GMAT Data Sufficiency Challenge: Master the 5 Answer Choices
GMAT Data Sufficiency Practice Test
15 Questions • 30 Minutes
Data Sufficiency Directions
Each data sufficiency problem consists of a question and two statements, labeled (1) and (2), that give data. You have to decide whether the data given in the statements are sufficient for answering the question.
You must indicate whether:
A: Statement (1) ALONE is sufficient
B: Statement (2) ALONE is sufficient
C: BOTH statements TOGETHER are sufficient
D: EACH statement ALONE is sufficient
E: Statements TOGETHER are NOT sufficient
Question 1/15
30:00
Question 1
What is the value of |x|?
(1) x = -|x|
(2) x² = 4
Correct Answer: B (Statement 2 alone is sufficient)
Step-by-Step Explanation:
1. We need to find the value of |x|
2. Statement (1): x = -|x| means x is negative or zero, but doesn't give a specific value for |x|. Not sufficient.
3. Statement (2): x² = 4 means x = 2 or x = -2, so |x| = 2 in either case. Sufficient.
Therefore, statement (2) alone is sufficient, but statement (1) alone is not.
Step-by-Step Explanation:
1. We need to find the value of |x|
2. Statement (1): x = -|x| means x is negative or zero, but doesn't give a specific value for |x|. Not sufficient.
3. Statement (2): x² = 4 means x = 2 or x = -2, so |x| = 2 in either case. Sufficient.
Therefore, statement (2) alone is sufficient, but statement (1) alone is not.
Strategy: For absolute value questions, remember that |x| is always non-negative. x² = a² means |x| = |a|.
Question 2
What percent of a group of people are women with red hair?
(1) Of the women in the group, 5 percent have red hair.
(2) Of the men in the group, 10 percent have red hair.
Correct Answer: E (Statements together are not sufficient)
Step-by-Step Explanation:
1. We need the percentage of the entire group that are women with red hair
2. Statement (1) gives the percentage of women who have red hair, but we don't know what percentage of the group are women
3. Statement (2) gives information about men with red hair, which is irrelevant to women with red hair
4. Even together, we don't know the gender distribution of the group
Therefore, neither statement alone or together is sufficient.
Step-by-Step Explanation:
1. We need the percentage of the entire group that are women with red hair
2. Statement (1) gives the percentage of women who have red hair, but we don't know what percentage of the group are women
3. Statement (2) gives information about men with red hair, which is irrelevant to women with red hair
4. Even together, we don't know the gender distribution of the group
Therefore, neither statement alone or together is sufficient.
Strategy: For percentage of subgroup problems, you need information about both the subgroup percentage and the proportion of the total.
Question 3
In a certain class, one student is to be selected at random to read. What is the probability that a boy will read?
(1) Two-thirds of the students in the class are boys.
(2) Ten of the students in the class are girls.
Correct Answer: A (Statement 1 alone is sufficient)
Step-by-Step Explanation:
1. The probability a boy is selected = number of boys / total students
2. Statement (1): If 2/3 of students are boys, then probability = 2/3. Sufficient.
3. Statement (2): Knowing there are 10 girls doesn't tell us how many boys there are or the total. Not sufficient.
Therefore, statement (1) alone is sufficient, but statement (2) alone is not.
Step-by-Step Explanation:
1. The probability a boy is selected = number of boys / total students
2. Statement (1): If 2/3 of students are boys, then probability = 2/3. Sufficient.
3. Statement (2): Knowing there are 10 girls doesn't tell us how many boys there are or the total. Not sufficient.
Therefore, statement (1) alone is sufficient, but statement (2) alone is not.
Strategy: For probability questions, if you know the proportion of the desired outcome, that's sufficient to determine probability.
Question 4
If n is an integer, is n + 1 odd?
(1) n + 2 is an even integer.
(2) n - 1 is an odd integer.
Correct Answer: D (Each statement alone is sufficient)
Step-by-Step Explanation:
1. n + 1 is odd if and only if n is even
2. Statement (1): n + 2 is even ⇒ n is even. Sufficient.
3. Statement (2): n - 1 is odd ⇒ n is even. Sufficient.
Therefore, each statement alone is sufficient.
Step-by-Step Explanation:
1. n + 1 is odd if and only if n is even
2. Statement (1): n + 2 is even ⇒ n is even. Sufficient.
3. Statement (2): n - 1 is odd ⇒ n is even. Sufficient.
Therefore, each statement alone is sufficient.
Strategy: For odd/even questions, remember that even ± even = even, odd ± odd = even, and even ± odd = odd.
Question 5
If 5 - 6/x = x, how many possible values does x have?
(1) x > 0
(2) x is an integer
Correct Answer: C (Both statements together are sufficient)
Step-by-Step Explanation:
1. Start with: 5 - 6/x = x
2. Multiply both sides by x: 5x - 6 = x²
3. Rearrange: x² - 5x + 6 = 0
4. Factor: (x - 2)(x - 3) = 0
5. Solutions: x = 2 or x = 3
6. Statement (1): x > 0 - both solutions satisfy this, so we still have 2 possible values
7. Statement (2): x is an integer - both solutions are integers, so we still have 2 possible values
8. Together: x > 0 and x is an integer - still 2 possible values (2 and 3)
9. Wait, the question asks "how many possible values does x have?"
10. With both statements, we know there are exactly 2 possible values
Therefore, both statements together are sufficient.
Step-by-Step Explanation:
1. Start with: 5 - 6/x = x
2. Multiply both sides by x: 5x - 6 = x²
3. Rearrange: x² - 5x + 6 = 0
4. Factor: (x - 2)(x - 3) = 0
5. Solutions: x = 2 or x = 3
6. Statement (1): x > 0 - both solutions satisfy this, so we still have 2 possible values
7. Statement (2): x is an integer - both solutions are integers, so we still have 2 possible values
8. Together: x > 0 and x is an integer - still 2 possible values (2 and 3)
9. Wait, the question asks "how many possible values does x have?"
10. With both statements, we know there are exactly 2 possible values
Therefore, both statements together are sufficient.
Strategy: For "how many values" questions, you need enough information to determine the exact number of solutions.
Question 6
What is the cube root of w?
(1) The 5th root of w is 64.
(2) The 15th root of w is 4.
Correct Answer: D (Each statement alone is sufficient)
Step-by-Step Explanation:
1. Cube root of w = w^(1/3)
2. Statement (1): w^(1/5) = 64 ⇒ w = 64^5
3. Then w^(1/3) = (64^5)^(1/3) = 64^(5/3) = (4^3)^(5/3) = 4^5 = 1024
4. Statement (1) alone is sufficient
5. Statement (2): w^(1/15) = 4 ⇒ w = 4^15
6. Then w^(1/3) = (4^15)^(1/3) = 4^5 = 1024
7. Statement (2) alone is sufficient
Therefore, each statement alone is sufficient.
Step-by-Step Explanation:
1. Cube root of w = w^(1/3)
2. Statement (1): w^(1/5) = 64 ⇒ w = 64^5
3. Then w^(1/3) = (64^5)^(1/3) = 64^(5/3) = (4^3)^(5/3) = 4^5 = 1024
4. Statement (1) alone is sufficient
5. Statement (2): w^(1/15) = 4 ⇒ w = 4^15
6. Then w^(1/3) = (4^15)^(1/3) = 4^5 = 1024
7. Statement (2) alone is sufficient
Therefore, each statement alone is sufficient.
Strategy: For exponent problems, use the property that (a^m)^n = a^(mn). Convert roots to fractional exponents.
Question 7
If n + k = m, what is the value of k?
(1) n = 10
(2) m + 10 = n
Correct Answer: B (Statement 2 alone is sufficient)
Step-by-Step Explanation:
1. We have n + k = m, and we need to find k
2. Statement (1): n = 10, so 10 + k = m, but we don't know m. Not sufficient.
3. Statement (2): m + 10 = n
4. From original equation: n + k = m
5. Substitute n from statement (2): (m + 10) + k = m
6. Simplify: m + 10 + k = m
7. Subtract m from both sides: 10 + k = 0, so k = -10
8. Statement (2) alone is sufficient
Therefore, statement (2) alone is sufficient, but statement (1) alone is not.
Step-by-Step Explanation:
1. We have n + k = m, and we need to find k
2. Statement (1): n = 10, so 10 + k = m, but we don't know m. Not sufficient.
3. Statement (2): m + 10 = n
4. From original equation: n + k = m
5. Substitute n from statement (2): (m + 10) + k = m
6. Simplify: m + 10 + k = m
7. Subtract m from both sides: 10 + k = 0, so k = -10
8. Statement (2) alone is sufficient
Therefore, statement (2) alone is sufficient, but statement (1) alone is not.
Strategy: For equation solving, substitute known values and simplify to solve for the unknown variable.
Question 8
Is x a negative number?
(1) 9x > 10x
(2) x + 3 is positive.
Correct Answer: A (Statement 1 alone is sufficient)
Step-by-Step Explanation:
1. Statement (1): 9x > 10x
2. Subtract 9x from both sides: 0 > x, so x is negative. Sufficient.
3. Statement (2): x + 3 > 0 ⇒ x > -3
4. x could be negative (if -3 < x < 0) or positive. Not sufficient.
Therefore, statement (1) alone is sufficient, but statement (2) alone is not.
Step-by-Step Explanation:
1. Statement (1): 9x > 10x
2. Subtract 9x from both sides: 0 > x, so x is negative. Sufficient.
3. Statement (2): x + 3 > 0 ⇒ x > -3
4. x could be negative (if -3 < x < 0) or positive. Not sufficient.
Therefore, statement (1) alone is sufficient, but statement (2) alone is not.
Strategy: For inequality questions, perform the same operations on both sides, but remember to flip the inequality if multiplying/dividing by a negative.
Question 9
If i and j are integers, is i + j an even integer?
(1) i < 10
(2) i = j
Correct Answer: B (Statement 2 alone is sufficient)
Step-by-Step Explanation:
1. i + j is even if both are even or both are odd
2. Statement (1): i < 10 tells us nothing about j or whether i and j have same parity. Not sufficient.
3. Statement (2): i = j ⇒ i + j = 2i, which is always even. Sufficient.
Therefore, statement (2) alone is sufficient, but statement (1) alone is not.
Step-by-Step Explanation:
1. i + j is even if both are even or both are odd
2. Statement (1): i < 10 tells us nothing about j or whether i and j have same parity. Not sufficient.
3. Statement (2): i = j ⇒ i + j = 2i, which is always even. Sufficient.
Therefore, statement (2) alone is sufficient, but statement (1) alone is not.
Strategy: For even/odd questions, if two numbers are equal, their sum is always even (2 × number).
Question 10
If a < x < b and c < y < d, is x < y?
(1) a < c
(2) b < c
Correct Answer: B (Statement 2 alone is sufficient)
Step-by-Step Explanation:
1. We know: a < x < b and c < y < d
2. Statement (1): a < c
3. This tells us the lower bound of x is less than the lower bound of y, but x could still be greater than y
4. Not sufficient
5. Statement (2): b < c
6. Since x < b and b < c and c < y, we have x < b < c < y, so x < y
7. Statement (2) alone is sufficient
Therefore, statement (2) alone is sufficient, but statement (1) alone is not.
Step-by-Step Explanation:
1. We know: a < x < b and c < y < d
2. Statement (1): a < c
3. This tells us the lower bound of x is less than the lower bound of y, but x could still be greater than y
4. Not sufficient
5. Statement (2): b < c
6. Since x < b and b < c and c < y, we have x < b < c < y, so x < y
7. Statement (2) alone is sufficient
Therefore, statement (2) alone is sufficient, but statement (1) alone is not.
Strategy: For inequality chains, if you can establish that the upper bound of one variable is less than the lower bound of another, then the first variable is always less than the second.
Question 11
If x and y are positive, is x/y greater than 1?
(1) xy > 1
(2) x - y > 0
Correct Answer: B (Statement 2 alone is sufficient)
Step-by-Step Explanation:
1. x/y > 1 means x > y (since y is positive)
2. Statement (1): xy > 1
3. This could be true with x < y (e.g., x=2, y=1) or x > y (e.g., x=3, y=1)
4. Not sufficient
5. Statement (2): x - y > 0 means x > y
6. Since x > y and y is positive, x/y > 1
7. Statement (2) alone is sufficient
Therefore, statement (2) alone is sufficient, but statement (1) alone is not.
Step-by-Step Explanation:
1. x/y > 1 means x > y (since y is positive)
2. Statement (1): xy > 1
3. This could be true with x < y (e.g., x=2, y=1) or x > y (e.g., x=3, y=1)
4. Not sufficient
5. Statement (2): x - y > 0 means x > y
6. Since x > y and y is positive, x/y > 1
7. Statement (2) alone is sufficient
Therefore, statement (2) alone is sufficient, but statement (1) alone is not.
Strategy: For fraction comparisons with positive numbers, x/y > 1 is equivalent to x > y.
Question 12
If x and y are integers, is xy even?
(1) x = y + 1
(2) x/y is an even integer.
Correct Answer: D (Each statement alone is sufficient)
Step-by-Step Explanation:
1. xy is even if at least one of x or y is even
2. Statement (1): x = y + 1
3. Consecutive integers: one must be even
4. So xy must be even
5. Statement (1) alone is sufficient
6. Statement (2): x/y is an even integer
7. Let x/y = 2k for some integer k
8. Then x = 2ky
9. So x is even (since it's 2 times an integer)
10. Therefore xy is even
11. Statement (2) alone is sufficient
Therefore, each statement alone is sufficient.
Step-by-Step Explanation:
1. xy is even if at least one of x or y is even
2. Statement (1): x = y + 1
3. Consecutive integers: one must be even
4. So xy must be even
5. Statement (1) alone is sufficient
6. Statement (2): x/y is an even integer
7. Let x/y = 2k for some integer k
8. Then x = 2ky
9. So x is even (since it's 2 times an integer)
10. Therefore xy is even
11. Statement (2) alone is sufficient
Therefore, each statement alone is sufficient.
Strategy: For even/odd questions, remember that a product is even if at least one factor is even. Consecutive integers always include an even number.
Question 13
If r and s are the roots of the equation x² + bx + c = 0, where b and c are constants, is rs < 0?
(1) b < 0
(2) c < 0
Correct Answer: B (Statement 2 alone is sufficient)
Step-by-Step Explanation:
1. For quadratic x² + bx + c = 0 with roots r and s:
- Sum of roots: r + s = -b
- Product of roots: rs = c
2. Statement (1): b < 0 means -b > 0, so r + s > 0
3. This tells us about the sum, but not the product. Not sufficient.
4. Statement (2): c < 0 means rs < 0
5. Statement (2) alone is sufficient
Therefore, statement (2) alone is sufficient, but statement (1) alone is not.
Step-by-Step Explanation:
1. For quadratic x² + bx + c = 0 with roots r and s:
- Sum of roots: r + s = -b
- Product of roots: rs = c
2. Statement (1): b < 0 means -b > 0, so r + s > 0
3. This tells us about the sum, but not the product. Not sufficient.
4. Statement (2): c < 0 means rs < 0
5. Statement (2) alone is sufficient
Therefore, statement (2) alone is sufficient, but statement (1) alone is not.
Strategy: For quadratic equations, remember Vieta's formulas: sum of roots = -b, product of roots = c.
Question 14
If x is an integer, is 9^x + 9^(-x) = b?
(1) 3^x + 3^(-x) = √(b + 2)
(2) x > 0
Correct Answer: A (Statement 1 alone is sufficient)
Step-by-Step Explanation:
1. We need to determine if 9^x + 9^(-x) = b
2. Statement (1): 3^x + 3^(-x) = √(b + 2)
3. Square both sides: (3^x + 3^(-x))² = b + 2
4. Expand: 9^x + 2 + 9^(-x) = b + 2
5. Subtract 2: 9^x + 9^(-x) = b
6. This is exactly what we're trying to prove
7. Statement (1) alone is sufficient
8. Statement (2): x > 0 doesn't give us information about b. Not sufficient.
Therefore, statement (1) alone is sufficient, but statement (2) alone is not.
Step-by-Step Explanation:
1. We need to determine if 9^x + 9^(-x) = b
2. Statement (1): 3^x + 3^(-x) = √(b + 2)
3. Square both sides: (3^x + 3^(-x))² = b + 2
4. Expand: 9^x + 2 + 9^(-x) = b + 2
5. Subtract 2: 9^x + 9^(-x) = b
6. This is exactly what we're trying to prove
7. Statement (1) alone is sufficient
8. Statement (2): x > 0 doesn't give us information about b. Not sufficient.
Therefore, statement (1) alone is sufficient, but statement (2) alone is not.
Strategy: For exponential equations, look for relationships between different bases and use algebraic manipulation.
Question 15
If n is a positive integer, is (1/10)^n < 0.01?
(1) n > 2
(2) (1/10)^(n-1) < 0.1
Correct Answer: D (Each statement alone is sufficient)
Step-by-Step Explanation:
1. (1/10)^n < 0.01 means 10^(-n) < 10^(-2), so -n < -2, so n > 2
2. Statement (1): n > 2 directly answers the question. Sufficient.
3. Statement (2): (1/10)^(n-1) < 0.1 means 10^(-(n-1)) < 10^(-1)
4. So -(n-1) < -1, so -n + 1 < -1, so -n < -2, so n > 2
5. Statement (2) alone is sufficient
Therefore, each statement alone is sufficient.
Step-by-Step Explanation:
1. (1/10)^n < 0.01 means 10^(-n) < 10^(-2), so -n < -2, so n > 2
2. Statement (1): n > 2 directly answers the question. Sufficient.
3. Statement (2): (1/10)^(n-1) < 0.1 means 10^(-(n-1)) < 10^(-1)
4. So -(n-1) < -1, so -n + 1 < -1, so -n < -2, so n > 2
5. Statement (2) alone is sufficient
Therefore, each statement alone is sufficient.
Strategy: For exponential inequalities with the same base, compare the exponents directly.
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