Comprehension
The following data sufficiency problems consist of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of counterclockwise), you must indicate whether:
(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.
NUMBERS: All numbers used are real numbers.
FIGURES: A figure accompanying a data sufficiency problem will conform to the information given in the question, but will not necessarily conform to the additional information given in statements (1) and (2). Lines shown as straight can be assumed to be straight, and lines that appear jagged can also be assumed to be straight. You may assume that the position of points, angles, regions, etc., exist in the order shown and that angle measures are greater than zero. All figures lie in a plane unless otherwise indicated.
NOTE: In data sufficiency problems that ask for the value of a quantity, the data given in the statements are sufficient only when it is possible to determine exactly one numerical value for the quantity.
(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.
NUMBERS: All numbers used are real numbers.
FIGURES: A figure accompanying a data sufficiency problem will conform to the information given in the question, but will not necessarily conform to the additional information given in statements (1) and (2). Lines shown as straight can be assumed to be straight, and lines that appear jagged can also be assumed to be straight. You may assume that the position of points, angles, regions, etc., exist in the order shown and that angle measures are greater than zero. All figures lie in a plane unless otherwise indicated.
NOTE: In data sufficiency problems that ask for the value of a quantity, the data given in the statements are sufficient only when it is possible to determine exactly one numerical value for the quantity.
Question: 1
A certain salad dressing made only of oil and vinegar is premixed and sold in the grocery store. What is the ratio of oil to vinegar in the mix?
(1)An 18.6-ounce bottle of the salad dressing contains 12.4 ounces of vinegar.
(2)In a 32-ounce bottle of the salad dressing, there is half as much oil as there is vinegar.
A certain salad dressing made only of oil and vinegar is premixed and sold in the grocery store. What is the ratio of oil to vinegar in the mix?
(1)An 18.6-ounce bottle of the salad dressing contains 12.4 ounces of vinegar.
(2)In a 32-ounce bottle of the salad dressing, there is half as much oil as there is vinegar.
(1)An 18.6-ounce bottle of the salad dressing contains 12.4 ounces of vinegar.
(2)In a 32-ounce bottle of the salad dressing, there is half as much oil as there is vinegar.
Show Hint
In data sufficiency problems, always assess each statement independently before combining them.
Updated On: Oct 3, 2025
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- EACH statement ALONE is sufficient.
- Statements (1) and (2) TOGETHER are NOT sufficient.
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The Correct Option is D
Solution and Explanation
Step 1: Analyze Statement (1).
Statement (1) tells us that in an 18.6-ounce bottle of salad dressing, 12.4 ounces are vinegar. This means: \[ \text{Amount of oil} = 18.6 - 12.4 = 6.2 \, \text{ounces} \] Thus, the ratio of oil to vinegar is: \[ \frac{6.2}{12.4} = 0.5 \] Statement (1) is sufficient to answer the question. Step 2: Analyze Statement (2).
Statement (2) tells us that in a 32-ounce bottle of salad dressing, there is half as much oil as there is vinegar. This means: \[ \text{Amount of oil} = \frac{1}{2} \times \text{Amount of vinegar} \] This allows us to determine the ratio of oil to vinegar is 1:2, which is sufficient to answer the question. Final Answer: \[ \boxed{D} \]
Statement (1) tells us that in an 18.6-ounce bottle of salad dressing, 12.4 ounces are vinegar. This means: \[ \text{Amount of oil} = 18.6 - 12.4 = 6.2 \, \text{ounces} \] Thus, the ratio of oil to vinegar is: \[ \frac{6.2}{12.4} = 0.5 \] Statement (1) is sufficient to answer the question. Step 2: Analyze Statement (2).
Statement (2) tells us that in a 32-ounce bottle of salad dressing, there is half as much oil as there is vinegar. This means: \[ \text{Amount of oil} = \frac{1}{2} \times \text{Amount of vinegar} \] This allows us to determine the ratio of oil to vinegar is 1:2, which is sufficient to answer the question. Final Answer: \[ \boxed{D} \]
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Question: 2
At a certain pancake festival with 600 attendees in which every attendee ate at least 1 pancake, how many people had only 1 pancake?
(1) At the festival there were 1,200 pancakes served, and no person had more than 3 pancakes.
(2) Seventy-two percent of the attendees at the festival had 2 or more pancakes.
At a certain pancake festival with 600 attendees in which every attendee ate at least 1 pancake, how many people had only 1 pancake?
(1) At the festival there were 1,200 pancakes served, and no person had more than 3 pancakes.
(2) Seventy-two percent of the attendees at the festival had 2 or more pancakes.
(2) Seventy-two percent of the attendees at the festival had 2 or more pancakes.
Show Hint
When a percentage is given, always calculate the exact number of people or items that the percentage applies to.
Updated On: Oct 3, 2025
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- EACH statement ALONE is sufficient.
- Statements (1) and (2) TOGETHER are NOT sufficient.
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The Correct Option is B
Solution and Explanation
Step 1: Analyze Statement (1).
We know that 1,200 pancakes were served and that no person had more than 3 pancakes. Since every person ate at least one pancake, and there were 600 people, the average number of pancakes per person is: \[ \frac{1200}{600} = 2 \] This means the average number of pancakes per person is 2, but it is not enough to determine how many people had exactly 1 pancake. Thus, Statement (1) alone is insufficient. Step 2: Analyze Statement (2).
Seventy-two percent of the 600 attendees had 2 or more pancakes, so: \[ 72% \times 600 = 432 \text{ people had 2 or more pancakes.} \] This implies: \[ 600 - 432 = 168 \text{ people had exactly 1 pancake.} \] Statement (2) is sufficient to determine the answer. Final Answer: \[ \boxed{B} \]
We know that 1,200 pancakes were served and that no person had more than 3 pancakes. Since every person ate at least one pancake, and there were 600 people, the average number of pancakes per person is: \[ \frac{1200}{600} = 2 \] This means the average number of pancakes per person is 2, but it is not enough to determine how many people had exactly 1 pancake. Thus, Statement (1) alone is insufficient. Step 2: Analyze Statement (2).
Seventy-two percent of the 600 attendees had 2 or more pancakes, so: \[ 72% \times 600 = 432 \text{ people had 2 or more pancakes.} \] This implies: \[ 600 - 432 = 168 \text{ people had exactly 1 pancake.} \] Statement (2) is sufficient to determine the answer. Final Answer: \[ \boxed{B} \]
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Question: 3
A rectangle is equal in area to a square with sides of length 12. Is the diagonal of the rectangle greater in length than 20?
(1) The rectangle has a length of 16.
(2) The rectangle has a width of 9.
A rectangle is equal in area to a square with sides of length 12. Is the diagonal of the rectangle greater in length than 20?
(1) The rectangle has a length of 16.
(2) The rectangle has a width of 9.
(2) The rectangle has a width of 9.
Show Hint
To determine the diagonal length, use the Pythagorean theorem when the length and width are known.
Updated On: Oct 3, 2025
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- EACH statement ALONE is sufficient.
- Statements (1) and (2) TOGETHER are NOT sufficient.
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The Correct Option is A
Solution and Explanation
Step 1: Analyze Statement (1).
The area of the square is: \[ 12^2 = 144 \] The area of the rectangle is also 144, and it has a length of 16. To find the width, we use the formula for area: \[ \text{Area} = \text{Length} \times \text{Width} \quad \Rightarrow \quad 144 = 16 \times \text{Width} \] Solving for the width: \[ \text{Width} = \frac{144}{16} = 9 \] Now, we can calculate the diagonal of the rectangle using the Pythagorean theorem: \[ \text{Diagonal} = \sqrt{16^2 + 9^2} = \sqrt{256 + 81} = \sqrt{337} \approx 18.36 \] Thus, the diagonal is less than 20, so the answer is "no." Step 2: Analyze Statement (2).
Statement (2) alone doesn't provide enough information to determine the length of the diagonal. Final Answer: \[ \boxed{A} \]
The area of the square is: \[ 12^2 = 144 \] The area of the rectangle is also 144, and it has a length of 16. To find the width, we use the formula for area: \[ \text{Area} = \text{Length} \times \text{Width} \quad \Rightarrow \quad 144 = 16 \times \text{Width} \] Solving for the width: \[ \text{Width} = \frac{144}{16} = 9 \] Now, we can calculate the diagonal of the rectangle using the Pythagorean theorem: \[ \text{Diagonal} = \sqrt{16^2 + 9^2} = \sqrt{256 + 81} = \sqrt{337} \approx 18.36 \] Thus, the diagonal is less than 20, so the answer is "no." Step 2: Analyze Statement (2).
Statement (2) alone doesn't provide enough information to determine the length of the diagonal. Final Answer: \[ \boxed{A} \]
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Question: 4
A type of candy comes in two flavors, sweet and sour, and in two colors, yellow and green. The color and flavor of the individual pieces of candy are not related. If in a certain box of this candy, \(\frac{1}{4}\) of the yellow pieces and \(\frac{5}{7}\) of the green pieces are sour, what is the ratio of the number of yellow pieces to the number of green pieces in the box?
(1)In the box, the number of sweet yellow pieces is equal to the number of sour green pieces.
(2)In the box, the number of green pieces is two less than the number of yellow pieces.
A type of candy comes in two flavors, sweet and sour, and in two colors, yellow and green. The color and flavor of the individual pieces of candy are not related. If in a certain box of this candy, \(\frac{1}{4}\) of the yellow pieces and \(\frac{5}{7}\) of the green pieces are sour, what is the ratio of the number of yellow pieces to the number of green pieces in the box?
(1)In the box, the number of sweet yellow pieces is equal to the number of sour green pieces.
(2)In the box, the number of green pieces is two less than the number of yellow pieces.
(2)In the box, the number of green pieces is two less than the number of yellow pieces.
Show Hint
When solving ratio problems, express each part of the ratio as a variable and solve using basic algebra.
Updated On: Oct 3, 2025
- Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
- Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
- BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
- EACH statement ALONE is sufficient.
- Statements (1) and (2) TOGETHER are NOT sufficient.
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The Correct Option is A
Solution and Explanation
Step 1: Analyze Statement (1).
In this statement, it is given that the number of sweet yellow pieces is equal to the number of sour green pieces. Let the total number of yellow pieces be \( y \), and the total number of green pieces be \( g \). Thus: \[ \frac{1}{4}y = \frac{5}{7}g \] Solving for \( y \) and \( g \): \[ y = \frac{5}{7} \times 4g = \frac{20}{7}g \] Thus, the ratio of yellow pieces to green pieces is: \[ \frac{y}{g} = \frac{\frac{20}{7}g}{g} = \frac{20}{7} \] Statement (1) alone is sufficient to answer the question. Final Answer: \[ \boxed{A} \]
In this statement, it is given that the number of sweet yellow pieces is equal to the number of sour green pieces. Let the total number of yellow pieces be \( y \), and the total number of green pieces be \( g \). Thus: \[ \frac{1}{4}y = \frac{5}{7}g \] Solving for \( y \) and \( g \): \[ y = \frac{5}{7} \times 4g = \frac{20}{7}g \] Thus, the ratio of yellow pieces to green pieces is: \[ \frac{y}{g} = \frac{\frac{20}{7}g}{g} = \frac{20}{7} \] Statement (1) alone is sufficient to answer the question. Final Answer: \[ \boxed{A} \]
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