a set of 10 GMAT-style Data Sufficiency questions with full step-by-step solutions merged directly into each section. Each question follows the official GMAT format to help you master statement evaluation.
How to Answer Data Sufficiency Questions
Each problem consists of a question and two statements, labeled (1) and (2).
- 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 alone is sufficient.
- D. Each statement alone is sufficient.
- E. The two statements together are not sufficient.
Practice Questions with Solutions
Question 1: Find the length of the rectangle.
Statements:
(1) The perimeter of the rectangle is \(120\).
(2) The width of the rectangle is \(20\).
Solution & Explanation:
Let \(L\) be the length and \(W\) the width of the rectangle. The perimeter \(P\) is given by:
\[ P = 2L + 2W = 120 \implies L + W = 60 \]This equation has two unknowns and cannot be solved without additional information. Statement (2) gives \(W = 20\), allowing us to find \(L\):
\[ L + 20 = 60 \implies L = 40 \]Both statements are required. Answer: C
Question 2: Find the value of \(x\) given \(x = 2t - z + 5w\), \(y = 2t - 1\), \(z = 2w + 1\).
Statements:
(1) \(y = 3\)
(2) \(z = -5\)
Solution & Explanation:
To determine \(x\), the values of \(t\), \(z\), and \(w\) are needed. From statement (2):
\[ z = 2w + 1 \implies -5 = 2w + 1 \implies 2w = -6 \implies w = -3 \]From statement (1):
\[ y = 2t - 1 \implies 3 = 2t - 1 \implies 2t = 4 \implies t = 2 \]Both statements are necessary to determine \(x\). Substituting \(t = 2\), \(z = -5\), and \(w = -3\) into the equation for \(x\):
\[ x = 2(2) - (-5) + 5(-3) = 4 + 5 - 15 = -6 \]Answer: C
Question 3: Find the area of an isosceles right triangle.
Statements:
(1) The length of one leg is \(10\) cm.
(2) The length of the hypotenuse is \(10\sqrt{2}\).
Solution & Explanation:
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:
\[ \text{Area} = \frac{1}{2}(10)(10) = 50 \]Using statement (2), let each leg be \(x\). By the Pythagorean theorem:
\[ x^2 + x^2 = (10\sqrt{2})^2 \implies 2x^2 = 200 \implies x^2 = 100 \implies x = 10 \]This allows computation of the area as well (\(\frac{1}{2}(10)(10) = 50\)).
Answer: D
Question 4: Is \(x = y\)?
Statements:
(1) \(|x| = |y|\)
(2) \(xy > 0\)
Solution & Explanation:
Statement (1) implies \(x = y\) or \(x = -y\), which is insufficient because \(x\) and \(y\) could have opposite signs. Statement (2) implies \(x\) and \(y\) have the same sign (both positive or both negative), but not necessarily equal (e.g., \(x=2, y=3\)), which is also insufficient.
Together, statement (1) means \(x = \pm y\), and statement (2) ensures that \(x\) and \(y\) share the same sign, eliminating the opposite sign case and leaving \(x = y\).
Answer: C
Question 5: Find total books in a library with Math, Biology, and Geography books.
Statements:
(1) The ratio of Mathematics to Biology books is \(5:7\).
(2) The ratio of Biology to Geography books is \(7:8\).
Additional Info: There are \(150\) more Geography books than Mathematics books.
Solution & Explanation:
Let \(x\), \(y\), and \(z\) represent the numbers of Mathematics, Biology, and Geography books. We seek the total:
\[ x + y + z \]Given:
\[ z = x + 150 \]From statement (1):
\[ \frac{x}{y} = \frac{5}{7} \implies y = \frac{7}{5}x \]From statement (2):
\[ \frac{y}{z} = \frac{7}{8} \implies z = \frac{8}{7}y = \frac{8}{7}\left(\frac{7}{5}x\right) = \frac{8}{5}x \]Equating the two expressions for \(z\):
\[ \frac{8}{5}x = x + 150 \implies \frac{3}{5}x = 150 \implies x = 250 \]Then \(y = 350\) and \(z = 400\), giving a total of \(1000\) books. All three equations/statements together are required.
Answer: C
Question 6: What is the value of \(x^2 + y^2\)?
Statements:
(1) \(x + y = 4\)
(2) \(x = \dfrac{2}{y}\)
Solution & Explanation:
From statement (1), squaring both sides:
\[ (x + y)^2 = 4^2 \implies x^2 + y^2 + 2xy = 16 \]Statement (2) gives \(xy = 2\). Substituting this into the expanded equation:
\[ x^2 + y^2 + 2(2) = 16 \implies x^2 + y^2 + 4 = 16 \implies x^2 + y^2 = 12 \]Answer: C
Question 7: What is the value of \(x\)?
Statements:
(1) \(2x + 0.5y = 7\)
(2) \(0.4x + 0.2y = 1.4\)
Solution & Explanation:
Statement (1) is a linear equation with two variables: \(2x + 0.5y = 7\). Statement (2) simplifies when multiplied by 5 to form an independent linear equation with different slopes, requiring both statements to uniquely determine \(x\) and \(y\).
Answer: C
Question 8: What is the perimeter of the quadrilateral?
Statements:
(1) The area of the quadrilateral is \(600\).
(2) One side of the quadrilateral is \(30\).
Solution & Explanation:
The type of quadrilateral is not specified. Area and one side length are insufficient to determine the perimeter, as different shapes can share the same area and side length.
Answer: E
Question 9: Value of \(m^2 - n^2\) where \(m\) and \(n\) are positive numbers with \(m < n\).
Statements:
(1) \(|m + n| = 14\)
(2) \(|m - n| = 2\)
Solution & Explanation:
Since \(m\) and \(n\) are positive, statement (1) gives \(m + n = 14\). Since \(m < n\), statement (2) gives \(n - m = 2 \implies m - n = -2\). Using the difference of squares formula \(m^2 - n^2 = (m - n)(m + n)\):
\[ m^2 - n^2 = (-2)(14) = -28 \]Thus, \(m^2 - n^2\) is determined only when both statements are used together.
Answer: C
Question 10: What is the volume of the sphere?
Statements:
(1) The surface area of the sphere is \(400\pi\).
(2) The radius of the sphere is \(10\).
Solution & Explanation:
Statement (1) gives surface area \(4\pi r^2 = 400\pi \implies r = 10\), which allows calculation of the volume. Statement (2) directly gives \(r = 10\). In both cases, the volume \(V = \frac{4}{3}\pi r^3\) can be calculated.
Answer: D