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SLIDE 1
Consolidation
Solve
? ?
? ?
? ?
SLIDE 2
? ?
? ?
? ?
SLIDE 3
Consolidation
When 24 students were added to a class, there were 5 times as many students as before. What was the original number of students?
SLIDE 4
When 24 students were added to a class, there were 5 times as many students as before. What was the original number of students?
Solution
Let the original number of students be x.
SLIDE 5
Consolidation
. What is the value of n?
SLIDE 6
. What is the value of n?
Solution
SLIDE 7
Consolidation
Find the distance x
SLIDE 8
It is to be noted that the units are different, but it is not necessary to convert in this question as the units cancel out in a ratio.
SLIDE 7B
Consolidation
Find the distance x
SLIDE 8B
It is to be noted that the units are different, but it is not necessary to convert in this question as the units cancel out in a ratio.
SLIDE 9
Consolidation
? . Find and .
? .
SLIDE 10
? . Find and .
?
SLIDE 11
Consolidation - positive and negative solutions
Find x, when
?
?
? and
SLIDE 12
?
?
? and
SLIDE 13
What is wrong with this argument?
SLIDE 14
The error in the argument took place between steps (3) and (4).
You are not allowed to divide by zero.
Alternatively, you cannot cancel through by zero.
SLIDE 15
What is wrong with this argument?
SLIDE 16
Substituting in , we get
But is impossible because then , and you cannot divide by zero.
SLIDE 17
Which of the following values of x are solutions to the above?
A
B
C
D
Hint. Substitute each value into the left-hand side and see if you get . Also, do not forget the lesson of the previous question.
SLIDE 18
A
B
C
D
SLIDE 19
Functions where the value is undefined
Since you cannot divide by zero, this function is undefined for .
FYI. This undefined value is also known as a singularity.
State the value where each of the following functions are undefined.
? ?
? ?
SLIDE 20
? ?
Undefined for Undefined for
? ?
Undefined for Undefined for
SLIDE 21
Example
Solve
? ?
? ?
SLIDE 22
? ?
? ?
SLIDE 23
State the value where each of the following functions are undefined.
? ?
SLIDE 24
? ?
SLIDE 25
Consolidation
A rational number is any number that can be written as a fraction.
For example, and are rational numbers.
All rational numbers have either finite or infinitely repeating decimal expansions. For example, .
An irrational number is a number that cannot be written as a fraction. and the square root of any prime number are irrational numbers.
A number cannot be both rational and irrational.
The real numbers is the collection (set) of all rational and irrational numbers.
The set of all real numbers is partitioned (divided exactly) into the set of rational and the set of irrational numbers.
SLIDE 26
Classify the following numbers as rational or irrational
SLIDE 27
SLIDE 28
Sets of rational numbers and real numbers
The set (collection) of all rational numbers is written .
The set (collection) of all real numbers is written .
Any real number, positive or negative, is written .
The use of an inequality indicates a real number
means, any real number greater than 0.
SLIDE 29
Solution sets
?
Solution
Since the product of any two numbers is a positive number, the square root of a number has both a positive and a negative solution. The writing of indicates that both solutions are required. Hence, the solution is
?
Solution
The writing of indicates that only the positive solution is asked for. The solution is
only
SLIDE 30
If , which of the following is equal to ?
A
B
C
D Either
E
SLIDE 31
If , which of the following is equal to ?
Solution
C
The question has specifically asked for only the positive solution.
SLIDE 32
? Solve where
? How many solutions does have in each of the following intervals?
? If , simplify
SLIDE 33
? Solve where
? How many solutions does have in each of the following intervals?
? If , simplify .
SLIDE 34
Puzzles on number facts
If x is an odd integer, then which of the following could be equal to ?
A 18 B 17 C 16 D 15 E 14
SLIDE 35
If x is an odd integer, then which of the following could be equal to ?
A 18 B 17 C 16 D 15 E 14
Solution
This is a number puzzle and there may be many approaches to solving it; the answer where may even be “obvious”. Points to consider: (1) as 1 and 6 are odd and even numbers, the product must be even. (2) Factorizing. (3) The product of two negatives is positive, so negative factors should be considered. (4) One can always just plug in some values of x and look to see what happens. If you start with positive values of x, you will soon realize you need to consider negative values too.
Answer E, 14
SLIDE 36
Number puzzles
If sum of consecutive integers from to x is 48. What is the value of x?
SLIDE 37
If sum of consecutive integers from to x is 48. What is the value of x?
Solution
The trick is to realize that the negative sum of the numbers from to 0 cancels out the positive sum of the numbers from to . The sum of these consecutive numbers is always negative until we reach when it is 0. After that it increases. Since the answer is .
Remarks
(1) It is essential to know the meaning of consecutive, which has been introduced before. (2) It is possible to solve this problem using the ideas of an arithmetical progression. The solution leads to a quadratic equation, and as quadratics have not been introduced as yet at this course level, here it is intended it be solved directly by the above approach. The quadratic approach is long-winded in this instance, and the solution requires factors of , which is itself “tricky”. (3) This question is modelled on an American SAT question, where such number puzzles are common.
SLIDE 38
The Little Gauss Problem
In the context of the previous type of number puzzle, involving sums of consecutive positive whole numbers, it is instructive to learn the story of Karl Gustav Gauss, who may be rated the all-time no.1 mathematician.
At the age of five, he was instructed by his school master to sum all the consecutive numbers from 1 to 100. He did it in less than a minute. How?
SLIDE 39
The Little Gauss Problem
At the age of five, he was instructed by his school master to sum all the consecutive numbers from 1 to 100. He did it in less than a minute. How?
Solution
Write the sum of the numbers from 1 to 100 in ascending order. Place under that the same sum in descending order. Add the terms column by column.
You always get the same number, 101. There are 100 such numbers. Thus, twice the sum of the consecutive from 1 to 100 is . The sum is 5050.
Remarks
(1) If you wish to do well in the American SAT or develop deeper understanding for success in the British system, then learn this example by heart. (2) The proof of the formula for the sum of an arithmetic progression is simply an algebraic version of this argument. We introduce the proof at a higher level.
SLIDE 40
Find the sum of the consecutive integers from 21 to 104 inclusive.
SLIDE 41
Puzzle
Find the sum of the consecutive integers from 21 to 104 inclusive.
Solution
This version of the Little Gauss Problem has been combined here with what we are calling
The Traffic Cone Question
There are six policemen standing between traffic cones along a street while a parade passes by. If each policeman is standing between two traffic cones, how many traffic cones are there?
Solution to the Traffic Cone Question
Let ?be a cone, and ?be a policeman
?????????????
There are six policemen but seven traffic cones. In such questions there is one more boundary than there is box.
In this question the number of consecutive integers is not but 84, because the numbers 21 and 104 are included. Then, as in the previous example, twice the sum is . The answer is half this number, 5,250.
SLIDE 42
Find the sum of the even integers from 16 to 98 inclusive.
SLIDE 43
Find the sum of the even integers from 16 to 98 inclusive.
Solution
We pair up the even numbers
The sum of each pair is . There are 21 such pairs. (This is a “traffic cone” question as well. but we must add 1 more, as we include the boundaries as well.) Hence the sum is
SLIDE 44
The number 24 is divided by a positive integer x. The remainder is 3. For how many different values of x is this true?
SLIDE 45
The number 24 is divided by a positive integer x. The remainder is 3. For how many different values of x is this true?
Solution
This puzzle tests your knowledge of factors and division. Basically, you have to check all eligible numbers.
We can eliminate all factors of 24, as their remainder is 0. These are the numbers, 1, 2, 3, 4, 6, 8 and 12. The only number above 12 with a remainder of 3 is 21. Count 1.
The only solutions are 7, and 21. Answer: 2.
Remark
This kind of problem belongs to that branch of Number Theory known as the Theory of Linear Congruences. This theory was introduced by Gauss (!) in 1801 in the first chapter of his Disquisitiones Arithmeticae.
SLIDE 46
25 tables have either 2 or 4 chairs. The total number of chairs is 86. How many tables have exactly 4 chairs?
SLIDE 47
25 tables have either 2 or 4 chairs. The total number of chairs is 86. How many tables have exactly 4 chairs?
Solution
This kind of problem can be solved by trial and error. Just guess a number and try it, then improve the guess if you are wrong. For example, let us guess 15 tables with 4 chairs. Then there are 10 tables with 2 chairs, and chairs. So there must be more than 15 chairs.
The problem can be solved by algebra. Let x be the number of tables with 4 chairs. Then the number of tables with 2 chairs is . The total number of chairs is 86, so
The answer is 18.
SLIDE 48
Number puzzles
Find all the three-digit numbers such that the sum of the first (hundreds) and last (units) digit is equal to the square of the second (tens) digit.
SLIDE 49
Find all the three-digit numbers such that the sum of the first (hundreds) and last (units) digit is equal to the square of the second (tens) digit.
Solution
The starting point is just to understand what the question is asking.
Let a three-digit number be where a, b and c are digits. The condition in the question translates to .
What we are being asked for is all the combinations that add up to a single digit square number. We cannot lead with a 0, as that would not define a three-digit number. The single-digit squares are 1, 4 and 9 giving respectively.
The numbers are 110, 123, 138, 222, 237, 321, 336, 420, 435, 534, 633, 732, 831 and 930.
SLIDE 50
Number puzzles
Find all three-digit numbers that are the product of three consecutive even numbers
SLIDE 51
Find all three-digit numbers that are the product of three consecutive even numbers
Solution
There is no equation that can be formed, as we do not know the number that is the product of the three numbers. We approach this through a combination of logic and trial and error.
Now as the numbers are consecutive even numbers, then the largest of the numbers cannot be much bigger than 10, since which is too big. Try . This works and is the largest possibility. Hence, all we have to do is list the combinations of smaller numbers, and stop when we get to a two-digit number.
There are three possibilities
? 8, 10, 12 ? 6, 8, 10 ? 2, 4, 6
SLIDE 52
Number puzzles with fractions
x is of y, and y is of z, where . What fraction is x of z?
SLIDE 53
x is of y, and y is of z, where . What fraction is x of z?
Solution
The information translates to
and . This is just a “plugging in” question.
SLIDE 54
Number puzzles with fractions
.
Which of the following is a possible value for a?
A B C D E
SLIDE 55
.
Which of the following is a possible value for a?
A B C D E
Solution
The answer is E because
SLIDE 56
Revision - the arithmetic mean
This is the same as the average. You add up all the values of the data points and divide by the number of data points. The arithmetical mean of 3 and 5 is .
Number puzzles with the arithmetic mean
9 students took a quiz. Their mean score was 76. The mean score of five of these students was 80. What was the mean score of the other four students?
SLIDE 57
9 students to a quiz. Their mean score was 76. The mean score of five of these students was 80. What was the mean score of the other four students?
Solution
The total score of the 9 students was
The total score of the five with a mean of 80 was
The total score of the other four was
The mean score of the other four was
SLIDE 58
Number puzzles with the arithmetic mean
The arithmetical mean of 6 numbers lies greater than 40 and less than 45. Find all possible sums of the 6 numbers.
SLIDE 59
The arithmetical mean of 6 numbers lies greater than 40 and less than 45. Find all possible sums of the 6 numbers.
Solution
Let be the arithmetical mean.
The information translates to . Since these are inequalities, rather than exact inequalities, this translates to
.
For each value the corresponding sum of the 6 numbers is
The possible sums are 246, 252, 258 and 264.
SLIDE 59
Set notation
A set or collection is shown by curly brackets.
Example
is the set or collection of numbers 1, 2, 3 or 4.
Write the following using curly brackets
? The set of numbers
? The set of positive odd numbers less than 10
? The set of square numbers less than or equal to 100
SLIDE 60
? The set of numbers
? The set of positive odd numbers less than 10
? The set of square numbers less than or equal to 100
SLIDE 61
In which of the following sets is the arithmetic mean less than the median?
A
B
C
D
SLIDE 62
In which of the following sets is the arithmetic mean less than the median?
A
B
C
D
The answer is B.
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