Question

The average (arithmetic mean) of the integers from 200 to 400, inclusive, is how much greater than the average of the integers from 50 to 100, inclusive?

• A. 150
• B. 175
• C. 200
• D. 225
• E. 300

Hint:

Remembering the shortcut for the average of an arithmetic progression (like consecutive integers) can save significant time. Instead of summing all the numbers and dividing by the count, just average the first and last numbers.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The question requires us to find the difference between the averages of two different sets of consecutive integers. For any evenly spaced set of numbers (like consecutive integers), the average is simply the mean of the first and last terms.
Step 2: Key Formula or Approach:
For a set of consecutive integers, the average (arithmetic mean) can be calculated using the formula: \[ \text{Average} = \frac{\text{First Term} + \text{Last Term}}{2} \] Step 3: Detailed Explanation:
First, let's calculate the average of the integers from 200 to 400, inclusive.
Here, the first term is 200 and the last term is 400.
\[ \text{Average}_1 = \frac{200 + 400}{2} = \frac{600}{2} = 300 \] Next, let's calculate the average of the integers from 50 to 100, inclusive.
Here, the first term is 50 and the last term is 100.
\[ \text{Average}_2 = \frac{50 + 100}{2} = \frac{150}{2} = 75 \] Finally, we need to find how much greater the first average is than the second average. This means we need to find their difference.
\[ \text{Difference} = \text{Average}_1 - \text{Average}_2 = 300 - 75 = 225 \] Step 4: Final Answer:
The average of the integers from 200 to 400 is 225 greater than the average of the integers from 50 to 100.