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
- Concept of Reciprocal or Multiplicative Inverse
- Rational Numbers on a Number Line
- Rational Numbers Between Two Rational Numbers
Linear Equations in One Variable
- Variable of Equation
- Concept of Equation
- Expressions with Variables
- 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
- Classification of Polygons
- Properties of a Quadrilateral
- Interior Angles of a Polygon
- Exterior Angles of a Polygon and Its Property
- Concept of Quadrilaterals
- 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 Geometric Tool
- 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 Circle-graph)
- 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
- Basic Concept of Percentage
- Increase Or Decrease as Percent
- Concept of Discount
- Estimation in Percentages
- Basic Concepts of Profit and Loss
- Sales Tax, Value Added Tax, and Good and Services Tax
- Calculation of 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) = a2-b2
- Expansion of (x + a)(x + b)
Mensuration
Visualizing Solid Shapes
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 Circle-graph)
- Graphical Representation of Data as Histograms
- Concept of a Line Graph
- Linear Graphs
- Some Application of Linear Graphs
Playing with Numbers
- Introduction
- Simple Interest
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 variable-rate 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:
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The money you borrow is known as sum borrowed or principal.
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This money would be used by the borrower for some time before it is returned.
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Example: Loan that you take from a bank is the principal.
2. Interest:
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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.
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How much is paid for the use of money (as a percent, or an amount)
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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:
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Simple interest is used commonly in variable-rate consumer lending and in mortgage loans where a borrower pays interest only on funds used.
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While calculating interest where the principal is not changed is known as simple interest.
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As the number of years increases the interest also increases.
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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.
