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Unit V Mathematical Reasoning And Aptitude
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Averages – Quiz 8
Averages Quiz 8 (10 MCQs)
This set of multiple-choice questions evaluates understanding of average calculation, sum manipulation, and solving equations. It covers concepts such as arithmetic mean, sum of scores, and time management. Students will practice calculating averages, solving for unknowns, and analyzing data to determine mean scores and total marks.
Quiz Instructions
Select an option to see the correct answer instantly.
1.
Calculate the average speed (in miles per hour) of Cassandra who runs a 18 mile marathon in 6 hours.
A) 24 m/hr.
B) 3 m/hr.
C) 0.333 m/hr.
D) 12 m/hr.
Show Answer
Correct Answer:
Correct answer is: (B) 3 m/hr.
Exam Relevance:
SAT, ACT, GRE, GMAT
Difficulty:
Easy
Concept notes:
The average speed is calculated by dividing the total distance by the total time taken.
Common Mistakes:
A common mistake is to confuse average speed with instantaneous speed or to misinterpret the units of measurement.
Explanations:
To find the average speed, we use the formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]
Given:
- Total Distance = 18 miles
- Total Time = 6 hours
Substitute the values into the formula:
\[ \text{Average Speed} = \frac{18 \text{ miles}}{6 \text{ hours}} = 3 \text{ miles per hour} \]
Thus, the average speed is 3 miles per hour.
Option Analysis:
Option A:
24 m/hr is incorrect because it is much higher than the calculated average speed.
Option B:
3 m/hr is the correct answer as it matches the calculated average speed.
Option C:
0.333 m/hr is incorrect because it is much lower than the calculated average speed.
Option D:
12 m/hr is incorrect because it is higher than the calculated average speed.
2.
Carl's semester math grade is based on the average of five unit tests. His scored were 88, 92, 79, and 95 on the first four tests. What does his score need to be on the next test in order to obtain a 90% as the final grade for the semester?
A) 96.
B) 89.
C) 85.
D) 90.
Show Answer
Correct Answer:
Correct answer is: (A) 96.
Exam Relevance:
SAT, ACT, GRE, GMAT
Difficulty:
Moderate
Concept notes:
The concept of averages is used to find the required score on the fifth test to achieve a final grade of 90%.
Common Mistakes:
A common mistake is to assume that the score needed on the fifth test is simply 90, without considering the average of all five tests.
Explanations:
To find the required score on the fifth test, we need to set up an equation based on the average of the five test scores. The average of the five tests should be 90.
Let \( x \) be the score on the fifth test. The sum of the scores of the first four tests is \( 88 + 92 + 79 + 95 = 354 \).
The average of the five tests is given by:
\[ \frac{354 + x}{5} = 90 \]
To solve for \( x \), we first multiply both sides of the equation by 5:
\[ 354 + x = 450 \]
Next, we isolate \( x \) by subtracting 354 from both sides:
\[ x = 450 - 354 \]
\[ x = 96 \]
Therefore, Carl needs to score 96 on the fifth test to achieve a final grade of 90%.
Option Analysis:
Option A:
96 is the correct score needed on the fifth test to achieve a final grade of 90%.
Option B:
89 is incorrect because it does not result in an average of 90%.
Option C:
85 is incorrect because it does not result in an average of 90%.
Option D:
90 is incorrect because it does not result in an average of 90%.
3.
What was Edmundo's mean score for a round of golf in August if his scores for each round were 78, 86, 82, 81, 82, and 77?
A) 77.
B) 78.
C) 81.
D) 82.
E) 84.
Show Answer
Correct Answer:
Correct answer is: (C) 81.
Exam Relevance:
SAT, ACT, GRE, GMAT
Difficulty:
Moderate
Concept notes:
The mean score is calculated by summing all the scores and dividing by the number of scores.
Common Mistakes:
A common mistake is to confuse the mean with the median or mode, or to miscalculate the sum or division.
Explanations:
To find the mean score, we first sum all the scores: 78 + 86 + 82 + 81 + 82 + 77 = 486.
Next, we divide the sum by the number of scores, which is 6: 486 / 6 = 81.
Thus, the mean score is 81.
Option Analysis:
Option A:
77 is incorrect because it is not the mean of the given scores.
Option B:
78 is incorrect because it is not the mean of the given scores.
Option C:
81 is correct because it is the mean of the given scores.
Option D:
82 is incorrect because it is not the mean of the given scores.
Option E:
84 is incorrect because it is not the mean of the given scores.
4.
What is the mean for these numbers?5,5,5,5
A) 5.
B) 20.
C) 15.
D) 100.
Show Answer
Correct Answer:
Correct answer is: (A) 5.
Exam Relevance:
SAT, ACT, GRE, GMAT
Difficulty:
Very Easy
Concept notes:
The mean of a set of numbers is calculated by summing all the numbers and then dividing by the count of numbers.
Common Mistakes:
A common mistake is to confuse the mean with other measures of central tendency, such as the median or mode.
Explanations:
To find the mean of the numbers 5, 5, 5, 5, we sum the numbers and divide by the count of numbers:
\[ \text{Sum} = 5 + 5 + 5 + 5 = 20 \]
\[ \text{Count} = 4 \]
\[ \text{Mean} = \frac{20}{4} = 5 \]
Thus, the mean is 5.
Option Analysis:
Option A:
Correct. The mean is 5.
Option B:
Incorrect. This is the sum of the numbers, not the mean.
Option C:
Incorrect. This is not the correct mean.
Option D:
Incorrect. This is the product of the numbers, not the mean.
5.
What is the average of the numbers 4, 8, and 12?
A) 6.
B) 8.
C) 10.
D) 12.
Show Answer
Correct Answer:
Correct answer is: (B) 8.
Exam Relevance:
SAT, ACT, GRE, GMAT
Difficulty:
Easy
Concept notes:
The average of a set of numbers is calculated by summing all the numbers and then dividing by the count of the numbers.
Common Mistakes:
A common mistake is to assume the average is the middle number in a sorted list, which is not always correct.
Explanations:
To find the average of the numbers 4, 8, and 12, we first sum the numbers: 4 + 8 + 12 = 24. Then, we divide the sum by the count of the numbers, which is 3. Therefore, the average is 24 / 3 = 8.
Option Analysis:
Option A:
6 is incorrect because it is not the result of the sum of the numbers divided by the count.
Option B:
8 is correct because it is the result of the sum of the numbers divided by the count.
Option C:
10 is incorrect because it is not the result of the sum of the numbers divided by the count.
Option D:
12 is incorrect because it is not the result of the sum of the numbers divided by the count.
6.
Daniel's grades for the first three tests in a year was 73, 56, and 83. He worked very hard for the last test in that year to improve his average mark to 75. How many marks did he score for the last test to achieve this?
A) 78.
B) 79.
C) 87.
D) 88.
Show Answer
Correct Answer:
Correct answer is: (D) 88.
Exam Relevance:
SAT, ACT, GRE, GMAT
Difficulty:
Moderate
Concept notes:
The concept of averages is used to find the mean value of a set of numbers. In this problem, we need to find the score of the last test that will bring the average of all four tests to 75.
Common Mistakes:
A common mistake is to assume that the average of the first three tests plus the fourth test should equal 75, without considering the total sum required to achieve the average.
Explanations:
To find the score of the last test, we first calculate the total sum of the scores needed to achieve an average of 75 over four tests. The formula for the average is:
\[ \text{Average} = \frac{\text{Sum of all scores}}{\text{Number of scores}} \]
Given the average is 75 and there are 4 tests, the total sum of the scores is:
\[ 75 \times 4 = 300 \]
The sum of the first three test scores is:
\[ 73 + 56 + 83 = 212 \]
Let \( x \) be the score of the fourth test. The equation to find \( x \) is:
\[ 212 + x = 300 \]
Solving for \( x \):
\[ x = 300 - 212 = 88 \]
Thus, Daniel scored 88 on the last test to achieve an average of 75.
Option Analysis:
Option A:
78 is incorrect because it does not bring the average to 75.
Option B:
79 is incorrect because it does not bring the average to 75.
Option C:
87 is incorrect because it does not bring the average to 75.
Option D:
88 is correct because it brings the average to 75.
7.
Find the mean:11, 10, 5, 2, 6, 10, 19
A) 63.
B) 9.
C) 7.
D) 17.
Show Answer
Correct Answer:
Correct answer is: (B) 9.
Exam Relevance:
SAT, ACT, GRE, GMAT
Difficulty:
Easy
Concept notes:
The mean (average) of a set of numbers is calculated by summing all the numbers and then dividing by the count of numbers.
Common Mistakes:
A common mistake is to confuse the mean with other measures of central tendency, such as the median or mode.
Explanations:
To find the mean of the numbers 11, 10, 5, 2, 6, 10, 19:
1. Sum the numbers: 11 + 10 + 5 + 2 + 6 + 10 + 19 = 63.
2. Count the numbers: There are 7 numbers.
3. Divide the sum by the count: 63 ÷ 7 = 9.
Thus, the mean is 9.
Option Analysis:
Option A:
63 is the sum of the numbers, not the mean.
Option B:
9 is the correct mean.
Option C:
7 is the count of the numbers, not the mean.
Option D:
17 is not the mean of the given numbers.
8.
Mike's history assignment is to read 420 pages. He planned to do the assignment in 6 hours. He read 160 pages in 2 hours. What is the mean number of pages he must read per hour during the next 4 hours in order to complete the assignment according to plan?
A) 60.
B) 65.
C) 70.
D) 75.
Show Answer
Correct Answer:
Correct answer is: (B) 65.
Exam Relevance:
SAT, ACT, GRE, GMAT
Difficulty:
Moderate
Concept notes:
The question involves calculating the average number of pages Mike needs to read per hour to complete his assignment on time.
Common Mistakes:
A common mistake is to assume the average reading speed remains constant, without considering the remaining pages and time.
Explanations:
1. Mike needs to read a total of 420 pages in 6 hours.
2. He has already read 160 pages in 2 hours.
3. Therefore, he has 420 - 160 = 260 pages left to read.
4. He has 6 - 2 = 4 hours left to complete the assignment.
5. To find the mean number of pages he must read per hour for the next 4 hours, we divide the remaining pages by the remaining hours: 260 / 4 = 65 pages per hour.
Option Analysis:
Option A:
60 pages per hour is incorrect because it does not account for the remaining pages and time accurately.
Option B:
65 pages per hour is correct as it matches the calculated mean.
Option C:
70 pages per hour is incorrect because it overestimates the required reading speed.
Option D:
75 pages per hour is incorrect because it overestimates the required reading speed.
9.
The average of four numbers is 27. A new number, x, is added to these four, and the new average of the five numbers is 25. What is the value of x?
A) 14.
B) 15.
C) 17.
D) 18.
Show Answer
Correct Answer:
Correct answer is: (C) 17.
Exam Relevance:
SAT, GRE, GMAT, ACT
Difficulty:
Moderate
Concept notes:
The concept of averages is used to find the value of a new number added to a set of numbers, given the initial and new averages.
Common Mistakes:
A common mistake is to incorrectly calculate the sum of the original numbers or the new sum after adding the new number.
Explanations:
1. The average of the four numbers is 27. Therefore, the sum of these four numbers is \(4 \times 27 = 108\).
2. When a new number \(x\) is added, the new average of the five numbers is 25. Therefore, the sum of the five numbers is \(5 \times 25 = 125\).
3. The sum of the original four numbers plus the new number \(x\) is \(108 + x = 125\).
4. Solving for \(x\), we get \(x = 125 - 108 = 17\).
Thus, the value of \(x\) is 17.
Option Analysis:
Option A:
14 is incorrect because it does not satisfy the equation \(108 + x = 125\).
Option B:
15 is incorrect because it does not satisfy the equation \(108 + x = 125\).
Option C:
17 is correct because it satisfies the equation \(108 + x = 125\).
Option D:
18 is incorrect because it does not satisfy the equation \(108 + x = 125\).
10.
The average marks of 13 papers is 40. The average marks of the first 7 papers are 42 and that of the last seven papers is 35. Find the marks obtained in the 7th paper.A. 23B. 38C. 19D. 57
A) 19.
B) 25.
C) 84.
D) 34.
Show Answer
Correct Answer:
Correct answer is: A) 19.
Exam Relevance:
GRE, GMAT, SAT, ACT
Difficulty:
Moderate
Concept notes:
The question involves calculating the marks obtained in the 7th paper using the concept of averages.
Common Mistakes:
A common mistake is to assume that the 7th paper's marks can be directly calculated from the given averages without considering the overlap in the first 7 and last 7 papers.
Explanations:
1. Calculate the total marks for the 13 papers: \( 13 \times 40 = 520 \).
2. Calculate the total marks for the first 7 papers: \( 7 \times 42 = 294 \).
3. Calculate the total marks for the last 7 papers: \( 7 \times 35 = 245 \).
4. The 7th paper is counted in both the first 7 and the last 7 papers, so the sum of the first 7 and the last 7 papers includes the 7th paper twice.
5. Therefore, the sum of the first 7 and the last 7 papers is \( 294 + 245 = 539 \).
6. Subtract the total marks for the 13 papers from this sum to find the marks of the 7th paper: \( 539 - 520 = 19 \).
Thus, the marks obtained in the 7th paper are 19.
Option Analysis:
Option A:
Correct. The marks obtained in the 7th paper are 19.
Option B:
Incorrect. The marks obtained in the 7th paper are not 25.
Option C:
Incorrect. The marks obtained in the 7th paper are not 84.
Option D:
Incorrect. The marks obtained in the 7th paper are not 34.
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Frequently Asked Questions
What is the difference between arithmetic average and arithmetic mean?
The arithmetic average and arithmetic mean are the same concept. Both are calculated by summing a set of numbers and dividing by the count of those numbers.
How do you calculate the average speed when traveling different distances at different speeds?
To calculate average speed, sum the total distance traveled and divide it by the total time taken. This gives the overall average speed for the entire journey.
Can you explain how to find the new average when a new score is added to a set of test scores?
To find the new average, add the new score to the sum of the existing scores and then divide by the total number of scores, including the new one.
How do averages apply to time management?
Averages can help in time management by providing a benchmark for how long tasks typically take, allowing for better planning and allocation of time.
What is the importance of understanding averages in calculating final grades?
Understanding averages is crucial for calculating final grades, as it involves summing all marks and dividing by the number of assessments to determine an overall grade.