Download IMPORTANT FACTS AND HANDY FACTS SUBJECT : MATHS

IMPORTANT FACTS AND HANDY FACTS
SUBJECT : MATHS
CLASS : VIII
CHAPTER : KNOWING OUR NUMBER
1.
Indian place value chart for 9-digit number
Crores
Lakhs
Thousands
TC
C
TL
L
TTh
Th
H
Ones
Tens Units
Period
Place
2.
Inserting commas : A comma is inserted after each period in both the systems.
1 lakh = 1,00,000
3.
International place value chart for a 9 digit number
Millions
Thousands
HM
TM
M
HTh
TTh
Th
H
1 million
Ten million
Hundred million
4.
5.
=
=
=
Ones
T
U
Period
Place
ten lakhs
1 crore
10 crores
In the Hindu-Arabic system, we use ten symbols namely 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 called
digits or figures to represent any number.
Place value of a digit is the product of the digit and its place. PLACE VALUE = FACE VALUE
X PLACE
6.
The number which is one more than the given number is called its successor.
7.
8.
The number which is one less than the given number is called its predecessor.
There are seven symbols to represent numbers of Hindu-Arabic system in Roman numeration.
They are
Roman
I
V
X
L
C
D
M
Numeral
Hindu1
5
10
50
100
500
1000
Arabic
Numeral
9.
While writing numbers in Roman system there are certain rules to be followed.
Rule – 1
The numerals I, X and C cannot be repeated to a maximum of 3 times.
Rule – 2
The numerals V, L and D cannot be repeated.
Rule – 3
10.
If a numeral of lesser value is written before a greater value, the resulting value is obtained by
finding their difference.
Rule – 4
If a numeral of lesser value is written after a greater value, the resulting value is obtained by
finding their sum.
I can be subtracted from V and X only.
CHAPTER 2 : PLAYING WITH NUMBERS
11.
A factor of a number is an exact divisor of that number.
12.
A number is said to be a multiple of any of its factors
Example : We know that 35 = 1 x 35 and 35 = 5 x 7.
This shows that each of the numbers 1, 5, 7, 35 exactly divide 35.
Therefore 1, 5, 7 and 35 are all factors of 35 and 35 is a multiple of each one of the numbers 1,
5, 7 and 35.
13.
All multiples of 2 are called even numbers.
Example : 2, 4, 6, 8, 10 etc.
14.
Numbers which are not multiples of 2 are called odd numbers.
Example :1, 3, 5, 7, 9, 11 etc.
15.
Numbers having more than two factors are known as composite numbers.
Examples : 4, 6, 8, 9, 10 etc.
Note :
(i)
1 is neither prime nor composite
(ii)
2 is the lowest prime number
(iii) 2 is the only even prime number
16.
Two consecutive odd prime numbers are known as twin-primes
Examples :(i) 3, 5 (ii) 5, 7
(iii) 11, 13 etc.
17.
A set of three consecutive prime numbers, differing by 2, is called a prime triplet.
The only prime triplet is (3,5,7)
18.
If the sum of all the factors of a number is twice the number then the number is called a perfect
number.
Examples :6 is a perfect number, since the factors of 6 are 1, 2, 3, 6 and (1 + 2 +3 + 6) = (2 x
6).
19.
Two numbers are said to be co-prime if they do not have a common factor, other
than 1.
Examples :(i) 2, 3 (ii) 3, 4
(iii) 8, 15
Note : (i) Two prime numbers are always co-prime.
(ii) Two co-primes need not be prime numbers.
Examples :6, 7 are co-primes, while 6 is not a prime number
20.
Every even number greater than 4 can be expressed as the sum of two odd prime numbers.
Examples :(i) 6 = 3 + 3
(ii) 8 = 3 + 5
21.
Highest Common Factor (HCF)
The highest common factor can be found by the following methods.
(i)
By listing factors:
Factors of 24 : 1 , 2, 3, 4,6,8,12,24,
Factors of 36 : 1, 2, 3, 4, 6, 9, 12, 18, 36
Common factors of 24 and 36
= 1, 2, 3, 4, 6, 12
Highest common factor of 24 & 36 = 12
(ii)
Division method :
24) 36 (1
24
12) 24 (2
24
0
HCFof 24 and 36 = 12.
(iii)
By prime factorization method
Note :
17.
(i)
The HCF of given numbers is not greater than any of the given numbers.
(ii)
The HCF of two co-primes is 1.
(iii) The LCM of given numbers is not less than any of the given numbers.
(iv) The LCM of two co-primes is equal to their product
(v)
The HCF of two given numbers is always a factor of their LCM.
Product of two numbers = product of their HCF and LCM
Example : Consider the numbers 24 and 36.
LCM of 24 and 36 = 72
HCF of 24 and 36 = 12
Therefore 24 x 36 = 12 x 72 = 864.
CHAPTER :INTEGERS
18.
Negative numbers are placed on left side of ‘0’ on the horizontal number line.
19.
To represent quantities like profit, income increase, rise high, north, east, above depositing,
climbing and so on, positive numbers.
20.
21.
22.
23.
24.
25.
26.
To represent quantities like loss, expenditure, decrease, fall, low, south, west, below,
withdrawing, sliding and so on, negative numbers are used.
Absolute value of a number is only its numerical value without considering the sign into
account.
|- 5| : absolute value is 5
| 20| : absolute value is 20
|+ 3| absolute value is 3
The smallest positive integer is 1.
The biggest negative integer is – 1
0 is greater than all negative integers. 0 is smaller than all positive integers.
Smallest negative integer does not exist as there is no end for integers.
Any rational number a when multiple by itself n times can be written in exponential form as a”.
CHAPTER :LAWS OF EXPONENTS
Let a and b be real numbers and m and n be positive integers. Then the following laws hold:
CHAPTER : PROFIT AND LOSS
1.
Cost Price:
The price, at which an article is purchased, is called its cost price, abbreviated as C.P.
2.
Selling Price:
The price, at which an article is sold, is called its selling prices, abbreviated as S.P.
3.
Profit or Gain:
If S.P. is greater than C.P., the seller is said to have a profit or gain.
4.
Loss:
If S.P. is less than C.P., the seller is said to have incurred a loss.
IMPORTANT FORMULAE
1. Gain = (S.P.) - (C.P.)
2. Loss = (C.P.) - (S.P.)
3. Loss or gain is always reckoned on C.P.
4. Gain Percentage: (Gain %)
Gain % =
Gain x 100
C.P.
5. Loss Percentage: (Loss %)
Loss % =
Loss x 100
C.P.
6. Selling Price: (S.P.)
SP =
(100 + Gain %)
x C.P
100
7. Selling Price: (S.P.)
SP =
(100 - Loss %)
x C.P.
100
8. Cost Price: (C.P.)
C.P. =
100
x S.P.
(100 + Gain %)
9. Cost Price: (C.P.)
C.P. =
100
x S.P.
(100 - Loss %)
10. If an article is sold at a gain of say 35%, then S.P. = 135% of C.P.
11. If an article is sold at a loss of say, 35% then S.P. = 65% of C.P.
CHAPTER :UNITARY METHOD
(i)
(ii)
If the cost of one object is given, we can find the cost of many objects by multiplying the
cost of one object with the number of objects.
If the cost of several objects are given, we can find the cost of one object by dividing the
cost of several objects by the number of objects.
CHAPTER :RATIO AND PROPORTION
1. Ratio: The ratio of two quantities a and b in the same units, is the fraction
a/b = a : b
In the ratio a : b, we call a as the first term which is also known as antecedent and b, the
second term which is also called consequent.
For example:4:5
Here 4 is antecedent and 5 is consequent
Important Rule :
The multiplication or division of each term of a ratio by the same non-zero number does
not affect the ratio.
2. Equality of two ratios is called Proportion – a : b :: c : d
Fourth Proportional:
If a : b = c : d, then d is called the fourth proportional to a, b, c.
Third Proportional:
a : b = c : d, then c is called the third proportion to a and b.
Thus , a x d = b x c
Product of extremes = Product if means
CHAPTER : SIMPLE INTEREST AND COMPOUND INTEREST
1. Simple interest is charged only on the principal amount. The following formula can be used to
calculate simple interest:
Simple Interest = (P × r × t) / 100
Where,
P is the principle amount;
r is the interest rate per period;
t is the time for which the money is borrowed or lent.
2. Compound Interest
Compound interest is charged on the principal plus any interest accrued till the point of time at which
interest is being calculated.
Compound Interest = P ×[ (1 +
)n–1]
Where,
P is the principle amount;
r is the compound interest rate per period;
n are the number of periods.
CHAPTER : MENSURATION
1. Area and Perimeter of a Rectangle
(i)
Area = l x b
(ii)
Perimeter = 2(l+b)
(iii) Diagonal = √ +
2. Area of 4 walls of a room
(i)
Area of the 4 walls = 2 h (l+b)
(ii)
Diagonal of the room = √ +
3. Area and Perimeter of a Square
+ ℎ
(i)
(ii)
Area = (side)2
Perimeter = 4 side
(i)
Area of a triangle = x b x h
(ii)
Area of a right triangle = x product of its legs
(iii)
Area of an equilateral triangle =
(iv)
Area of an isosceles triangle with base b units and each equal side is a units is
(iii) Diagonal = √2 a
4. Triangle
xbx
(v)
√
(side)2
sq units.
If a, b, c are the sides of a triangle ABC, then area is
sq
units where s is the semi perimeter and is
5. Areas of parallelograms and Rhombus
(i)
Area of a parallelogram = base x height
(ii)
Area of a rhombus = x product of diagonals
6. Circumference and Area of circle
(i)
Circumference = 2 r
(ii)
Area = r2
GEOMETRY
PARALLEL LINES Two lines in a plane which do not meet even when produced indefinitely in
either direction, are known as parallel lines.
If I and m are two parallel lines, we write l || m
and read it as l is parallel to m.
Clearly, when l || m, we have, m|| l.
l
Two lines are said to be parallel if
(i)
they both lie in the same plane, and,
(ii)
they do not intersect (or cross each other)
1. pairs of adjacent angles always add up to 180degrees, as you can easily see from the figure.
Thus ∠1 + ∠2 = 180degrees
∠2 + ∠33 = 180 degrees
2.
3.
4.
5.
∠3 + ∠4 = 180 degrees
∠5 + ∠6 = 180 degrees
Angles in the same relative position around the two intersection points are called corresponding
angles. Thus
∠1 and ∠5 are corresponding angles,
∠4 and ∠8, ∠2 and ∠6, and also ∠3 and ∠7.
Corresponding angles are equal.
∠3 and ∠5 are called alternate interior angles.
∠4 and ∠6 are also alternate interior angles.
Alternate interior angles are equal.
∠2 and ∠8 are called alternate exterior angles. ∠1 and ∠7 are also alternate exterior angles.
Alternate exterior angles are equal.
Angles on the same side of the transversal are supplementary.
∠4 + ∠6 = 180degrees
∠5 + ∠3 = 180 degrees
∠1 + ∠7= 180 degrees
∠2 + ∠8 = 180 degrees
UNITS OF MEASURING AN ANGLE
The standard unit of measuring an angle is degree, to be denoted by ‘o’.
TYPES OF ANGLES
RIGHT ANGLE
A quarter turn of a ray #####$
!" about O describes an angle called a right angle.
The measure of a right angle is 90
In the adjoining figure, ∠ AOB = 900.
B
0
1 right angle = 90 .
10 = 60minutes, written
as 60’.
1’ = 60 seconds, written
as 60”.
O
(A right angle)
A
PERPENDICULAR LINES Two lines l and m are said to be perpendicular to each other if one of the
angles formed by them is a right angle, and we write ⊥ & (read as l is perpendicular to m).
POLYGONS A simple closed figure formed of three or more line segments is called a polygon.
TRIANGLE A polygon of 3 sides is called a triangle.
QUADRILATERAL A polygon of 4 sides is called a quadrilateral.
PENTAGON A polygon of 5 sides is called a pentagon.
HEXAGON A polygon of 6 sides is called a hexagon.
HEPTAGON A polygon 7 sides is called a heptagon.
OCTAGON A polygon of 8 sides is called an actagon.
Triangle Let A, B and C be three noncollinear points.
Then, the figure formed by the three line segments AB, BC and CA is called a triangle
with verticles A, B and C.
Such a triangle is denoted by the symbol ∆"().
(I) EQUILATERAL TRIANGLE A triangle having all sides equal is called an
equilateral triangle.
A
B
Equilateral Triangle
C
A
B
C
In the figure given above, ∆"() is an equilateral triangle in which AB = BC = CA.
(II) ISOSCELES TRIANGLE A triangle having two sides equal is called an isosceles triangle.
(III) SCALENE TRIANGLE A triangle having three sides of different lengths is called a
scalene triangle.
ON THE BASIS OF ANGLES
(I) ACUTE TRIANGLE A triangle each of whose angles measures less than 900 is called an acuteangle triangle or simply an acute triangle.
(II) RIGHT TRIANGLE A triangle whose one angle measures 900 is called a right-angled triangle
or simply a right triangle.
(III) OBTUSE TRIANGLE A triangle one of whose angles measures more than 900 is called an
obtuse-angled triangle or simply an obtuse triangle.
RESULT 1 Each angle of an equilateral triangle measures 600.
RESULT 2 The angles opposite to equal sides of an isosceles triangle are equal.
RESULT 3 A scalene triangle has no two angles equal.
The sum of the angles of a triangle is 1800, or 2 right angles.
(i)
a triangle cannot have more than one right angle.
(ii)
a triangle cannot have more than one obtuse angle.
(iii) In a right triangle, the sum of the acute angle is 900.
CONGRUENT TRIANGLES
Any triangle is defined by six measures (three sides, three angles). But you don't need to know all of
them to show that two triangles are congruent. Various groups of three will do. Triangles are congruent
if:
1. SSS (side side side)
All three corresponding sides are equal in length.
SAS (side angle side)
A pair of corresponding sides and the included angle are equal.
2. ASA (angle side angle)
A pair of corresponding angles and the included side are equal.
3. AAS (angle angle side)
A pair of corresponding angles and a non-included side are equal.
4. RHS(Right angle , hypotenuse and leg of a right triangle)
Two right triangles are congruent if the hypotenuse and one leg are equal.
5. Two triangles are said to be congruent if their respective sides and angles are equal.
Here we will learn about the congruence of triangles and the axioms and rules for two triangles
to be congruent.
PROPERTIES OF TRIANGLES
Triangle Angle Sum Theorem
The sum of the measures of the interior angles of a triangle is
180 degrees
Exterior Angle
The angle formed by one side of a triangle with the extension of another side is called an exterior
angle of the triangle.
EXTERIOR ANGLE THEOREM
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote
interior angles.
PYTHAGORAS THEOREM
The longest side of the triangle is called the "hypotenuse", so the definition is:
In a right angled triangle:
the square of the hypotenuse is equal to
the sum of the squares of the other two sides
a 2 + b2 = c
SIDE SUM PROPERTY
Sum of Two Sides of the Triangle is always Greater than the Third Side."
Similarly, we can check and prove that "The difference of two sides of a triangle is smaller than
the third side"
Thus , In a given triangle ABC
A
B
AB+BC > AC
AB+AC> BC
AC+BC> AB
C
QUADRILATERAL : A simple closed figure bounded by four line segments is called a quadrilateral.
CONVEX QUADRILATERAL A quadrilateral in which the measure of each angle is less than 1800
is called a convex quadrilateral.
C
D
A
(Convex quadrilateral)
B
DIMENSIONAL FIGURES
33-DIMENSIONAL FIGURES
NETS OF 33-DIMENSIONAL FIGURES