209313998_Math_Let_Reviewer.ppt.pdf

NUMERATION SYSTEM
A system of reading and writing numbers is a numeration
system. This consists of symbols and rules or principles on
how to use these symbols. Our system of reading and writing
numbers is the decimal system or the Hindu-Arabic system.
Ten digits are used- 0,1,2,3,4,5,6,7,8, and 9. This system
is based on groups of ten. It uses the place value concept.

MIXED DECIMALS
• In reading mixed decimals the whole number is read first,
followed by the word “and” to locate the decimal point,
then the decimal. In reading decimal, read the numbers to
the right of the decimal point as if it were whole number.
Then say the name of the position of the last digit on the
right as the same of the decimal. We read the number in
the second place value chart as “sixty-eight thousand, three
hundred fourteen and three hundred sixty-eight million
nine hundred forty-two thousand fifteen billionths. The ths
in the name of a digit to the right of the decimal point
distinguishes it from the name of the corresponding digit to
the left of the decimal point. There are no commas in the
decimal.

ROUNDING OFF NUMBERS
When portion of the dropped begins with 0, 1,2,3,4 or less than 5, the last digit to be retained is
unchanged.
Example: 34, 214.018 34, 214 rounded off to the nearest ones
34, 210 rounded off to the nearest tens
34, 200 rounded off to the nearest hundreds
When the portion to be dropped begins with 6,7,8,9, or a digit greater than 5, the last digit to be
retained in increased by 1.
Example: P98. 726 P 98.73 rounded off to the nearest centavo
P 99 rounded off to the nearest pesos
P 100 rounded off to the nearest ten pesos
When the portion to be dropped is exactly 5, 50, 500 .5 etc. or exactly half of the preceding place value,
we round it off to the nearest even number.
Example: P 47.235 P 47.24 rounded off to the nearest centavos
P 25,000 P 20,000 rounded off to the nearest ten thousands (half of the preceding
place value)
The digits dropped in the whole number are replaced by zero or zeros. Using the examples above.
Examples: 34, 214.018 34, 210 rounded off to the nearest ones
34, 200 rounded off to the nearest hundreds
34, 000 rounded off to the nearest thousands

SIMPLE AVERAGES
To get the simple average, we get the sum of all given
values and divide the sum by the number of values.
Example: Find the average of the following grades:
85%, 78%, 87%, 80%, 83%, 90%.
Solution: 85%
78% 503 / 6 = 83.5%
87%
80%
83%
90%_
503%

WEIGHTED AVERAGE
To get the weighted average, we multiply the quantities by the
measures involved. Then we divide the sum of the products by the
sum of the quantities.
Example:
A sewer made 10 bags. On 3 bags, she spent 2 hours each; on 4
bags, 3 hours each; and on 3 bags, 1 hour each. What was the
average time spent on each bag?
Solution:
3 bags x 2 hours = 6 hours
4 bags x 3 hours = 12 hours
3 bags x 1 hour = 3 hours
10 bags = 21 hours
21 hours / 10 bags = 2.1 hours or 2 hours and 6 minutes.

FRACTIONS
• A fraction is one or more of the equal parts into
which a whole is divided. The terms of the
fraction are numerator and denominator. The
numerator is the number above the line showing
how many of the equal parts are expressed or
taken. The denominator is the number below the
line showing into how many equal parts the
whole is divided. The line between the numerator
and the denominator is the “vinculum” which
means “divided by”.

KINDS OF FRACTIONS
• Proper fraction
• Improper fraction
• Mixed number
• Similar fractions
• Dissimilar fractions
• Decimal fraction

OTHER TERMS
• Least common denominator (LCD)
• Lowest-terms fraction
• Reciprocal of a fraction

LAWS OF FRACTIONS
• The value of a fraction does not change if its
terms are multiplied by the same number
except zero. Example: ½ x 3/3 = 3/6
• The value of a fraction does not change if its
terms are divided by the same number except
zero. Example: 3/9 / 3/3 = 1/3

CONVERSION OF FRACTIONS
• Improper fraction to a whole or mixed numbers. To change the improper
fraction to a whole or mixed number, we divide the numerator of the
fraction by its denominator. The remainder becomes the numerator and
the divisor the denominator of the fraction.
• Mixed number to an improper fraction. To change a mixed number to an
improper fraction, we multiply the whole number by the denominator of
the fraction. Then, we add the product to the numerator of the fraction
over the denominator.
• Lower terms fraction to higher terms fraction. To raise a fraction from
lower to higher terms, we multiply both numerator and denominator by
the number that will result to the specified denominator of bigger value.
• Higher terms fraction to lowest terms fraction. To reduce a fraction from
higher to lowest terms, we divide both the numerator and denominator
by the greatest common factor.
• Dissimilar fractions to similar fractions. To change dissimilar fractions we
find the LCD and divide it be each denominator. Then we multiply each
quotient by the numerator and place it over the LCD

ADDITION OF FRACTIONS
• In adding similar fractions, we add the numerator
and write the sum over the common
denominator.
• In adding dissimilar fractions, we make the
fractions similar before addition.
• In adding mixed numbers, we add the whole
numbers first then combine the two. If the whole
numbers are not very large we can change the
mixed numbers to improper fractions; then we
add.

SUBTRACTION OF FRACTIONS
• In subtracting similar fractions, we change the
fraction to similar fractions first before we subtract.
• In subtracting dissimilar fractions, we change the
fraction to similar fractions first before we subtract.
• In subtracting mixed numbers, we subtract the
whole numbers first and then the fractions. We can
also subtract them by changing the mixed numbers
to improper fractions. This will be difficult if the
whole numbers are very large.

• When the value of the fraction in the subtrahend
is bigger than the value in the minuend, we
borrow 1 unit from the whole number of the
minuend, change it to similar fractions and add it
to the fraction. Then we proceed to subtraction.
• If we change both mixed numbers to improper
fractions, we can proceed to subtraction at once
without borrowing.
• When the fractions in the mixed numbers are
dissimilar, we change them first to similar
fractions; then we subtract.

MULTIPLICATION OF FRACTIONS
• In multiplying fractions, we multiply the
numerators together to form a new
numerator. Then we multiply the
denominators to form new denominator.

FRACTION AND A WHOLE
NUMBER
• In multiplying a fraction and a whole number,
we multiply the whole number by the
numerator of the fraction and write the
product over the denominator of the fraction.

FRACTION AND MIXED NUMBER
• In multiplying a fraction and a mixed number,
we change the mixed number to an improper
fraction and then we proceed to
multiplication.

A MIXED NUMBER AND A WHOLE
NUMBER
• In multiplying a mixed numbers, we change
the mixed numbers to improper fractions;
then, we proceed to multiplication.

MIXED NUMBER AND MIXED
NUMBER
• In multiplying mixed numbers, we change the
mixed numbers to improper fractions; then we
proceed to multiplication.

CANCELLATION
• In canceling factors, we divide a pair of
numerator and denominator by the same
number just as when we reduce a fraction to
lowest terms.

DIVISION OF FRACTIONS
• In dividing fractions, we invert the divisor and
multiply. All rules pertaining to multiplication
of fractions will be applied.

RATIO
• A ratio is a relation between two like numbers
or quantities expressed as a quotient or
fraction.
• We can also reduce ratios to their lowest
terms in the same manner that fractions are
reduced.

PROPORTIONS
• A proportion is a statement that two ratios are
equal. There are four terms in a proportion: the first
and fourth terms are called extremes; the second
and the third are known as the means. Hence, the
ratio 6:24 which is equal to the ratio 1:4 can be
expressed as a proportion 6: 24 = 1:4, 6 and 4 are
the extremes while 24 and 1 are means. To check
whether our proportion is correct, the rule is: the
product of the means equals the product of the
extremes.
•

CONVERSION TECHNIQUES
• To reduce a decimal to a common fraction, we
write the given decimal number disregarding
the decimal point as the numerator of a
common fraction with a denominator of the
power of 10 of the given decimal.
• Examples: .7 = 7/10 There is one decimal
place so the denominator is 10.
• To reduce a common fraction to a decimal, we
divide the numerator by the denominator.
• Example: ½ = 0.5

• To change percent to a decimal, we move the
decimal point two places to the left and drop
the percent sign. If the percent is in fractional
units, we change first the fraction to decimal
before moving the decimal point.
• Example: 40% = .40
• To change a decimal to a percent, we move
the decimal point two places to the right and
add the percent sign.
• Example: .23 = 23%

• To change a percent to a fraction, we drop the percen
sign and replace it by 100 as denominator. If the
percent is in decimal, we move the decimal point two
places to the left after dropping the percent sign. The
we convert the decimal to its fractional equivalent. If
the percent is in fraction, divide it by 100 and drop th
percent sign.
• Example: 27% = 27/100

• To change a fraction to a percent, we divide
the numerator by the denominator, then we
move the decimal point of the quotient two
places to the right and add the percent sign.
For a mixed number, we change it first to an
improper fraction before performing the
indicated division.
• Example: 3/5 = .60 = 60%

PERCENTAGE PROBLEMS
• A percentage is the result obtained by
taking a certain percent of a number.
Percentage is equal to the base times the rate.
The base is the number on which the
percentage is computed. The rate is the
number indicating how many percent or
hundredths are taken.

PERCENTAGE FORMULAS
• P = R X B R = P/B B = P/R
• The base (B) is usually preceded by the
preposition “of” in word problems. “Of”
indicates multiplication. The word “is” is
symbolized by the equal sign (=). Other words
may be used instead “of” as “as many as” , “as
large as”, “as great as”, “ as much as”.

PERCENTAGE FORMULAS cont..
• The rate ( R ) is identifiable because it is
usually in the form of a percent. However, it
can also be in decimal, or in a fraction.
• The percentage (P) refers to the equal
quantity or number of items represented by
the rate.

Sample Problems:
• What number is 5% of 480
P = R x B
P = .05 x 480
P = 24

• Percentage is unknown. 5% is changed to a
decimal (.05%) before multiplying it to 480.
Another way is to change 5% to fraction ( 5 /
100 or 1/20) before multiplying it to 480 and
we arrive at the same product.

P 26 is what part of P 130?
P = R x B
R = P / B
R = 26 / 130
R = 1 / 5

PROBLEMS INVOLVING PERCENTAGE
INCREASE OR DECREASE
• Note: We don’t multiply or divide a number by
percent. We always change the percent to a
decimal or a fraction first before multiplying or
dividing.

INCREASE OR DECREASE Cont…
• To find the percentage of increase or
decrease, we multiply the base by the rate
and add the product to the base if it is an
increase but subtract the product from the
base if it is a decrease.
B + ( B X R ) = PERCENTAGE OF INCREASE
B – ( B X R ) = PERCENTAGE OF DECREASE

• Example: P 120 increased by 12% is how
much?
B R = P of increase
P 120 + ( P 120 x .12) = P 134.40

• To find the rate of increase or decrease, get
the difference between the two given related
values and divide it by the base or original
quantity. Change the fraction to percent if it is
needed.
LARGER VALUE – SMALLER VALUE = RATE OF INCREASE OR DECREASE
BASE OR ORIGINAL QUANTITY

• Example : Find what percent, more than 25 is 30?
• 30 - 25 = 5 = 20%
25 25
• To determine the base when a number that is a
fractional part or percent greater than or smaller
than that of the unknown value, we divide the given
number or percentage by the sum (if greater than) or
the difference (if smaller than) between 1 and the
given fraction or 100% and the given rate.

• P = BASE OF INCREASE
1 + FRACTION
1 - FRACTION
100% + Given %
100% - Given %

Example: 2/5 greater than amount is P 70?
P 70= P 70 = P 70 = P 70 x 5/7 = P 50
1 + 2/5 5/5 + 2/5 7/5

DIVISIBILITY RULES
• Rule 1. Divisibility by 0 or 1
• Any number is divisible by 1. Any number
divided by 1 is the number itself. 0 divided by
any number is 0. Any number divided by 0 is
undefined.

DIVISIBILITY RULES
• Rule 2. Divisibility by 2.
• A number is divisible by 2 if, and only if, the
last digit is even, that is, if it is 0,2,4,6,8,.
• A number is divisible by 3 and if only if its
digital root (sum of the digits) is divisible by 3.

DIVISIBILITY RULES
• Rule 4. Divisibility by 4
• A number is divisible by 4 if, and if only of, the
last two digits form a number which is
divisible by 4.
• A number that ends in 0 or 5 is divisible by 5.

DIVISIBILITY RULES
• Rule 6. Divisible by 6.
• A number that is divisible both by 2 and 3 is
divisible by 6.

DIVISIBILITY RULES
To test if the number is divisible by 7 follow these steps.
• For three or more than three digit number.
– Multiply the last digit (unit digits) by 2.
– Get the difference of these product ( from step 1) and the
remaining number (without the units digits)
– Repeat step 1 and 2 until you reached a two-digit
difference. If this difference is divisible by 7, then the
number is divisible by 7.

DIVISIBILITY RULES
• If the number consist of more than 3 digits you
can use these alternative rule.
– Divide the number into periods of three digits
each beginning with the last digits.
– Add together the odd numbered periods, and
then all the even numbered periods.
– Take the difference of the two sums. If this
difference is divisible by 7, then the number is
divisible by 7.

DIVISIBILITY RULES
• A number is divisible by 8 if and if only if the
last three digits form a number that is divisible
by 8.

DIVISIBILITY RULES
• A number is divisible by 9 if, and only if the
digital root ( sum of the digits) is 0 or 9. When
the digital root of a number is not 0 or 9, that
digital root is the remainder when the number
is divided by 9.

DIVISIBILITY RULES
• A number ending in 0 is always divisible by 10.

• To test if the number is divisible by 11,
alternatively subtract and add the digits from
the last digits to the first, that is from right to
left. If the end result is divisible by 11, then
the number is divisible by 11, otherwise the
end result is the remainder when the number
is divisible by 11. OR
DIVISIBILITY RULES

DIVISIBILITY RULES
Cont…
• A number is divisible by 11 if the difference
(large minus smaller) between the sum of the
odd-numbered digits and the sum of the even
numbered digits is divisible by 11.

DIVISIBILITY RULES
• A number is divisible by 12 if, and only if, it is
divisible by 3 and 4.

DIVISIBILITY RULES
• Rule 13. Divisibility by 11,7 and 13
• To test if the number with more than three
digits is divisible by 7, 11 and 13, follow the
step by step procedure:
• Starting from the right end of the number,
divide the number into periods of three digits.

DIVISIBILITY RULES
• Rule 13. Divisibility by 11,7 and 13 Cont…
• Add the odd numbered periods together and
the even numbered periods together.
• Find the difference of the two sums. If that
difference is divisible by 7 or 11 or 13, then
the number is divisible by whichever divisor
that divides it evenly.

DIVISIBILITY RULES
• Rule 14. Divisibility by 25; 75 and 125
• A number is divisible by 25 if, and only if, the
last two digits form a number divisible by 25.
A number divisible by 75 if it is divisible by 3
and 25. A number is divisible by 125 have the
last three digits divisible by 125.

FACTORS, MULTIPLIES, GCF, LCM,
PRIME OR COMPOSITE
• Any counting number a can be expressed as the product of b
and c, which are both counting numbers.
• That is, a = b x c
Then a is a multiple of b and of c.
Consider 42 and 68. Then,
42 = 6 x 7
= 2 x 21
= 3 x 14
= 1 x 42

PRIME OR COMPOSITE
• Thus, 42 is a multiple of 1, 2, 3, 6, 7, 14, 21
and 42.
• 68 = 2 x 34
= 4 x 17
= 1 x 68

PRIME OR COMPOSITE
• What are the factors of 42 and 68?
• We mentioned previously that 1 and the
number itself are factors of the given number.
This time we will be considering the kinds of
numbers according to their factors.
• Consider the following counting numbers:
43 10 81 12 97

PRIME OR COMPOSITE
• Let us list the factors of the above counting
numbers by expressing them as the product of
two counting numbers.
43 = 43 x 1
10 = 2 x 5 = 10 x 1
81 = 9 x 9 = 27 x 3 = 81 x 1
12 = 4 x 3 = 2 x 6 = 12 x 1
97 = 97 x 1

PRIME OR COMPOSITE
• We see that 43 and 97 have factors 1 and themselves.
While 10, 81, and 12 have different factors other than
1 and themselves. Thus we come up with the
following definitions:
• if a counting number has 1 and itself as its only
factors, then the counting number is called a prime
number.
• If a counting number has factors other than 1 and
itself, then the counting number is called a composite
number.

PRIME OR COMPOSITE
• Example: Factors the following number: 13, 9,
2, and 21.
• 13 = 13 x 1
• 9 = 3 x 3 = 9 x 1
• 2 = 2 x 1
• 21 = 3 x 7 = 21 x 1

PRIME OR COMPOSITE
• Note that 13 and 2 are prime numbers, while
9 and 21 are composite numbers.
• A prime number is a number greater than one,
which has only two factors, one and itself.
• The prime numbers less than 50 are 2,
3,5,7,11,13,17,19,23,29,31,37,41,43 and 47.

Composite Numbers and Prime
Factors
• A whole number which is not prime, and is
greater than 1 is called a composite number. It
can be expressed as a product of two smaller
factors. Examples are 4,6,8,10.
• The number 0 and 1 are special numbers
which are neither prime nor composite.

Greatest Common Factor (gcf) or
greatest common divisor (gcd)
• of two or more whole number is the greatest
whole number that is a factor or a divisor of
the given numbers. This is less than or equal
to the smallest number given. If the greatest
common factor of the given number is 1, then
the numbers are said to be relatively prime.

Greatest Common Factor (gcf) or
greatest common divisor (gcd) cont…
• Consider all the factors of 12 and 36.
12 = 1, 2, 3, 4, 6, 12
36 = 1, 2, 3, 4, 6, 9, 12, 18, 36
• The factors of 12 and which are also the
factors of 36 are called common factors. So 1,
2, 3, 4, 6 and 12 are the common factors of 12
and 36
• The largest common factor, 12, is called the
Greatest Common Factor or GCF

Greatest Common Factor (gcf) or greatest
common divisor (gcd) cont…
• Remarks:
• The number 1 is neither prime or composite.
• The number 2 is the smallest prime number,
and it is the only even prime number.

• The first ten prime number are 2, 3, 5, 7, 11,
13, 17, 19, 23, and 29. Since we can always
determine more prime numbers (although this
may be quite a chore), we say there are an
infinite number of primes.
• Prime and composite are terms that relate
only to counting numbers.

• There are couple of ways to determine the prime factorization
of a number. One of these methods is described as follows:
• Method 1: Factor Tree
• Find two numbers whose product is the given number.
• if each of these factors is prime, then you have found the
prime factors of the given number.
• If the factors in step 2 are not prime numbers, find two
numbers whose product is equal to the factors in step 2.
• Repeat step 3 until all the factors are prime numbers.

• Example: Write the prime factorization of 24.
• Solution:
– Find the numbers that when multiplied, give 24.
Two possible numbers are 4 and 6.
– Since neither factor is prime, we continue with
step 3
– Two numbers that give a product of 4 are 2 and 2.
Go to step 2.

• Method 2: Decomposition
• Repeatedly divide the given number by prime
numbers like 2,3,5, and so on, until a quotient
of 1 is obtained. All remainders obtained in
these division must be 0.
• Write the prime factorization by using all the
divisors from step 1 as factors.

• GCF
• If c is the greatest whole number that can
divide a and b, such that the quotient is again
a whole number, then c is the greatest
common factor (GCF) of a and b. We denote
GCF of a and b as (a, b) = c.

• LCM
The least common multiple (LCM) of any two
or more whole numbers is the smallest whole
that they can devide such that the quotient is
again a whole number. We denote the LCM of
a and b as (a,b) = c.

TOPICS ON ANGLES AND LINES
• Definition
• Betweeness: Let A, B, and C be the three points on a
line, B is between A and C if a<b<c or c>b>a. B is also
between A and C if /AB/ +/ BC/ = /AC/.
• Line segment. The line segment determined by the
points P and Q is the set consisting of P and Q and all
the points between them. The length of the line
segment is the distance between points P and Q.

Cont…
• Ray PQ. This is the union of the line segment
PQ and all the points on the line PQ, such that
Q is between P and these points. Point P is the
endpoint of the ray.
• Ray opposite ray PQ: This is the set consisting
of the point P and all the points of the line PQ
which do not belong to ray PQ. Opposite rays
is always collinear.

• Collinear: Two or more points are collinear if
and only if there is a line containing all these
points.
• Coplanar: Two or more points are coplanar if
and only if there is a plane containing these
points.
• Midpoint: R is the midpoint of line segment
PQ if R is on line segment PQ and PR = RQ
Cont…

Cont…
• Angle: This is a set of points in the union of
two noncollinear rays which intersect at their
endpoints. The rays are the sides of the angle
and the common endpoint is the vertex.
• Right,Acute and Obtuse angles: An angle is
right if it measures 90 degrees. It is acute if its
measure is less than 90. It is obtuse if its
measure is more than 90.

Cont…
• Vertical angles: These angles are vertical
when their union forms two intersecting lines.
• Complementary angles: Two angles having 90
as the sum of their measures.
• Supplementary angles: Two angles having 180
as the sum of their measures.

• Adjacent angles: Two angles which are
coplanar, have common vertex and a common
side, and the intersection of their interiors is
empty.
• Congruent angles: Angles with the same
measures.
• Congruent line segments: Segments with the
same length.
Cont…

• Perpendicular: A line, ray, or segment is
perpendicular to another line, ray or segment
if they intersect and form a right angle. The
point of intersection is called the foot of the
perpendicular.
• Parallel lines: Two lines which are coplanar
and do not intersect.
Cont…

• Skew lines: Two lines which do not lie on one
plane.
• Transversals: This is a line that intersects two
other at different points. Note: Different kinds
of angle formed: alternate interior angles,
alternate exterior angles, and corresponding
angles.
• Concurrent lines: Three or more lines that meet
at point.
Cont…

Topics on Triangle
• Triangle: Triangle ABC (denoted by ABC) is the
union of segments AB, BC, and AC. The points
A,B, and C are the vertices, the segments
AB,BC, and AC are the sides, <A,<B,<C are the
angles.
• Sides Included between two angles: This is
the side whose endpoints are the vertices of
the two angles.

• Angle Included between two sides: This is the
angle formed by the two sides.
• Right triangle: A triangle with one right angle.
The sides that are perpendicular (sides
forming the right angle) are called the LEGS,
and the side opposite the right angle is
hypotenuse. The hypotenuse is longer than
any of the legs of the right triangle.
Topics on Triangle

• Obtuse triangle: A triangle with one obtuse
angle.
• Acute triangle: A triangle which has three acute
angles.
• Isosceles triangle: A triangle with two sides
congruent. The congruent sides are called the
LEGS and the third side, the BASE. The VERTEX
angle is the angle formed by the legs; the two
remaining angles are the Base angles.
Topics on Triangle

• Scalene triangle: Triangle having no sides
congruent.
• Equilateral triangle: A triangle with all sides
congruent.
• Equiangular triangle: A triangle with three
congruent angles. The measure of each is 60
degrees.
Topics on Triangle

• Altitude of a triangle: This is a line segment
drawn from the vertex of the triangle
perpendicular to the line containing the
opposite side.
• Median of the triangle: This is a line segment
from the vertex to the midpoint of the
opposite side.
Topics on Triangle

• Triangle:
A = ½ b h
P = a + b + c
• Square:
A = s2
P = 4s
FORMULAS for Areas and
Perimeters of geometrical figures:

• Rectangle:
A = L x w
P = 2L + 2w
• Trapezoid:
A = ½ h (b1 + b2)

• Rhombus:
A = ½ d1d2
• Circle: A = π r2
C = 2π r
• Area of equilateral triangle
A = s2/4

Algebra
Basic Rules:
• Commutative Property of addition
a + b = b + a
• Commutative Property of Multiplication
ab = ba
• Associative Property of Addition
(a + b) + c = a + ( b + c)
• Associative Property of Multiplication
( ab) c = a ( bc )

• Distributive property
a ( b + c ) = ab + bc ( a + b) c = ac + bc
• Additive Identity Property
a + 0 = 0 + a = a
• Multiplicative Identity Property
a .1 = 1 . a = 1
Algebra
Basic Rules:

• Additive Inverse Property
• a + (-a) = 0
• Multiplicative Inverse Property
• a . 1/a = 1, a ≠ 0
Algebra
Basic Rules:

Properties of Exponents
• xm . x n = xm+n
• ( xy)n = xnyn
• (xm)n = xmn
• xm = xm – n ; m > n
ym
• (x/y)n = xnyn
• x0 = 1
• x-n = 1/ xn ; x ≠ 0

INTRODUCTION to PROBABILITY
THEORY
• Techniques in Counting:
– The Fundamental Principle. If an event E1 can
happen in n1 number of ways and another event E2
can happen in n2 number of ways, then the
number of ways both events can happen in the
specified order = n1 n2 ways.
• Permutation is an arrangement or objects
wherein order is taken into account.

• Permutation of objects taken all at a time.
n P n = n!
• Permutation of n objects taken r at a time.
n P r =
• Circular Permutation ( one position must be
fixed)
( n-1) P (n-1) = (n – 1) !
THEORY

Permutation of n objects not all
distinct
THEORY
Combination is a selection of objects with
no attention given to the order of the
objects.

Combination of n objects taken all at
the same time
n C n = 1
Combination of n objects taken r at a time.
n C r =
THEORY

• Combination in a series
n C 1 + n C 2 + n C 3 + … n C n = 2n – 1
– Classical Definition of Probability
• The probability of the occurrence of an event (called
success) is
Pr (E) =
Where
n = no. of successes
N = total number of possible outcomes
THEORY

Scientific Notation
• A number is expressed in scientific
notation when it is written in the form
a x 10n
where 1 is less than/equal to a < 10 and n is
an integer.

RADICAL EXPRESSIONS
• If n is a positive integer greater than 1, a
and b are real numbers such that
• xn = a, then x is the nth root of a.
• Let n be a positive integer greater than 1,
and let a be a real number such that is
denoted in real numbers. Then, a1/n =

Radicals
• If n is a positive integer greater than 1, and a
and b are real numbers such that
bn =a, then b is an nth root of a.
Example:
• 7 is a square root of 49 because 72 = 49;
furthermore
• -7 is also a square root of 49 because (-7)2 = 49

Radicals
• Let n be a positive integer greater than 1, and a be a
real number. The principal nth root of a is denoted
by , and has the following defining properties:
• If a then is the positive nth root of a.
• If a < 0 and n is odd, then is the negative nth root
of a , and = 0
• If a and n is even, then has no meaning in
real numbers.

209313998_Math_Let_Reviewer.ppt.pdf

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