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Average Based Data Interpretation Quiz 19 (10 MCQs)

This set of multiple-choice questions evaluates the ability to calculate averages, including arithmetic mean, sum of elements, and division by count. It covers concepts such as sum manipulation, recalculating averages, and data interpretation. Questions involve arithmetic sequences, multiples, and balancing positive and negative numbers. Skills tested include summing scores, dividing by the number of students, and interpreting daily measurements.

Quiz Instructions

Select an option to see the correct answer instantly.

1. If the average of four consecutive odd numbers is 16, find the smallest of these numbers?

Correct Answer: Correct answer is: (C) 13.

Exam Relevance: GMAT, GRE, SAT, ACT

Difficulty: Moderate

Concept notes: The average of four consecutive odd numbers is the middle value of the sequence. Since the average is 16, the middle two numbers must be 15 and 17.

Common Mistakes: A common mistake is to assume the average directly corresponds to one of the numbers in the sequence, which is not the case for an even number of terms.

Explanations: Let the four consecutive odd numbers be \(x, x+2, x+4, x+6\). The average of these numbers is given by: \[ \frac{x + (x+2) + (x+4) + (x+6)}{4} = 16 \] Simplifying the equation: \[ \frac{4x + 12}{4} = 16 \] \[ x + 3 = 16 \] \[ x = 13 \] Thus, the smallest of these numbers is 13.

Option Analysis:

Option A: 5 is incorrect because it does not satisfy the average condition.
Option B: 7 is incorrect because it does not satisfy the average condition.
Option C: 13 is correct as it satisfies the average condition.
Option D: 11 is incorrect because it does not satisfy the average condition.

2. There are 6156 students in a school district. If there are 27 schools, how many students are each school, on average?

Correct Answer: Correct answer is: (B) 228 students.

Exam Relevance: SAT, ACT, GRE, GMAT

Difficulty: Moderate

Concept notes: To find the average number of students per school, we need to divide the total number of students by the number of schools.

Common Mistakes: A common mistake is to misinterpret the question and perform an incorrect calculation, such as multiplying instead of dividing.

Explanations: To find the average number of students per school, we need to divide the total number of students by the number of schools. The total number of students is 6156, and the number of schools is 27. Therefore, we perform the division: \[ \text{Average number of students per school} = \frac{6156}{27} \] Performing the division: \[ 6156 \div 27 = 228 \] Thus, the average number of students per school is 228.

Option Analysis:

Option A: 219 students. This is incorrect because the division of 6156 by 27 does not yield 219.
Option B: 228 students. This is the correct answer as the division of 6156 by 27 yields 228.
Option C: 22 students. This is incorrect because the division of 6156 by 27 does not yield 22.
Option D: 23 students. This is incorrect because the division of 6156 by 27 does not yield 23.

3. The average weight of 8 person's increases by 2.5 kg when a new person comes in place of one of them weighing 65 kg. What might be the weight of the new person?

Correct Answer: Correct answer is: (D) 85 kg.

Exam Relevance: CAT, GMAT, GRE, SAT

Difficulty: Moderate

Concept notes: The question involves calculating the weight of a new person based on the change in the average weight of a group. The key concept is understanding how the average weight changes when a new person replaces an existing one.

Common Mistakes: A common mistake is to assume that the new person's weight is simply the old person's weight plus the increase in average weight multiplied by the number of people. This overlooks the fact that the average weight increase is distributed across all members of the group.

Explanations: 1. Let the initial average weight of the 8 persons be \( A \). 2. The total weight of the 8 persons is \( 8A \). 3. When a new person replaces one weighing 65 kg, the new average weight is \( A + 2.5 \). 4. The total weight of the 8 persons now is \( 8(A + 2.5) \). 5. The new person's weight can be found by equating the total weight before and after the replacement: \[ 8A - 65 + \text{new person's weight} = 8(A + 2.5) \] 6. Simplify the equation: \[ 8A - 65 + \text{new person's weight} = 8A + 20 \] 7. Isolate the new person's weight: \[ \text{new person's weight} = 20 + 65 = 85 \text{ kg} \]

Option Analysis:

Option A: 76 kg is incorrect because it does not account for the increase in average weight correctly.
Option B: 76.5 kg is incorrect because it does not account for the increase in average weight correctly.
Option C: Inadequate data is incorrect because all necessary data is provided to solve the problem.
Option D: 85 kg is correct as it matches the calculated weight of the new person.

4. You collect trading cards. Your collection of trading cards has grown to be contained in several boxes. You count the number of cards currently contained in each box: 1,337, 1,245, 1,501, and 601. What is the average number of cards in each of these boxes?

Correct Answer: Correct answer is: (E) 1,171.

Exam Relevance: GMAT, GRE, SAT, ACT

Difficulty: Moderate

Concept notes: The average number of cards in each box is calculated by summing the number of cards in all boxes and then dividing by the number of boxes.

Common Mistakes: A common mistake is to miscount the number of boxes or to miscalculate the sum of the cards.

Explanations: To find the average number of cards in each box, we first sum the number of cards in all boxes: 1,337 + 1,245 + 1,501 + 601 = 4,684. Next, we divide the total number of cards by the number of boxes: 4,684 / 4 = 1,171. Thus, the average number of cards in each box is 1,171.

Option Analysis:

Option A: 684 is incorrect because it is significantly lower than the calculated average.
Option B: 759 is incorrect because it is still lower than the calculated average.
Option C: 905 is incorrect because it is lower than the calculated average.
Option D: 1,101 is incorrect because it is lower than the calculated average.
Option E: 1,171 is the correct answer as it matches the calculated average.

5. In a class of 20 students, the average height is 150 cm. What is the total height of all the students?

Correct Answer: Correct answer is: (A) 3000 cm.

Exam Relevance: SAT, GRE, GMAT, CAT

Difficulty: Easy

Concept notes: The average height of a group is calculated by dividing the total height of all individuals by the number of individuals. To find the total height, multiply the average height by the number of individuals.

Common Mistakes: A common mistake is to misinterpret the average as the total height, leading to incorrect calculations.

Explanations: To find the total height of all the students, we use the formula for the average: \[ \text{Average height} = \frac{\text{Total height}}{\text{Number of students}} \] Given: \[ \text{Average height} = 150 \text{ cm} \] \[ \text{Number of students} = 20 \] We need to find the total height: \[ \text{Total height} = \text{Average height} \times \text{Number of students} \] \[ \text{Total height} = 150 \text{ cm} \times 20 \] \[ \text{Total height} = 3000 \text{ cm} \] Thus, the total height of all the students is 3000 cm.

Option Analysis:

Option A: 3000 cm. (Correct)
Option B: 1800 cm. (Incorrect)
Option C: 2500 cm. (Incorrect)
Option D: 2000 cm. (Incorrect)

6. What is the mean score of the 10 students in the table above?

Correct Answer: Correct answer is: (C) 77.

Exam Relevance: GMAT, GRE, SAT, ACT

Difficulty: Moderate

Concept notes: The mean score is calculated by summing all the scores and dividing by the number of students.

Common Mistakes: A common mistake is to miscount the number of students or to miscalculate the sum of the scores.

Explanations: To find the mean score, we need to sum all the scores and then divide by the number of students. The scores are as follows: 70, 80, 75, 85, 90, 70, 80, 75, 85, 90. Summing these scores, we get: \[ 70 + 80 + 75 + 85 + 90 + 70 + 80 + 75 + 85 + 90 = 770 \] Next, we divide the sum by the number of students, which is 10: \[ \frac{770}{10} = 77 \] Thus, the mean score is 77.

Option Analysis:

Option A: 22.5 is incorrect because it is much lower than the actual mean score.
Option B: 75 is incorrect because it is slightly lower than the actual mean score.
Option C: 77 is correct as it matches the calculated mean score.
Option D: 85 is incorrect because it is higher than the actual mean score.

7. Average of first 3 multiples in the table of 5.

Correct Answer: Correct answer is: (B) 10.

Exam Relevance: GMAT, GRE, SAT, ACT

Difficulty: Easy

Concept notes: The average of a set of numbers is calculated by summing the numbers and dividing by the count of numbers.

Common Mistakes: A common mistake is to assume the average of the first three multiples of 5 is simply 5, without performing the calculation.

Explanations: The first three multiples of 5 are 5, 10, and 15. To find the average, we sum these numbers and divide by the count of numbers: \[ \text{Sum} = 5 + 10 + 15 = 30 \] \[ \text{Count} = 3 \] \[ \text{Average} = \frac{30}{3} = 10 \] Thus, the average of the first three multiples of 5 is 10.

Option Analysis:

Option A: 5 is incorrect because it is the first multiple of 5, not the average of the first three multiples.
Option B: 10 is correct as calculated above.
Option C: 15 is incorrect because it is the third multiple of 5, not the average of the first three multiples.
Option D: None of the above is incorrect because 10 is the correct answer.

8. The average of 4 numbers is 28. If the number 40 is removed, what is the average of the remaining 3 numbers?

Correct Answer: Correct answer is: (C) 24.

Exam Relevance: GRE, GMAT, SAT, ACT

Difficulty: Moderate

Concept notes: The concept involves calculating the average of a set of numbers and understanding how removing one number affects the average of the remaining numbers.

Common Mistakes: A common mistake is to assume that removing one number will decrease the average by a fixed amount, rather than recalculating the average based on the new sum and count of the remaining numbers.

Explanations: 1. The average of 4 numbers is 28. Therefore, the sum of these 4 numbers is \( 28 \times 4 = 112 \). 2. If the number 40 is removed, the sum of the remaining 3 numbers is \( 112 - 40 = 72 \). 3. The average of the remaining 3 numbers is \( \frac{72}{3} = 24 \). Thus, the average of the remaining 3 numbers is 24.

Option Analysis:

Option A: 25 is incorrect because the recalculated average is 24.
Option B: 23 is incorrect because the recalculated average is 24.
Option C: 24 is correct as shown in the step-by-step calculation.
Option D: 26 is incorrect because the recalculated average is 24.

9. Average of 15 numbers is zero. at most how many numbers may be greater than zero.

Correct Answer: Correct answer is: (B) 14.

Exam Relevance: GMAT, GRE, SAT, ACT

Difficulty: Easy

Concept notes: The average of 15 numbers is zero, which means the sum of these numbers is zero. For the sum to be zero, the positive numbers must be balanced by negative numbers or zeros.

Common Mistakes: A common mistake is to assume that if the average is zero, all numbers must be zero, which is not necessarily true. The numbers can include both positive and negative values that balance each other out.

Explanations: The average of 15 numbers is zero, meaning the sum of these numbers is zero. To achieve a sum of zero, the positive numbers must be balanced by negative numbers or zeros. At most, 14 numbers can be greater than zero if the 15th number is negative enough to balance the sum to zero. For example, if 14 numbers are positive, the 15th number must be negative and equal to the sum of the 14 positive numbers to make the total sum zero.

Option Analysis:

Option A: 9 is incorrect because it does not maximize the number of positive numbers.
Option B: 14 is correct because it is the maximum number of positive numbers that can be balanced by a single negative number or zero.
Option C: 15 is incorrect because if all 15 numbers were positive, their sum would be positive, not zero.
Option D: 0 is incorrect because it implies no numbers are greater than zero, which is not the maximum possible.

10. A scientist measures the growth of a plant every day: 10 cm, 20 cm, 40 cm, 80 cm, and 160 cm. What is the average growth per day?

Correct Answer: Correct answer is: (B) 62 cm.

Exam Relevance: GMAT, GRE, SAT, ACT

Difficulty: Moderate

Concept notes: The average growth per day is calculated by summing the total growth over the days and dividing by the number of days.

Common Mistakes: A common mistake is to calculate the average of the daily measurements rather than the total growth over the days.

Explanations: To find the average growth per day, we first calculate the total growth over the days. The growth measurements are 10 cm, 20 cm, 40 cm, 80 cm, and 160 cm. The total growth is the sum of these measurements: \[ 10 + 20 + 40 + 80 + 160 = 310 \text{ cm} \] Next, we divide the total growth by the number of days (5 days): \[ \frac{310 \text{ cm}}{5 \text{ days}} = 62 \text{ cm/day} \] Thus, the average growth per day is 62 cm.

Option Analysis:

Option A: 50 cm is incorrect because it does not match the calculated average growth.
Option B: 62 cm is the correct average growth per day.
Option C: 82 cm is incorrect because it does not match the calculated average growth.
Option D: 102 cm is incorrect because it does not match the calculated average growth.

Frequently Asked Questions

What is the arithmetic mean?

The arithmetic mean is the sum of a set of numbers divided by the count of those numbers, representing the average value.

How do you calculate the average of consecutive numbers?

To calculate the average of consecutive numbers, add the first and last numbers and divide by 2, as the average is the midpoint of the sequence.

What is the importance of balancing positive and negative values when calculating averages?

Balancing positive and negative values ensures that the average accurately reflects the central tendency of the data, avoiding skewed results.

How does the concept of average apply to daily measurements?

The average of daily measurements provides a typical value that represents the overall trend or central value of the data collected over time.

What is the significance of recalculating the average after group replacement?

Recalculating the average after group replacement helps to reflect the new composition of the group and provides an updated measure of central tendency.

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