Comprehension
The following information about student enrollment in a certain small college.
DISTRIBUTION OF ENROLLMENT
BY CLASS AND SEX
(Total enrollment: 1,400)
PERCENT OF TOTAL ENROLLMENT MAJORING IN EACH OF THE FOLLOWING ACADEMIC AREAS
(No student is majoring in more than one area.)
Question: 1
The ratio of the number of male freshmen to the number of female sophomores is approximately
The ratio of the number of male freshmen to the number of female sophomores is approximately
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When a question asks for an "approximate" ratio, round the numbers to values that are easy to divide. This is usually sufficient to identify the correct multiple-choice option.
Updated On: Nov 27, 2025
- 2 to 1
- 3 to 1
- 3 to 2
- 4 to 1
- 5 to 3
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept:
This question asks for an approximate ratio between two values found in the "DISTRIBUTION OF ENROLLMENT" table.
Step 2: Detailed Explanation:
1. Find the required values from the table. - Number of male freshmen: Look at the row for "Freshmen" and the column for "Males". The value is 303. - Number of female sophomores: Look at the row for "Sophomores" and the column for "Females". The value is 109.
2. Form the ratio and approximate. - The ratio is 303 to 109, or \(\frac{303}{109}\). - To approximate this, we can round the numbers. 303 is very close to 300, and 109 is very close to 100. - The ratio is approximately \(\frac{300}{100} = 3\). - This corresponds to a ratio of 3 to 1. - For a more precise check, \(303 \div 109 \approx 2.78\). This value is much closer to 3 than to 2, 1.5, 4, or 1.67.
Step 3: Final Answer:
The ratio is approximately 3 to 1.
This question asks for an approximate ratio between two values found in the "DISTRIBUTION OF ENROLLMENT" table.
Step 2: Detailed Explanation:
1. Find the required values from the table. - Number of male freshmen: Look at the row for "Freshmen" and the column for "Males". The value is 303. - Number of female sophomores: Look at the row for "Sophomores" and the column for "Females". The value is 109.
2. Form the ratio and approximate. - The ratio is 303 to 109, or \(\frac{303}{109}\). - To approximate this, we can round the numbers. 303 is very close to 300, and 109 is very close to 100. - The ratio is approximately \(\frac{300}{100} = 3\). - This corresponds to a ratio of 3 to 1. - For a more precise check, \(303 \div 109 \approx 2.78\). This value is much closer to 3 than to 2, 1.5, 4, or 1.67.
Step 3: Final Answer:
The ratio is approximately 3 to 1.
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Question: 2
How many of the enrolled students are not majoring in humanities, social sciences, or physical sciences?
How many of the enrolled students are not majoring in humanities, social sciences, or physical sciences?
Show Hint
When working with percentages from a table that don't add up to 100%, the remaining percentage represents the "other" or "unlisted" category. Calculate this remaining percentage first before applying it to the total.
Updated On: Nov 27, 2025
- 87
- 122
- 182
- 230
- 322
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Concept:
This question requires us to use the second table ("PERCENT OF TOTAL ENROLLMENT") to find the number of students who fall outside the listed categories. These could be students with undeclared majors or majors in other fields not listed.
Step 2: Detailed Explanation:
1. Find the total percentage of students in the listed majors. - Humanities: 33% - Social Sciences: 30% - Physical Sciences: 24% - Total percentage in these majors = \(33% + 30% + 24% = 87%\).
2. Find the percentage of students NOT in these majors. - The total of all students is 100%. - Percentage not in these majors = \(100% - 87% = 13%\).
3. Calculate the number of students corresponding to this percentage. - The total enrollment is given as 1,400 students. - Number of students not in these majors = 13% of 1,400. \[ 0.13 \times 1400 = 13 \times \frac{1400}{100} = 13 \times 14 \] \[ 13 \times 14 = 13 \times (10 + 4) = 130 + 52 = 182 \] Step 3: Final Answer:
There are 182 students who are not majoring in one of the three listed academic areas.
This question requires us to use the second table ("PERCENT OF TOTAL ENROLLMENT") to find the number of students who fall outside the listed categories. These could be students with undeclared majors or majors in other fields not listed.
Step 2: Detailed Explanation:
1. Find the total percentage of students in the listed majors. - Humanities: 33% - Social Sciences: 30% - Physical Sciences: 24% - Total percentage in these majors = \(33% + 30% + 24% = 87%\).
2. Find the percentage of students NOT in these majors. - The total of all students is 100%. - Percentage not in these majors = \(100% - 87% = 13%\).
3. Calculate the number of students corresponding to this percentage. - The total enrollment is given as 1,400 students. - Number of students not in these majors = 13% of 1,400. \[ 0.13 \times 1400 = 13 \times \frac{1400}{100} = 13 \times 14 \] \[ 13 \times 14 = 13 \times (10 + 4) = 130 + 52 = 182 \] Step 3: Final Answer:
There are 182 students who are not majoring in one of the three listed academic areas.
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Question: 3
Which of the following can be inferred from the tables?
I. The number of males majoring in physical sciences is greater than the number of females majoring in that area.
II. Students majoring in either social sciences or physical sciences constitute more than 50 percent of the total enrollment.
III. The ratio of the number of males to the number of females in the senior class is less than 2 to 1.
Which of the following can be inferred from the tables?
I. The number of males majoring in physical sciences is greater than the number of females majoring in that area.
II. Students majoring in either social sciences or physical sciences constitute more than 50 percent of the total enrollment.
III. The ratio of the number of males to the number of females in the senior class is less than 2 to 1.
I. The number of males majoring in physical sciences is greater than the number of females majoring in that area.
II. Students majoring in either social sciences or physical sciences constitute more than 50 percent of the total enrollment.
III. The ratio of the number of males to the number of females in the senior class is less than 2 to 1.
Show Hint
Be wary of making assumptions about how data is distributed. If a table gives totals for two different categories (like sex and major), you cannot infer anything about the overlap between those categories unless a combined table is provided.
Updated On: Nov 27, 2025
- I only
- II only
- I and II
- I and III
- II and III
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The Correct Option is
Solution and Explanation
Step 1: Understanding the Concept:
This is an inference question. We must evaluate each statement to see if it can be proven true using only the data provided in the two tables.
Step 2: Detailed Explanation:
Statement I: The number of males majoring in physical sciences is greater than the number of females majoring in that area.
- Total physical science majors = 24% of 1400 = \(0.24 \times 1400 = 336\) students. - Total males = 860; Total females = 540. - The tables do not provide a breakdown of majors by sex. We cannot determine how the 336 physical science majors are split between males and females. It's possible that all 336 are male, or that all 336 are female (since 336 \textless 540). Since we cannot prove it, the statement cannot be inferred. - Therefore, Statement I is not necessarily true.
Statement II: Students majoring in either social sciences or physical sciences constitute more than 50 percent of the total enrollment.
- Percentage of social science majors = 30%. - Percentage of physical science majors = 24%. - Total percentage = \(30% + 24% = 54%\). - Since \(54% \textgreater 50%\), this statement is true. - Therefore, Statement II can be inferred.
Statement III: The ratio of the number of males to the number of females in the senior class is less than 2 to 1.
- From the first table, number of senior males = 160. - From the first table, number of senior females = 84. - The ratio is \(\frac{160}{84}\). - A ratio of 2 to 1 is equivalent to \(\frac{2}{1}\). - To compare, we can check if \(\frac{160}{84} \textless 2\). Multiplying by 84 gives \(160 \textless 2 \times 84\), which is \(160 \textless 168\). This is true. - Therefore, Statement III can be inferred.
Step 3: Final Answer:
Statements II and III can be inferred from the tables, but Statement I cannot. The correct option is (E).
This is an inference question. We must evaluate each statement to see if it can be proven true using only the data provided in the two tables.
Step 2: Detailed Explanation:
Statement I: The number of males majoring in physical sciences is greater than the number of females majoring in that area.
- Total physical science majors = 24% of 1400 = \(0.24 \times 1400 = 336\) students. - Total males = 860; Total females = 540. - The tables do not provide a breakdown of majors by sex. We cannot determine how the 336 physical science majors are split between males and females. It's possible that all 336 are male, or that all 336 are female (since 336 \textless 540). Since we cannot prove it, the statement cannot be inferred. - Therefore, Statement I is not necessarily true.
Statement II: Students majoring in either social sciences or physical sciences constitute more than 50 percent of the total enrollment.
- Percentage of social science majors = 30%. - Percentage of physical science majors = 24%. - Total percentage = \(30% + 24% = 54%\). - Since \(54% \textgreater 50%\), this statement is true. - Therefore, Statement II can be inferred.
Statement III: The ratio of the number of males to the number of females in the senior class is less than 2 to 1.
- From the first table, number of senior males = 160. - From the first table, number of senior females = 84. - The ratio is \(\frac{160}{84}\). - A ratio of 2 to 1 is equivalent to \(\frac{2}{1}\). - To compare, we can check if \(\frac{160}{84} \textless 2\). Multiplying by 84 gives \(160 \textless 2 \times 84\), which is \(160 \textless 168\). This is true. - Therefore, Statement III can be inferred.
Step 3: Final Answer:
Statements II and III can be inferred from the tables, but Statement I cannot. The correct option is (E).
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Question: 4
How many students are either juniors or males or both?
How many students are either juniors or males or both?
Show Hint
For "A or B or both" questions with overlapping categories, using the direct addition method (Count all of A, then add the part of B that is NOT in A) can be faster and less prone to error than the formal inclusion-exclusion formula.
Updated On: Nov 27, 2025
- 678
- 766
- 948
- 1,130
- 1,312
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The Correct Option is C
Solution and Explanation
Step 1: Understanding the Concept:
This question asks for the total number of individuals in the union of two sets: the set of juniors and the set of males. We must be careful not to double-count the individuals who are in both sets (male juniors).
Step 2: Key Formula or Approach:
The principle of inclusion-exclusion for two sets states: \[ |A \cup B| = |A| + |B| - |A \cap B| \] Here, Set A = Juniors, and Set B = Males. Alternatively, we can add the total number of males to the number of juniors who are not male (i.e., female juniors).
Step 3: Detailed Explanation:
Method 1: Using Inclusion-Exclusion
1. Find the total number of juniors. - From the table: Male Juniors = 182, Female Juniors = 88. - Total Juniors = \(182 + 88 = 270\).
2. Find the total number of males. - From the table's total row: Total Males = 860.
3. Find the number of students who are both male and junior (the intersection). - From the table: Male Juniors = 182.
4. Apply the formula. - Number of (Juniors or Males) = (Total Juniors) + (Total Males) - (Male Juniors) - Number = \(270 + 860 - 182\) - Number = \(1130 - 182 = 948\)
Method 2: Direct Addition
The group "juniors or males" consists of two distinct groups: (1) all males, and (2) all juniors who are not male (i.e., female juniors).
Total number of males (this includes all male freshmen, sophomores, juniors, and seniors): 860
Number of female juniors: 88
Total = \(860 + 88 = 948\)
Step 4: Final Answer:
There are 948 students who are either juniors or males or both.
This question asks for the total number of individuals in the union of two sets: the set of juniors and the set of males. We must be careful not to double-count the individuals who are in both sets (male juniors).
Step 2: Key Formula or Approach:
The principle of inclusion-exclusion for two sets states: \[ |A \cup B| = |A| + |B| - |A \cap B| \] Here, Set A = Juniors, and Set B = Males. Alternatively, we can add the total number of males to the number of juniors who are not male (i.e., female juniors).
Step 3: Detailed Explanation:
Method 1: Using Inclusion-Exclusion
1. Find the total number of juniors. - From the table: Male Juniors = 182, Female Juniors = 88. - Total Juniors = \(182 + 88 = 270\).
2. Find the total number of males. - From the table's total row: Total Males = 860.
3. Find the number of students who are both male and junior (the intersection). - From the table: Male Juniors = 182.
4. Apply the formula. - Number of (Juniors or Males) = (Total Juniors) + (Total Males) - (Male Juniors) - Number = \(270 + 860 - 182\) - Number = \(1130 - 182 = 948\)
Method 2: Direct Addition
The group "juniors or males" consists of two distinct groups: (1) all males, and (2) all juniors who are not male (i.e., female juniors).
Total number of males (this includes all male freshmen, sophomores, juniors, and seniors): 860
Number of female juniors: 88
Total = \(860 + 88 = 948\)
Step 4: Final Answer:
There are 948 students who are either juniors or males or both.
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Question: 5
If the total enrollment is 12 percent greater than it was five years ago, what was the total enrollment five years ago?
If the total enrollment is 12 percent greater than it was five years ago, what was the total enrollment five years ago?
Show Hint
A common mistake in reverse percentage problems is to subtract 12% of the new amount (e.g., 12% of 1400). This is incorrect because the percentage increase was calculated based on the smaller, original number. Always set up the algebraic equation \( \text{Original} \times (1 + \text{percent increase}) = \text{Final} \) and solve for the "Original" value.
Updated On: Oct 1, 2025
- 1,180
- 1,192
- 1,220
- 1,232
- 1,250
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The Correct Option is
Solution and Explanation
Step 1: Understanding the Concept:
This is a reverse percentage problem. We are given the final value after a percentage increase (the current enrollment) and asked to find the original value (the enrollment five years ago).
Step 2: Key Formula or Approach:
Let \(E_0\) be the enrollment five years ago (the original value). The current enrollment, \(E_c\), is 12% greater. The relationship can be expressed with the formula: \[ E_c = E_0 + (0.12 \times E_0) = E_0 \times (1 + 0.12) = 1.12 \times E_0 \] To find the original enrollment, we rearrange the formula: \(E_0 = E_c / 1.12\).
Step 3: Detailed Explanation:
1. Identify the given values. - The current enrollment, \(E_c\), is 1,400 (from the information provided for the data interpretation questions). - The percentage increase is 12%, or 0.12.
2. Set up the equation. - Using the formula, we have: \(1400 = 1.12 \times E_0\).
3. Solve for the original enrollment, \(E_0\). - To isolate \(E_0\), we divide both sides by 1.12. \[ E_0 = \frac{1400}{1.12} \] - To make the division easier, remove the decimal by multiplying the numerator and denominator by 100: \[ E_0 = \frac{140000}{112} \] - Simplify the fraction. We can see both are divisible by 14: \(112 = 14 \times 8\) and \(140000 = 14 \times 10000\). \[ E_0 = \frac{10000}{8} \] - Perform the final division: \[ E_0 = 1250 \] Step 4: Final Answer:
The total enrollment five years ago was 1,250.
This is a reverse percentage problem. We are given the final value after a percentage increase (the current enrollment) and asked to find the original value (the enrollment five years ago).
Step 2: Key Formula or Approach:
Let \(E_0\) be the enrollment five years ago (the original value). The current enrollment, \(E_c\), is 12% greater. The relationship can be expressed with the formula: \[ E_c = E_0 + (0.12 \times E_0) = E_0 \times (1 + 0.12) = 1.12 \times E_0 \] To find the original enrollment, we rearrange the formula: \(E_0 = E_c / 1.12\).
Step 3: Detailed Explanation:
1. Identify the given values. - The current enrollment, \(E_c\), is 1,400 (from the information provided for the data interpretation questions). - The percentage increase is 12%, or 0.12.
2. Set up the equation. - Using the formula, we have: \(1400 = 1.12 \times E_0\).
3. Solve for the original enrollment, \(E_0\). - To isolate \(E_0\), we divide both sides by 1.12. \[ E_0 = \frac{1400}{1.12} \] - To make the division easier, remove the decimal by multiplying the numerator and denominator by 100: \[ E_0 = \frac{140000}{112} \] - Simplify the fraction. We can see both are divisible by 14: \(112 = 14 \times 8\) and \(140000 = 14 \times 10000\). \[ E_0 = \frac{10000}{8} \] - Perform the final division: \[ E_0 = 1250 \] Step 4: Final Answer:
The total enrollment five years ago was 1,250.
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Question: 6
If the ratio of the number of English books to the number of all other books on a bookshelf is 4 to 1, what percent of the books on the bookshelf are English books?
If the ratio of the number of English books to the number of all other books on a bookshelf is 4 to 1, what percent of the books on the bookshelf are English books?
Show Hint
Be careful to distinguish between a part-to-part ratio (A:B) and a part-to-whole ratio (A:Total). When given a part-to-part ratio, the first step is always to find the total (A+B) to create the whole.
Updated On: Oct 1, 2025
- 20%
- 25%
- 50%
- 75%
- 80%
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The Correct Option is
Solution and Explanation
Step 1: Understanding the Concept:
This question requires converting a part-to-part ratio into a part-to-whole fraction, and then expressing that fraction as a percentage.
Step 2: Detailed Explanation:
1. Understand the ratio. - The ratio of (English books) to (all other books) is 4 to 1. - This is a part-to-part ratio. It means for every 4 English books, there is 1 non-English book.
2. Calculate the 'whole'. - To find the percentage of English books, we need to know what fraction they represent of the total. - We can think in terms of "ratio units". Total ratio units = (units for English books) + (units for other books) = \(4 + 1 = 5\).
3. Form the part-to-whole fraction. - Out of a total of 5 "ratio units", 4 belong to English books. - The fraction of books that are English is \(\frac{\text{Part}}{\text{Whole}} = \frac{4}{5}\).
4. Convert the fraction to a percentage. - To convert a fraction to a percentage, we multiply by 100%. \[ \text{Percentage} = \frac{4}{5} \times 100% = 0.8 \times 100% = 80% \] Step 3: Final Answer:
80 percent of the books on the bookshelf are English books.
This question requires converting a part-to-part ratio into a part-to-whole fraction, and then expressing that fraction as a percentage.
Step 2: Detailed Explanation:
1. Understand the ratio. - The ratio of (English books) to (all other books) is 4 to 1. - This is a part-to-part ratio. It means for every 4 English books, there is 1 non-English book.
2. Calculate the 'whole'. - To find the percentage of English books, we need to know what fraction they represent of the total. - We can think in terms of "ratio units". Total ratio units = (units for English books) + (units for other books) = \(4 + 1 = 5\).
3. Form the part-to-whole fraction. - Out of a total of 5 "ratio units", 4 belong to English books. - The fraction of books that are English is \(\frac{\text{Part}}{\text{Whole}} = \frac{4}{5}\).
4. Convert the fraction to a percentage. - To convert a fraction to a percentage, we multiply by 100%. \[ \text{Percentage} = \frac{4}{5} \times 100% = 0.8 \times 100% = 80% \] Step 3: Final Answer:
80 percent of the books on the bookshelf are English books.
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