Quite a lot of my students have problems figuring out this type of problems so here are the notes.

**#1: How many consecutive numbers from 1 to 5 inclusive?**

**1 _ 2 _ 3 _ 4 _ 5**There are 5 numbers if you just list them and count them out; however, what if

the question is:

**#2: How many consecutive numbers from 34 to 200 (SAT type problem)?**

Most students would think it's 200 - 34 = 166, but it's not.

Using #1 case, if you do 5 - 1 = 4, you are only getting how many spaces between those consecutive numbers.

Thus for question #2, the correct answer is 200 - 34 + 1 or 200 - 33 = 167

**#3: What about how many consecutive numbers from 5 to 100 exclusive?**

Inclusive means including the first and the last numbers; exclusive means not including the first and the last numbers, so for this question, you do 100 - 5 - 1 = 94.

Use # 1 case to help you figure out and really understand the concepts involved.

Here are other questions to help you practice the skills.

Word problems: Answers below.

#1: How many numbers from 45 to 100 inclusive?

#2: How many numbers from 17 to 127 inclusive?

#3: How many numbers from 12 to 34 exclusive?

#4: How many multiples of 9 from 1 to 200 inclusive?

#5: The dimension of the square on the left is 20 feet by 20 feet. If you place a post every four feet, starting at one corner, how many posts will be placed?

#6: The distance from exit 13 to 21 is 216 miles. How many miles is the distance between two exits if all exits are equally spaced?

#7: How many multiples of 5 from 120 to 218 exclusive?

#8: Who is right? The teacher or the student? Try this question.

#9: How many numbers from -12, -11, -10.........56 inclusive?

What is their sum?

#10: How many numbers are in the list: 17.25, 18.25, 19, 25...111.25?

#10: How many numbers are in the list: 17.25, 18.25, 19, 25...111.25?

Solutions: To excel at Mathcounts state/national, you need to practice all these questions mentally.

#1: 100 - 45 + 1 = 100 - 44 =

**56**

#2: 127-17 + 1 = 127 - 16 =

**111**

#3: Exclusive: 34 -12 -1 = 34 - 13 =

**21**

#4: Multiples of 9 from 1 to 200 starts with 9 and ends in 198.

Solution I: 9 , 18, 27...198 = 9 (1, 2, 3, ...22) The answer is

**22**.

Solution II: \(\frac{(198 - 9)}{9} + 1 = 22"\)

#5: Just observe one side first. Exclude the 4 corners, the other posts

are similar to those exclusive type problems.

There are posts on each side so 4 * 4 + 4 (corner posts)

=

**20**

#6: There are 21 - 13 = 8 space so miles. The answer is

**27 miles**.

#7: Multiples of 5 from 120 to 218 start with 120 and end in 215.

Since it's asking exclusive, 120, 125, ...215 = 5(24, 25, ...43)

43 - 24 - 1 = 43 - 25 =

**18**

#8: You only need two cuts to get 3 pieces so 2 * 10 =

**20 minutes**. The student is right.

#9: 56 - ( -12) + 1 = 56 + 13 =

**69**

The sum is from 13 to 56 since up to 12 it got cancelled with the negative equivalent numbers.

Use average * the term you got . The sum is

**1518**.

#10:111.25 - 117.25 + 1 = 111.25 - 116.25 =

**95**