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Integers (Multiplication ((+) x (+) = + (-) x (-) = + (+) x (-) = - (…
Integers
Multiplication
(+) x (+) = +
(-) x (-) = +
(+) x (-) = -
(-) x (+) = -
Closure
a x b is also an integer
Commutative
a x b = b x a
Associative
(a x b) x c = a x (b x c)
Multiplicative identity
a x 1 = 1 x a = a
Multiplication by zero
a x 0 = 0 x a = 0
Distributive property of multiplication over addition
a x (b + c) = (a x b) + (a x c)
Division
(+) / (+) = (+)
(-) / (-) = (+)
(+) / (-) = (-)
(-) / (+) = (-)
Closure
a / b is not always integer
Commutative
a / b notequal b / a
Associative
a / (b / c) not equal (a / b) / c
Property of one
a / 1 = a
Property of zero
0 / a = 0
a / 0 = not defined
Addition
Closure Property
The sum of two integers is always an integer.
a & b are integers,
a + b is an Integer.
Commutative
a & b are integers
a + b = b + a
Associative
for any three integers a, b & c
a + (b+c) = (a+b) + C
existence of additive Identity
a + 0 = a = 0 + a
Existence of additive inverse
a + (-a) = 0 = (-a) + a
Subtraction
Closure Property
a - b is also an integer
Commutatuive
a - b not equal b - a
Associative
(a-b) - c notequal a - (b-c)
Define
Collection of numbers
.....-4,-3,-2,-1,0,1,2,3,4,..
1,2,3,....are called positive integers
-1,-2,-3,.... are called Negative Integers
Algebraic Expressions
Addition & Subtraction
You can Add or subtract only like terms
Horizontal Method : all expressions are written in a horizontal line & different like terms are arranged into their individual groups & then added
Column Method : expression to be added is written in a separate row such that their like terms are arranged one below the other in a column
A combination of constants & variables connected by any of the symbols +, - , X or / is called an algebraic expression
Perimeter Formulae (P) : sum of lengths of its sides
Geometrically X2 = X x X represents the area of a square of side x units.
expressions : 13 x2y - 7 z3
Terms : 13x2y, -7z3
Factors : 13,x,x,y -7,z,z,z,
In 9ab2, number 9 is the coefficient of ab2, a is coefficient of 9b2 and b2 is the coefficient of 9a
In the algebraic expression 12x2 + 15y2 -16, the constant term is -16
Like Terms : terms having the same algebraic factors.
e.g. : 2x2, 3x2, -18x2
Unlike Terms : Terms having different algebraic factors are unlike terms
e.g. 4p, 5q
Monomial : expression containing only one term
Binomial : expression containing two unlike terms
Trinomial : which contains three unlike terms
Quadrinomial : which contains four unlike terms
Multinomial : two or more terms is called a polynomial : in two or more variables with every variable in it having only positive integral powers
Fractions
Multiplication
Product of two or more fractions = product of numerators / product of denominators
e.g. (3/4) x (5/6) = (3 x 5)/ (4 x 6) = 15/ 24= 5/8
The product of two proper fractions is smaller than each of the two fractions
The product of two improper fractions is greater than each of the two fractions
The product of an improper & a proper fraction is greater than the proper fraction & less than the improper fraction.
Division
Reciprocal of a number is also known as its multiplicative inverse
Division of fraction by a natural number
e.g. (5/6) / 5 = (5/6) X (1/5) = 1/6
Division of a fraction by another fraction
(1/3) / (5/6) = (1/3) X (6/5) = 2/5
Proper
numerator is less than its denominator
e.g. 5/6, 15/23...
Improper
numerator is greater than or equal to its denominator
e.g. 8/3, 19/7..
Mixed
whole number & a proper fraction
e.g. 2 3/4, 51/6,..
equivalent fractions
multiply its numerator & denominator by the same nonzero number
Like
which have same denominator
e.g. 5/9, 6/9, 8/9..
Unlike
which have different denominators
e.g. 3/4, 7/8, 5/12.
Simplification of Expression
Use BODMAS rule
for different brackets
Line Bracket
_
Common Bracket ( )
Curly Brackets { }
rectangular Brackets [ ]
Simple Interest
Lender
: Financial body or individual from whom the money is borrowed for a specified period of time.
Borrower
: individual who borrows the money
Principal (P) : the amount of money borrowed
Interest (
I
) : additional money paid by the borrower
Amount (A)
: total amount paid by the borrower
Rate (r)
: interest on the borrowed money is paid according to a certain percentage of the principal.
A = P + I
SI = (P x r x t) / 100
t= number of years
When the principal remains the same for the entire loan period, the interest charged is called simple interest
Angles / Lines
Line that intersects two or more lines at distinct points is called a
transversal
.
Interior angles
Exterior angles
Co-interior angles
Corresponding angles
Alternate interior angles
Alternate exterior angles
Angles made by a transversal to two parallel lines
Each pair of corresponding angles are equal in measures.
The "F" - shape shows corresponding angles
Each pair of alternate interior angles are equal in measure
The "Z" - shape shows alternate interior angles
Co-interior angles are supplementary
Complementary
: Two angles are said to be complementary if the sum of their measures is 90
In right angled triangle, two acute angles will be complementary as sum of all the angles of triangle is 180
Supplementary
: Two angles are supplementary if their sum is 180
Two angles are said to be adjacent if they have
a. a common vertex
b. a common arm and
c. the other two arms on opposite sides if the common arm.
Linear pair
is the pair of adjacent angles whose non-common arms are opposite rays
the sumof angles forming a linear pair is 180
The sum of all angles around a point is 360
If two lines intersect, then the vertically opposite angles are equal
Line segments or rays are said to be
intersecting
if they have a common point
The distance between two parallel lines is the same everywhere
Perimeter & Area
Area & Perimeter of a Rectangle
For a rectangle with length L units & breadth B units
Perimeter of rectangle = 2 x (L + B) Units
Area of rectangle A = L x B sq Units
Length of rectangle = Area / Breadth = A / B units
Breadth of rectangle = Area / Length = A / L Units
Area & Perimeter of a Square
for a square whose each side is of measure 's' units
Perimeter of square = 4 x side = 4s units
Area of square = side X side = S2 sq. units
Area o square = (diagonal)2 / 2
Area of 4 walls of a room= 2 x (L x H) + 2 x (B X H)
Area of parallelogram = Base X corresponding height
Base = Area of parallelogram / corresponding height
Height = Area of the parallelogram / corresponding base
Area of a triangle = 1/2 x ( Base x Height )
Area of a rectangular path = Area of outer rectangle - Area of inner rectangle
I cm2 = 100 mm2, 1 m2 = 10000 cm2
Decimals
Multiplication of Decimals
Multiplication by 10,100,1000...
0.09 X 10 = 0.9
0.09 X 100 = 9
0.09 X 1000 = 90
two decimals or whole number
Multiply the decimals as if they are whole numbers. then mark the decimal point in the product leaving as many as places to the right of decimals as is the sum of such places in the given decimals.
e.g. 0.6 X 0.4 = 0.24
Division
Division by 10, 100, 1000...
63.5 / 10 = 6.35
63.5 / 100 = 0.635
63.5 / 1000 = 0.0635
Two Decimal or whole number
convert the decimals to fractions first & then cancel the common factors
Comparison
Change decimal numbers to like decimal numbers
Compare whole number parts.
number having whole number part is larger
If whole number part is equal
compare the tenth digits
If tenth digits are same than compare 100th place...so on
LIke
same number decimal palces
e.g. 5.375, 0.024..
Unlike
different number of decimal palces
e.g. 6.2, 8.73, 29.184
Suffixing zeros to the extreme right of the decimal part of a decimal number doesn't change its value
Ratio
Proportion
ratios which are equivalent
a : b = c : d
Equivalence of ratios can also be written as
a : b :: c : d
a x d = b x c
Product of extremes = product of means
three numbers x,y, z are said to be in continued proportion, if x : y : : y : z
Here y is called mean proportional between x and z
y2 = xz
The ratio of two quantities with the
same unit
is the fraction that one quantity is of the other
ratio a is to b written a : b
a is called antecedent
b is called consequent
ratio remains unchanged if both its terms are multiplied by the same nonzero number
Two ratio can also be compared by making their denominators same
Profit & Loss
Cost Price that is abbreviated as
CP
Total Cost Price = Purchase price + Overhead charges
Selling Price that is abbreviated as
SP
Profit = SP - CP ( SP > CP)
Loss = CP - SP (CP > SP)
Profit or loss % is always calculated on the cost price of an article
Profit % = (Profit/ CP) x 100
Loss % = (Loss/CP) x 100
Two Useful Formulae
SP/CP = (100 + profit %) / 100 , when there is profit
SP/CP = (100 - loss %) / 100 , when there is loss
Points to Remember
Percentages are numerators of fractions with denominators 100.
Percent is represented by the symbol % and it means hundredth too.
We can convert fraction, decimals & ratios to percentages & vice versa
Percentage increase / decrease =
[ (Amount of change / Original amount) x 100 ] %
If any two lines are cut by a transversal such that
a. any pair of corresponding angles are equal or
b. any pair of alternate interior angles are equal
c. co-interior angles are supplementary, then the two lines are parallel.