Numbers and Number Sense
There are 435 representatives and 100 senators serving in the United States Congress. How many
identical groups with the same numbers of representative and senators could be formed from all of
Congress, if we want the largest groups possible??
5 groups, each with 87 representatives and 20 senators.
Measurement
Geometry
If all the angles of a polygon with 40 sides are congruent, then
find the measure of each angle of the polygon.
Solution:
n = 40
let y be the measure of each angle of the polygon.
Then the sum of the angles of the polygon on one hand is:
S = 40y ………. Equation 1
On the other hand, the sum of the angles is given by the formula:
S = (n – 2) × 180o …….. Equation 2
Equating equation(1) and Equation(2) we get:
40y = (n – 2) × 180o
40y = (40 – 2) × 180o
40y = 38 × 180o
y=
y = 171o
so each of the 40 sides has a measure of 171o
Pattern and Algebra
A student take a mathematics test scores of 64 and 78. What score on a third test
will give the student an average of 80?
Solution:
The unknown quantity is the score on the third test, so you let x = score on the third test.
The average scores will be calculated on 64, 78 and x. Thus average score = 3
64 + 78 + x 3
64 + 78 + x = 80
64 + 78+x = 80 × 3 …………Multiplying both sides by 3
142+x= 240………Simplify
x = 98……… x is solved
The sum of the ages of a man and his wife is 96 years. The man is 6 years older than his wife. How old
is his wife?
Solution: let m = man age and w = wife age, then
m+w = 96 …….Translated equation(1)
m = 6+w ……..Translated equation (2)
6+w+w = 96……..Substituting equation (2) into(1)
6+2w = 96-6……Combine like terms
2w+6-6 = 96-6………Subtracting 6 both sides.
2w = 90 ………Simplifying
2
90
22w = ……………Divides both sides by 2
w = 45 …….w is solved
Therefore, the age of his wife is 45 years old.
Statistics and Probability
The mean of three numbers is 10, and the mean of four other
numbers is 16. What is the mean of all seven numbers?
Solution: For the first set of data
We get, sum of values = mean × number of values
Sum of values = 10 × 3
Sum of values = 30
For the second set of data
Similarly sum of values = mean × number of values
= 16 × 4
= 64
Thus total sum = 30 + 64 = 94
And total number of data = 3 + 4 = 7.
Therefore, mean of all seven numbers =
=
= 13.4
Therefore, the mean of all seven number is 13.4.
Example 10. The mean of four numbers is 9. Three of the numbers are 8, 16
and 6. Find the value of the other number.
Solution: let x be the missing number.
Thus mean =
9=
30 + x = 36
x=6
The missing number is 6
The Pigeon Hole Principle
Three people are running for student government. There are 202 people who vote. What is the
minimum number of votes needed for someone to win the election?
Solution: By pigeonhole, there exists a person who has gotten at least d202/3e = 68 votes. So, someone
could win with a 67 − 67 − 68 split.
There are 38 different time periods during which classes at a university can be scheduled. If there are
677 different classes, what is the minimum number of different rooms that will be needed?
Solution: There exists a time period will have at least d677/38e = 18 classes during it. So 18 different
rooms will be needed.
The Invariance Principle;
Show that when a 6 × 6 square floor is tiled using 1 × 2 rectangular tiles, there is always a straight line
which crosses the floor without cutting through any of the tiles.
Solution: There are 10 lines which could possibly cross through the square without cutting any tiles (5
horizontal lines and 5 vertical lines). Notice that each line must cut through an even number of tiles,
since if it cut through an odd number of tiles then, excluding those tiles, the area of the remaining
portion of the floor on each side of the line would be odd and could not be tiled by 1 × 2 tiles. Suppose,
for a contradiction, that each of the 10 lines crosses a tile. Then (since each line crosses an even number
of tiles), each of the 10 lines crosses at least 2 tiles and so (since each tile is crossed by a unique line)
there are at least 20 tiles. But the floor is 6 × 6 so there are only 18 tiles.
Induction Principle
Example 2: Show that 1 + 3 + 5 + … + (2n−1) = n2
Solution:
Step 1: Result is true for n = 1
That is 1 = (1)2 (True)
Step 2: Assume that result is true for n = k
1 + 3 + 5 + … + (2k−1) = k2
Step 3: Check for n = k + 1
i.e. 1 + 3 + 5 + … + (2(k+1)−1) = (k+1)2
We can write the above equation as,
1 + 3 + 5 + … + (2k−1) + (2(k+1)−1) = (k+1)2
Using step 2 result, we get
k2 + (2(k+1)−1) = (k+1)2
k2 + 2k + 2 −1 = (k+1)2
k2 + 2k + 1 = (k+1)2
(k+1)2 = (k+1)2
L. H.S. and R.H.S. are same.
M. So the result is true for n = k+1
By mathematical induction, the statement is true.We see that the given statement is also true for n=k+1.
Hence we can say that by the principle of mathematical induction this statement is valid for all natural
numbers n.
Pattern and conjectures
Make a diagram
Organize data and used logic
Modify problem
In a school 50% of the students are younger than 10, 1/20 are 10 years old and 1/10 are older than 10
but younger than 12, the remaining 70 students are 12 years or older. How many students are 10 years
old?
Solution
Let us write the fraction for each group of students
Group A: younger than 10: 50% = 50/100 = 1/2
Group B: 10 years old: 1/20
Group C: older that 10 but younger than 12: 1/10
Group D: 12 years or older: 70 students
The fraction for group A, B and C together is given by
1 / 2 + 1 / 20 + 1 / 10 = 10 / 20 + 1 / 20 + 2 / 20 , common denominator
= 13 / 20 , add numerators
The fraction for group D is given by
20 / 20 - 13 / 20 = 7 / 20 and corresponds to 70 students
If X is the total number of students, then
7 / 20 of X = 70
or (7 / 20) × X = 70
Solve for X
X = 70 × (20 / 7) = 200
Students who are 10 years old have a fraction of 1 / 20 of the total X and their number is equal to
(1 / 20) of 200 = 10 students
Divide into cases
Example 1:
A team won 3 times as many matches as it lost. If it won 15 matches, how many games did it lose?
Solution:
Step 1: Sentence: A team won 3 times as many matches as it lost.
Assign variables:
Let
x = matches lost
3x = matches won
Sentence: It won 15 matches
3x = 15
Step 2: Solve the equation
Isolate variable x
x=5
Answer: The team lost 5 games.
Consider extreme cases
Example:
John is twice as old as his friend Peter. Peter is 5 years older than Alice. In 5 years, John will be three
times as old as Alice. How old is Peter now?
Solution:
Step 1: Set up a table.
Algebra Age Problems
Step 2: Fill in the table with information given in the question.
John is twice as old as his friend Peter. Peter is 5 years older than Alice. In 5 years, John will be three
times as old as Alice. How old is Peter now?
Let x be Peter’s age now. Add 5 to get the ages in 5 yrs.
Algebra Age Problems
Write the new relationship in an equation using the ages in 5 yrs.
In 5 years, John will be three times as old as Alice.
2x + 5 = 3(x – 5 + 5)
2x + 5 = 3x
Isolate variable x
x=5
Answer: Peter is now 5 years old.
Look for symmetry
Let’s first check for symmetry about the xx-axis. To do this we need to replace
all the yy’s with –yy.
x=4(−y)6−(−y)2→x=4y6−y2x=4(−y)6−(−y)2→x=4y6−y2
The resulting equation is identical to the original equation and so is equivalent
to the original equation. Therefore, the equation is has symmetry about
the xx-axis.
Hide Step 2
Next, we’ll check for symmetry about the yy-axis. To do this we need to
replace all the xx’s with –xx.
−x=4y6−y2−x=4y6−y2
The resulting equation is not equivalent to the original equation (i.e. it is not
same nor is it the same equation except with opposite signs on every term).
Therefore, the equation is does not have symmetry about the yy-axis.
Hide Step 3
Finally, a check for symmetry about the origin. For this check we need to
replace all the yy’s with –yy and to replace all the xx’s with –xx.
−x=4(−y)6−(−y)2→−x=4y6−y2−x=4(−y)6−(−y)2→−x=4y6−y2
The resulting equation is not equivalent to the original equation (i.e. it is not
same nor is it the same equation except with opposite signs on every term).
Therefore, the equation does not have symmetry about the origin.
Choose effective notation
Work backward
Example:
The sum of the least and greatest of 3 consecutive integers is 60. What are the values of the 3 integers?
Solution:
Step 1: Assign variables:
Let x = least integer
x + 1 = middle integer
x + 2 = greatest integer
Translate sentence into an equation.
Sentence: The sum of the least and greatest is 60.
Rewrite sentence:
x + (x + 2) = 60
Step 2: Solve the equation
Combine like terms
2x + 2 = 60
Isolate variable x
2x = 58
equation
Step 3: Check your answer
29 + 29 + 2 = 60
The question wants all the 3 consecutive numbers: 29, 30 and 31
Answer: The 3 consecutive numbers are 29, 30 and 31.
Pursue parity
Example 3:
John has a board that is 5 feet long. He plans to use it to make 4 shelves whose lengths are to be a series
of consecutive even numbers. How long should each shelf be in inches?
Solution:
Step 1: Being consecutive even numbers we need to add 2 to the previous number.
Assign variables:
Let x = length of first shelf
x + 2 = length of second shelf
x + 4 = length of third shelf
x + 6 = length of fourth shelf
Step 2: Convert 5 feet to inches
5 × 12 = 60
Step 3: Sum of the 4 shelves is 60
x + x + 2 + x + 4 + x + 6 = 60
Combine like terms
4x + 12 = 60
Isolate variable x
4x = 60 – 12
4x = 48
x = 12
Step 3: Check your answer
12 + 12 + 2 + 12 + 4 + 12 + 6 = 60
The lengths of the shelves should be 12, 14, 16 and 18.
Answer: The lengths of the shelves in inches should be 12, 14, 16 and 18.