The average of the first five multiples of 9 is:

Question

The average of the first five multiples of 9 is:

A. 20
B. 27
C. 28
D. 30

Core Concept

The core concept being tested is Averages and Multiples. This question involves calculating the average of the first five multiples of 9.

Definition of Average:

The average (or mean) of a set of numbers is the sum of the numbers divided by the number of items in the set. Mathematically, the average is expressed as:

$$\text{Average} = \frac{\text{Sum of the numbers}}{\text{Number of numbers}}$$

In this problem, we need to find the average of the first five multiples of 9.

Step-by-Step Solution

Step 1: Identify the First Five Multiples of 9

The first five multiples of 9 are:

$$9, 18, 27, 36, 45$$

Step 2: Calculate the Sum of These Numbers

Now, we need to find the sum of these five multiples:

$$9 + 18 + 27 + 36 + 45 = 135$$

Step 3: Calculate the Average

The average is calculated by dividing the sum by the number of numbers (which is 5 in this case):

$$\text{Average} = \frac{135}{5} = 27$$

Thus, the average of the first five multiples of 9 is 27.

Correct Answer and Option

The average of the first five multiples of 9 is 27.

Correct Answer: B

Answer Reasoning

We first identified the first five multiples of 9, which are 9, 18, 27, 36, and 45. After calculating the sum of these numbers, we found it to be 135. Dividing this sum by the total number of multiples (5), we get the average as 27.

Explanation of Incorrect Options

• Option A (20): This is incorrect because it underestimates the sum of the multiples. The actual sum (135) results in an average of 27.
• Option C (28): This is incorrect as it exceeds the correct average. A mistake might arise from incorrectly adding the multiples or an assumption of a higher average.
• Option D (30): This is incorrect because the correct sum of the multiples (135) does not yield this value when divided by 5. It is a possible error from miscalculating the sum.

Common Mistakes to Avoid

1. Incorrectly Adding the Multiples: Ensure that the multiples are correctly identified and added together. Miscalculating the sum could lead to an incorrect average.
2. Dividing by the Wrong Number of Values: Double-check that the number of terms (multiples) is 5, and divide the sum by the correct number.

Simplest Way to Solve (Shortcut or Insight)

The simplest way to solve this is to recognize that the average of an evenly spaced set of numbers (like multiples of 9) is the middle number. Since we are working with the first five multiples of 9, the third multiple (27) is the average.

Visual Aid Suggestion

A number line can be a useful visual aid for this problem. Plot the first five multiples of 9 on a number line: 9, 18, 27, 36, and 45. The middle point on this line is 27, which visually confirms that the average is 27.

Practice Questions

1. Question 1:
   The average of the first five multiples of 7 is:
   A. 12
   B. 14
   C. 16
   D. 18
   (Answer: B)
2. Question 2:
   The average of the first six multiples of 5 is:
   A. 15
   B. 18
   C. 20
   D. 25
   (Answer: A)
3. Question 3:
   The average of the first four multiples of 8 is:
   A. 12
   B. 14
   C. 16
   D. 18
   (Answer: C)
4. Question 4:
   The average of the first six multiples of 6 is:
   A. 15
   B. 16
   C. 18
   D. 20
   (Answer: B)
5. Question 5:
   The average of the first seven multiples of 4 is:
   A. 12
   B. 14
   C. 16
   D. 18
   (Answer: B)