Question

The average of 12 numbers is 15 and the average of first two is 14. What is the average of the rest?

• A. 14
• B. 76/5
• C. 66/5
• D. 55/5

Hint:

To solve average problems quickly, always work with the sums of the quantities.
Total Sum ($12 \times 15 = 180$) minus Known Sum ($2 \times 14 = 28$) equals Remaining Sum ($152$).
Then, simply divide by the remaining count: $152 / 10 = 15.2$, which is $76/5$.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The average (arithmetic mean) of a set of observations is defined as the sum of all observations divided by the total number of observations.
Using this relationship, we can determine the sum of any subset of numbers by multiplying their count by their average.
Key Formula or Approach:
The basic formula to compute the sum of observations is:
\[ \text{Sum of observations} = \text{Average} \times \text{Number of observations} \]

Step 2: Detailed Explanation:
Let us perform the calculations step-by-step:
1. Find the total sum of all 12 numbers:
Given that the average of 12 numbers is 15:
\[ \text{Sum of all 12 numbers} = 12 \times 15 = 180 \]
2. Find the sum of the first two numbers:
Given that the average of the first two numbers is 14:
\[ \text{Sum of the first 2 numbers} = 2 \times 14 = 28 \]
3. Determine the sum of the remaining numbers:
The count of the remaining numbers is:
\[ 12 - 2 = 10 \text{ numbers} \]
The sum of these remaining 10 numbers is found by subtracting the sum of the first two from the total sum:
\[ \text{Sum of the remaining 10 numbers} = 180 - 28 = 152 \]
4. Calculate the average of the remaining 10 numbers:
\[ \text{Average of the rest} = \frac{\text{Sum of the remaining 10 numbers}}{10} \]
\[ \text{Average of the rest} = \frac{152}{10} \]
Simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:
\[ \text{Average of the rest} = \frac{76}{5} = 15.2 \]
This matches Option B.

Step 3: Final Answer:
The average of the remaining numbers is 76/5.