- What was the score on the fourth exam?
Answer: 93
To find the score on the fourth exam, we can work backward from the average. The sum of the four scores must be the average multiplied by the number of scores: 89 * 4 = 356. The sum of the first three scores is 86 + 93 + 84 = 263. To find the fourth score, subtract the sum of the first three from the total sum: 356 - 263 = 93.
- How many students play neither soccer nor basketball?
Answer: 50
To find the number of students who play at least one sport, add the number who play each sport and subtract the number who play both (to avoid double-counting): 65 (Soccer) + 55 (Basketball) - 20 (Both) = 100 students. This is the number of students who play either soccer or basketball or both. To find the number who play neither, subtract this number from the total number of students: 150 (Total) - 100 (Play at least one) = 50 students.
- What fraction of the books are either Biography or Reference books, expressed in simplest form?
Answer: 13/40
First, find the total number of books: 150 (Fiction) + 120 (Non-Fiction) + 90 (Biography) + 40 (Reference) = 400 books. Next, find the number of books that are either Biography or Reference: 90 + 40 = 130 books. Now, form the fraction of these books out of the total: 130/400. To simplify, divide both the numerator and the denominator by their greatest common factor, which is 10. 130/10 = 13 and 400/10 = 40. The simplified fraction is 13/40.
- If a diner chooses one item from each category at random, what is the probability they will choose a meal that includes salad and fish?
Answer: 1/6
First, determine the total number of possible meal combinations. This is found by multiplying the number of choices in each category: 2 (appetizers) 3 (main courses) 2 (desserts) = 12 total combinations. Next, determine the number of favorable outcomes. A meal with salad and fish can be paired with either cake or pie for dessert. This means there are 2 favorable outcomes: (Salad, Fish, Cake) and (Salad, Fish, Pie). The probability is the number of favorable outcomes divided by the total number of outcomes: 2/12, which simplifies to 1/6.
- The company advertises its average salary as $80,000. Which statement best describes the data?
Answer: The median salary is significantly lower than the advertised average.
The advertised average is the mean: ($40k+$42k+$45k+$53k+$220k)/5 = $400k/5 = $80,000. However, the high salary of $220,000 is an outlier that skews the mean. The median is the middle value, which is $45,000. This is significantly lower than the mean of $80,000 and is more representative of a typical employee's salary. The mode doesn't exist as no salary is repeated. The range is $220,000 - $40,000 = $180,000, which is greater than the average.
- By what percent did the number of notebooks sold increase from September to October?
Answer: 40%
The formula for percent increase is ((New Value - Original Value) / Original Value) 100%. The increase in sales is 112 - 80 = 32 notebooks. The percent increase is (32 / 80) 100%. Simplifying the fraction, 32/80 = 4/10 = 0.4. Multiplying by 100% gives 40%.
- Which of the following statements accurately compares the two players' performances?
Answer: The players have the same mean, but Player B has a greater range.
First, calculate the mean for both players. Mean A: (18+20+22+24+26)/5 = 110/5 = 22. Mean B: (10+15+20+30+35)/5 = 110/5 = 22. The means are the same. Next, calculate the range. Range A: 26 - 18 = 8. Range B: 35 - 10 = 25. Player B has a greater range. Therefore, the statement that the players have the same mean, but Player B has a greater range is accurate.
- Based on this description, what conclusion can be drawn about the relationship between a car's age and its value?
Answer: There is a negative correlation, meaning older cars are less valuable.
When you encounter scatterplot questions, focus on understanding what the trend of the points tells you about correlation. A scatterplot reveals the relationship between two variables by showing how they change together.
The key detail here is that points trend "from the top left to the bottom right." This creates a downward slope, which indicates a negative correlation. As one variable increases (age, moving right on the x-axis), the other variable decreases (value, moving down on the y-axis). So as cars get older, their resale value decreases.
Answer D correctly identifies this negative correlation and accurately describes what it means: older cars are less valuable. This matches the downward trend described in the passage.
Answer A misinterprets the correlation direction. A positive correlation would show points trending from bottom left to top right, indicating that both variables increase together. That's not what's described here.
Answer B makes a common error by confusing correlation with causation. While the scatterplot shows these variables are related, it doesn't prove that age directly causes the decrease in value. Correlation doesn't equal causation.
Answer C contradicts the given information entirely. If there were no correlation, the points would be scattered randomly with no clear trend pattern.
Remember this pattern: downward-trending points = negative correlation, upward-trending points = positive correlation. Also, always distinguish between correlation (variables are related) and causation (one variable directly causes changes in another) on quantitative reasoning questions.
- What is the student's new mean grade for all 6 quizzes?
Answer: 86
First, find the total points from the first 5 quizzes: 5 * 84 = 420 points. Next, add the score from the sixth quiz to this total: 420 + 96 = 516 points. Finally, calculate the new mean by dividing the new total points by the new number of quizzes: 516 / 6 = 86.
- What was the average rate of change in the number of visitors per year from 2010 to 2020?
Answer: 300 visitors per year
When you encounter questions about average rate of change, you're looking for how much something changes per unit of time over a specific interval. This requires focusing on the starting and ending values, not the values in between.
To find the average rate of change from 2010 to 2020, you need the change in visitors divided by the change in time. The number of visitors went from 8,000 in 2010 to 11,000 in 2020. That's a change of 11,000−8,000=3,000 visitors over 2020−2010=10 years. The average rate of change is 10 years3,000 visitors=300 visitors per year.
Choice A (200 visitors per year) might result from incorrectly calculating the change in visitors or the time period. Choice B (800 visitors per year) could come from using only the first half of the data—from 2010 to 2015, the rate was 54,000=800 visitors per year. This is a common trap since that period showed steady growth. Choice C (500 visitors per year) might result from calculation errors or confusion about which values to use.
The key insight is that average rate of change only cares about endpoints—the 2015 data point of 12,000 visitors is irrelevant to this calculation. Don't let intermediate data points distract you. Always identify your starting point, ending point, and total time elapsed to avoid getting sidetracked by fluctuations in between. - How many more students chose Summer than chose Spring?
Answer: 45
First, find the percentage of students who chose Spring: 100% - 35% - 30% - 15% = 20%. The percentage difference between Summer and Spring is 35% - 20% = 15%. To find the number of students, calculate 15% of the total number of students: 0.15 300 = 45. Alternatively, calculate the number for each: Summer is 0.35 300 = 105; Spring is 0.20 * 300 = 60. The difference is 105 - 60 = 45.
- How many girls were surveyed in total?
Answer: 45
First, find the number of students who preferred each mascot. Tiger: (2/3) 120 = 80 students. Bear: 120 - 80 = 40 students. Next, find the number of girls in each preference group. Girls who preferred tiger: (1/4) 80 = 20 girls. Girls who preferred bear: (5/8) * 40 = 25 girls. Finally, add the number of girls from both groups to find the total: 20 + 25 = 45 girls.
- What is Alex's final grade in the class?
Answer: 80
This is a weighted average calculation. Multiply each score by its weight (as a decimal) and sum the results. Homework: 90 × 0.20 = 18. Quizzes: 80 × 0.30 = 24. Final Exam: 76 × 0.50 = 38. The final grade is the sum of these weighted scores: 18 + 24 + 38 = 80.
- What is the difference between the median and the mode of the number of books read?
Answer: 0
First, find the mode, which is the most frequent number of books read. Five students read 2 books, which is more than any other number, so the mode is 2. Next, find the median. There are 15 students, so the median is the 8th value when the data is listed in order: 1, 1, 1, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5. The 8th value is 2. The difference between the median (2) and the mode (2) is 2 - 2 = 0.
- What is the smallest possible integer in this data set?
Answer: 1
Let the 5 integers in ascending order be a, b, c, d, e. The median is the middle number, so c = 12. The mode is the most frequent number, and since it must be different from the median, it must appear at least twice. To be the mode, the two largest numbers must be 15, so d=15 and e=15. The data set is a, b, 12, 15, 15. The mean is 10, so the sum of the numbers is 5 * 10 = 50. Therefore, a + b + 12 + 15 + 15 = 50, which simplifies to a + b + 42 = 50, so a + b = 8. Since the numbers are positive integers in ascending order (a ≤ b < 12) and not equal to 15, we need to find pairs (a, b) that sum to 8. Possible pairs are (1, 7), (2, 6), (3, 5). The pair (4, 4) is not possible because 15 is the only mode. To find the smallest possible integer, we look at the smallest possible value for 'a', which is 1.
- What is the median height of the students in the group?
Answer: 61 inches
First, list all the data points from the stem-and-leaf plot in order: 58, 59, 60, 61, 61, 64, 67, 70, 72. There are 9 data points in total. The median is the middle value. Since there is an odd number of data points, the median is the (9 + 1) / 2 = 5th value in the ordered list. The 5th value is 61.
