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. $300$

Answer 1

The correct option is D $225$

sum of the integers 200 to 400 is $200+201+...+400$

The above series in arithmetic progression with the first term is $a=200$ ,$n=201$ and $d=201-200=1$

Therefore, $sum=\frac{n}{2}(2a+(n-1\left)d\right)=\frac{201}{2}(400+200)=60300$

Average of integers from $200$ to $400$ is $\frac{60300}{201}=300$ -----(1)

sum of the integers $50$ to $100$ is $50+51+...+100$

The above series in arithmetic progression with the first term is $a=50$ ,$n=51$ and $d=51-50=1$

Therefore, $sum=\frac{n}{2}(2a+(n-1\left)d\right)=\frac{51}{2}(100+50)=3825$

Average of integers from $50$ to $100$ is $\frac{3825}{51}=75$ ------(2)

From (1) and (2)

The average of integers from $200$ to $400$ is $300-75=225$ greater than the average of integers from $50$ to $100$.