Topics
Rational Numbers
 Rational Numbers
 Closure Property of Rational Numbers
 Commutative Property of Rational Numbers
 Associative Property of Rational Numbers
 Distributive Property of Multiplication Over Addition for Rational Numbers
 Identity of Addition and Multiplication of Rational Numbers
 Negative Or Additive Inverse of Rational Numbers
 Reciprocal Or Multiplicative Inverse of Rational Numbers
 Rational Numbers on a Number Line
 Rational Numbers Between Two Rational Numbers
Linear Equations in One Variable
 The Idea of a Variable
 Expressions with Variables
 Concept of Equation
 Balancing an Equation
 The Solution of an Equation
 Linear Equation in One Variable
 Solving Equations Which Have Linear Expressions on One Side and Numbers on the Other Side
 Some Applications Solving Equations Which Have Linear Expressions on One Side and Numbers on the Other Side
 Solving Equations Having the Variable on Both Sides
 Some More Applications on the Basis of Solving Equations Having the Variable on Both Sides
 Reducing Equations to Simpler Form
 Equations Reducible to the Linear Form
Understanding Quadrilaterals
 Concept of Curves
 Different Types of Curves  Closed Curve, Open Curve, Simple Curve.
 Concept of Polygons  Side, Vertex, Adjacent Sides, Adjacent Vertices and Diagonal
 Classification of Polygons
 Angle Sum Property of a Quadrilateral
 Interior Angles of a Polygon
 Exterior Angles of a Polygon and Its Property
 Concept of Quadrilaterals  Sides, Adjacent Sides, Opposite Sides, Angle, Adjacent Angles and Opposite Angles
 Properties of Trapezium
 Properties of Kite
 Properties of a Parallelogram
 Properties of Rhombus
 Property: The Opposite Sides of a Parallelogram Are of Equal Length.
 Property: The Opposite Angles of a Parallelogram Are of Equal Measure.
 Property: The adjacent angles in a parallelogram are supplementary.
 Property: The diagonals of a parallelogram bisect each other. (at the point of their intersection)
 Property: The diagonals of a rhombus are perpendicular bisectors of one another.
 Property: The Diagonals of a Rectangle Are of Equal Length.
 Properties of Rectangle
 Properties of a Square
 Property: The diagonals of a square are perpendicular bisectors of each other.
Practical Geometry
 Introduction to Practical Geometry
 Constructing a Quadrilateral When the Lengths of Four Sides and a Diagonal Are Given
 Constructing a Quadrilateral When Two Diagonals and Three Sides Are Given
 Constructing a Quadrilateral When Two Adjacent Sides and Three Angles Are Known
 Constructing a Quadrilateral When Three Sides and Two Included Angles Are Given
 Some Special Cases
Data Handling
 Concept of Data Handling
 Interpretation of a Pictograph
 Interpretation of Bar Graphs
 Drawing a Bar Graph
 Interpretation of a Double Bar Graph
 Drawing a Double Bar Graph
 Organisation of Data
 Frequency Distribution Table
 Graphical Representation of Data as Histograms
 Concept of Pie Graph (Or a Circlegraph)
 Interpretation of Pie Diagram
 Chance and Probability  Chance
 Basic Ideas of Probability
Squares and Square Roots
 Concept of Square Number
 Properties of Square Numbers
 Some More Interesting Patterns of Square Number
 Finding the Square of a Number
 Concept of Square Roots
 Finding Square Root Through Repeated Subtraction
 Finding Square Root Through Prime Factorisation
 Finding Square Root by Division Method
 Square Root of Decimal Numbers
 Estimating Square Root
Cubes and Cube Roots
Comparing Quantities
 Concept of Ratio
 Concept of Percent and Percentage
 Increase Or Decrease as Percent
 Concept of Discount
 Estimation in Percentages
 Concepts of Cost Price, Selling Price, Total Cost Price, and Profit and Loss, Discount, Overhead Expenses and GST
 Sales Tax, Value Added Tax, and Good and Services Tax
 Concept of Principal, Interest, Amount, and Simple Interest
 Concept of Compound Interest
 Deducing a Formula for Compound Interest
 Rate Compounded Annually Or Half Yearly (Semi Annually)
 Applications of Compound Interest Formula
Algebraic Expressions and Identities
 Algebraic Expressions
 Terms, Factors and Coefficients of Expression
 Types of Algebraic Expressions as Monomials, Binomials, Trinomials, and Polynomials
 Like and Unlike Terms
 Addition of Algebraic Expressions
 Subtraction of Algebraic Expressions
 Multiplication of Algebraic Expressions
 Multiplying Monomial by Monomials
 Multiplying a Monomial by a Binomial
 Multiplying a Monomial by a Trinomial
 Multiplying a Binomial by a Binomial
 Multiplying a Binomial by a Trinomial
 Concept of Identity
 Expansion of (a + b)2 = a2 + 2ab + b2
 Expansion of (a  b)2 = a2  2ab + b2
 Expansion of (a + b)(a  b)
 Expansion of (x + a)(x + b)
Visualizing Solid Shapes
Mensuration
Exponents and Powers
Direct and Inverse Proportions
Factorization
 Factors and Multiples
 Factorising Algebraic Expressions
 Factorisation by Taking Out Common Factors
 Factorisation by Regrouping Terms
 Factorisation Using Identities
 Factors of the Form (x + a)(x + b)
 Dividing a Monomial by a Monomial
 Dividing a Polynomial by a Monomial
 Dividing a Polynomial by a Polynomial
 Concept of Find the Error
Introduction to Graphs
 Concept of Bar Graph
 Interpretation of Bar Graphs
 Drawing a Bar Graph
 Concept of Double Bar Graph
 Interpretation of a Double Bar Graph
 Drawing a Double Bar Graph
 Concept of Pie Graph (Or a Circlegraph)
 Graphical Representation of Data as Histograms
 Concept of a Line Graph
 Linear Graphs
 Linear Graphs
 Some Application of Linear Graphs
Playing with Numbers
 Simple Interest for one year
 Simple Interest for multiple years
Definition
 Sum borrowed or Principal: The money you borrow is known as sum borrowed or principal.
 Interest: For keeping sum borrowed or principal money for some time the borrower has to pay some extra money to the bank. This is known as Interest.
 Amount: The amount you have to pay at the end of the year by adding the sum borrowed and the interest.
 Simple interest: Simple interest is used commonly in variablerate consumer lending and in mortgage loans where a borrower pays interest only on funds used.
Formula
 Amount = Principal + Interest.
 Simple Interest = `("P" xx "R" xx "T")/100`.
Notes
Concept of Principal, Interest, Amount, and Simple Interest:
1. Sum borrowed or Principal:

The money you borrow is known as sum borrowed or principal.

This money would be used by the borrower for some time before it is returned.

Example: Loan that you take from a bank is the principal.
2. Interest:

For keeping sum borrowed or principal money for some time the borrower has to pay some extra money to the bank. This is known as Interest.

How much is paid for the use of money (as a percent, or an amount)

Interest is generally given in percent for a period of one year. It is written as say 10% per year or per annum or in short as 10 % p.a. (per annum).
3. Amount:
 The amount you have to pay at the end of the year by adding the sum borrowed and the interest.
 Amount = Principal + Interest.
4. Simple Interest:

Simple interest is used commonly in variablerate consumer lending and in mortgage loans where a borrower pays interest only on funds used.

While calculating interest where the principal is not changed is known as simple interest.

As the number of years increases the interest also increases.

Simple Interest = `(P × R × T)/100`
P = Principal Amount
R = Interest rate
T = Time
(i) Interest for one year:
 We can write a general relation to finding an interest for one year.
 Take P as the principal or sum and R % as Rate percent per annum.
 Now on every Rs. 100 borrowed, the interest paid is Rs. R
 Therefore, on Rs. P borrowed, the interest paid for one year would be `(R xx P)/100 = (P xx R)/100`.
(ii) Interest for multiple years:
If the amount is borrowed for more than one year the interest is calculated for the period the money is kept for.
For example, if Anita returns the money at the end of two years and the rate of interest is the same then she would have to pay twice the interest i.e., Rs. 750 for the first year and Rs. 750 for the second. This way of calculating interest where the principal is not changed is known as simple interest. As the number of years increase the interest also increases. For Rs. 100 borrowed for 3 years at 18%, the interest to be paid at the end of 3 years is 18 + 18 + 18 = 3 × 18 = Rs. 54.
We can find the general form for simple interest for more than one year.
We know that on a principal of Rs. P at R% rate of interest per year, the interest paid for one year is `(R xx P)/100`. Therefore, interest I paid for T years would be
`(T xx R xx P)/100 = (P xx R xx T)/100 or "PRT"/100`.
And the amount you have to pay at the end of T years is A = P + I
Example
Anita takes a loan of Rs. 5,000 at 15% per year as the rate of interest. Find the interest she has to pay at the end of one year.
The sum borrowed = Rs. 5,000, Rate of interest = 15% per year.
This means if Rs. 100 is borrowed, she has to pay Rs. 15 as interest for one year.
If she has borrowed Rs. 5,000, then the interest she has to pay for one year.
= Rs. `15/100 × 5000 = Rs. 750.`
So, at the end of the year she has to give an amount of Rs. 5,000 + Rs. 750 = Rs. 5,750.
Example
If Manohar pays an interest of Rs. 750 for 2 years on a sum of Rs. 4,500, find the rate of interest.
I = `(P × T × R)/(100)`
750 = `(4500 × 2 × R)/(100)`
`(750)/(45 × 2) = R`
Therefore, Rate = `8 1/3 %`
Example
Vinita deposited Rs. 15000 in a bank for one year at an interest rate of 7 p.c. p.a. How much interest will she get at the end of the year?
Let us suppose that the interest on the principal of Rs. 15000 is x.
On principal Rs. 100, the interest is Rs. 7.
`x/(15000) = 7/100`
`x/(15000) xx 15000 = 7/100 xx 15000`.......(Multiplying both sides by 15000)
x = 1050
Vinita will get an interest of Rs. 1050.
Example
Vilasrao borrowed Rs. 20000 from a bank at a rate of 8 p.c.p.a. What is the amount he will return to the bank at the end of the year?
Let interest on principal 20000 rupees be x rupees.
Interest on principal 100 rupees is 8 rupees.
`x/(20000) = 8/100`
`x/(20000) xx 20000 = 8/100 xx 20000`.....(Multiplying both sides by 20000)
x = 16000
Amount to be returned to the bank = principal + interest
= 20000 + 1600
= Rs. 21600
Example
Sandeepbhau borrowed 120000 rupees from a bank for 4 years at the rate of `8 1/2` p.c.p.a. for his son’s education. What is the total amount he returned to the bank at the end of that period?
Principal = 120000, P = 120000, R = 8.5, T = 4
∴ Total interest = `(P xx R xx T)/100`
= `(120000 xx 8.5 xx 4)/100`
= `(120000 xx 85 xx 4)/(100 xx 10)`
= 120 × 85 × 4
= 40800
The total amount returned to the bank = 120000 + 40800 = 160800 rupees.