How Many Numbers (Terms)? Space, Inclusive and Exclusive Notes

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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? 

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

Problem Solving Strategy: Probability, Counting, Grid

Check out Mathcounts here, the best competition math program for middle school students.
Download this year's Mathcounts handbook here. (It's free.)


#5 1993 Mathcounts National Target : Find the probability that four randomly selected points on the geoboard below will be the vertices of a square? Express your answer as a common fraction.

#5 2004 AMC 10A: A set of three points is chosen randomly from the grid shown. Each three-point set has the same probability of being chosen. What is the probability that the points lie on the same straight line?

Solution:

#5 National Target: There are 16C4 = (16 x 15 x 14 x 13)/ 4 x 3 x 2 = 1820 ways to select 4 points on the geoboard.

There are 3 x 3 = 9  one by one squares and 2 x 2 = 4 two by two squares and 1 x 1 = 1 three by three squares. (Do you see the pattern?)

                                                         

There are 4 other squares that have side length of √ 2
and 2 other larger squares that have side length of √ 5.

9 + 4 + 1 + 4 + 2 = 20 and 20/1820 = 1/91

#5 AMC-10A: There are 9C3 = (9 x 8 x 7) / 3 x 2 x 1 = 84 ways to chose the three dots and 8 of the lines connecting the three dots will form straight lines. (Three verticals, three horizontals and two diagonals.) so 

8/84 = 2/21