Topics
Compound Interest
- Compound Interest as a Repeated Simple Interest Computation with a Growing Principal
- Use of Compound Interest in Computing Amount Over a Period of 2 Or 3-years
- Use of Formula
- Finding CI from the Relation CI = A – P
GST (Goods and Services Tax)
- Sales Tax, Value Added Tax, and Good and Services Tax
- Computation of Tax
- Concept of Discount
- List Price
- Basic Concepts of Profit and Loss
- Basic/Cost Price Including Inverse Cases.
- Selling Price
- Dealer
- Goods and Service Tax (Gst)
- Gst Tax Calculation
- Input Tax Credit (Itc)
Commercial Mathematics
Algebra
Banking
Geometry
Shares and Dividends
Linear Inequations
- Linear Inequations in One Variable
- Solving Algebraically and Writing the Solution in Set Notation Form
- Representation of Solution on the Number Line
Mensuration
- Circumference of a Circle
- Surface Area of a Right Circular Cone
- Surface Area of a Sphere
- Circle - Direct Application Problems Including Inner and Outer Area
- Surface Area of a Right Circular Cone
- Surface Area of a Sphere
- Volume of a Cylinder
- Volume of a Combination of Solids
- Surface Area of Cylinder
Symmetry
Quadratic Equations
- Quadratic Equations
- Solutions of Quadratic Equations by Factorization
- Nature of Roots of a Quadratic Equation
- Equations Reducible to Quadratic Equations
Trigonometry
- Trigonometric Ratios of Complementary Angles
- Trigonometric Identities
- Heights and Distances - Solving 2-D Problems Involving Angles of Elevation and Depression Using Trigonometric Tables
- Trigonometry
Statistics
- Median of Grouped Data
- Graphical Representation of Data as Histograms
- Ogives (Cumulative Frequency Graphs)
- Concepts of Statistics
- Graphical Representation of Data as Histograms
- Graphical Representation of Ogives
- Finding the Mode from the Histogram
- Finding the Mode from the Upper Quartile
- Finding the Mode from the Lower Quartile
- Finding the Median, upper quartile, lower quartile from the Ogive
- Calculation of Lower, Upper, Inter, Semi-Inter Quartile Range
- Concept of Median
- Mean of Grouped Data
- Mean of Ungrouped Data
- Median of Ungrouped Data
- Mode of Ungrouped Data
- Mode of Grouped Data
- Mean of Continuous Distribution
Solving (Simple) Problems (Based on Quadratic Equations)
- Problems Based on Numbers
- Problems Based on Time and Work
- Problems Based on Geometrical Figures
- Problems Based on Distance, Speed and Time
- Problems on C.P. and S.P.
- Miscellaneous Problems
Ratio and Proportion
- Concept of Ratio
- Concept of Proportion
- Componendo and Dividendo Properties
- Alternendo and Invertendo Properties
- Direct Applications
Probability
Factorization
- Factor Theorem
- Remainder Theorem
- Factorising a Polynomial Completely After Obtaining One Factor by Factor Theorem
Matrices
Arithmetic Progression
- Arithmetic Progression - Finding Their General Term
- Sum of First ‘n’ Terms of an Arithmetic Progressions
- Simple Applications of Arithmetic Progression
- Arithmetic mean
- Properties of an Arithmetic Progression
Geometric Progression
- Geometric Progression - Finding Their General Term.
- Geometric Progression - Finding Sum of Their First ‘N’ Terms
- Simple Applications - Geometric Progression
Reflection
- Reflection Examples
- Reflection Concept
- Reflection of a Point in a Line
- Reflection of a Point in the Origin.
- Invariant Points.
Co-ordinate Geometry Distance and Section Formula
- Co-ordinates Expressed as (x,y)
- Distance Formula
- Section Formula
- The Mid-point of a Line Segment (Mid-point Formula)
- Points of Trisection
- Centroid of a Triangle
Co-ordinate Geometry Equation of a Line
- Slope of a Line
- Concept of Slope
- Equation of a Line
- Various Forms of Straight Lines
- General Equation of a Line
- Slope – Intercept Form
- Two - Point Form
- Geometric Understanding of ‘m’ as Slope Or Gradient Or tanθ Where θ Is the Angle the Line Makes with the Positive Direction of the x Axis
- Geometric Understanding of c as the y-intercept Or the Ordinate of the Point Where the Line Intercepts the y Axis Or the Point on the Line Where x=0
- Conditions for Two Lines to Be Parallel Or Perpendicular
- Simple Applications of All Co-ordinate Geometry.
- Collinearity of Three Points
Similarity
- Similarity of Triangles
- Axioms of Similarity of Triangles
- Areas of Similar Triangles Are Proportional to the Squares on Corresponding Sides
- Conditions for Similarity of Two Triangles: (Sas, Aa Or Aaa and Sss)
- Basic Proportionality Theorem with Applications
- Relation Between the Areas of Two Triangles
- Similarity as a Size Transformation
- Direct Applications Based on the Above Including Applications to Maps and Models
Loci
Circles
- Concept of Circle
- Areas of Sector and Segment of a Circle
- Tangent Properties - If a Line Touches a Circle and from the Point of Contact, a Chord is Drawn, the Angles Between the Tangent and the Chord Are Respectively Equal to the Angles in the Corresponding Alternate Segments
- Tangent Properties - If a Chord and a Tangent Intersect Externally, Then the Product of the Lengths of Segments of the Chord is Equal to the Square of the Length of the Tangent from the Point of Contact to the Point of Intersection
- Tangent to a Circle
- Number of Tangents from a Point on a Circle
- Chord Properties - a Straight Line Drawn from the Center of a Circle to Bisect a Chord Which is Not a Diameter is at Right Angles to the Chord
- Chord Properties - the Perpendicular to a Chord from the Center Bisects the Chord (Without Proof)
- Theorem: Equal chords of a circle are equidistant from the centre.
- Theorem : The Chords of a Circle Which Are Equidistant from the Centre Are Equal.
- Chord Properties - There is One and Only One Circle that Passes Through Three Given Points Not in a Straight Line
- Arc and Chord Properties - the Angle that an Arc of a Circle Subtends at the Center is Double that Which It Subtends at Any Point on the Remaining Part of the Circle
- Theorem: Angles in the Same Segment of a Circle Are Equal.
- Arc and Chord Properties - Angle in a Semi-circle is a Right Angle
- Arc and Chord Properties - If Two Arcs Subtend Equal Angles at the Center, They Are Equal, and Its Converse
- Arc and Chord Properties - If Two Chords Are Equal, They Cut off Equal Arcs, and Its Converse (Without Proof)
- Arc and Chord Properties - If Two Chords Intersect Internally Or Externally Then the Product of the Lengths of the Segments Are Equal
- Cyclic Properties
- Tangent Properties - If Two Circles Touch, the Point of Contact Lies on the Straight Line Joining Their Centers
Constructions
- Circumscribing and Inscribing a Circle on a Regular Hexagon
- Circumscribing and Inscribing a Circle on a Triangle
- Construction of Tangents to a Circle
- Circumference of a Circle
- Circumscribing and Inscribing Circle on a Quadrilateral
Definition
Ratio: The quantitative relation between two amounts showing the number of times one value contains or is contained within the other. This method is known as ratio.
Notes
Ratio:
i) Comparison by taking difference:
In our daily life, many times we compare two quantities of the same type. If the height of Rahim is 150 cm and that of Avnee is 140 cm then, we may say that the height of Rahim is 150 cm – 140 cm = 10 cm more than Avnee. This is one way of comparison by taking the difference.
ii) Comparison by division:
In many situations, a more meaningful comparison between quantities is made by using division, i.e. by seeing how many times one quantity is to the other quantity. This method is known as comparison by ratio.
Consider an example. The cost of a car is 2,50,000 and that of a motorbike is 50,000. If we calculate the difference between the costs, it is 2,00,000 and if we compare by division; i.e. `(2,50, 000)/(50, 000)` = 5: 1.
Thus, in certain situations, comparison by division makes better sense than comparison by taking the difference. The comparison by division is the Ratio.
- The quantitative relation between two amounts showing the number of times one value contains or is contained within the other. This method is known as ratio.
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Mathematical numbers used in comparing two things that are similar to each other in terms of units are ratios.
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We denote ratio using symbol ‘:’
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A ratio can be written in three different ways viz, m to n, m: n, and mn but read as the ratio of m is to n. Example: 2: 3 = ratio of 2 is to 3.
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Note that the ratio 3: 2 is different from 2: 3. Thus, the order in which quantities are taken to express their ratio is important.
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For comparison by ratio, the two quantities must be in the same unit. If they are not, they should be expressed in the same unit before the ratio is taken.
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A ratio may be treated as a fraction, thus the ratio 10 : 3 may be treated as `10/3`.
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A ratio can be expressed in its lowest form. For example, ratio 50: 15 is treated as `50/15`; in its lowest form `50/15 = 10/3`. Hence, the lowest form of ratio 50: 15 is 10: 3.
Same ratio in different situations:
Ratios can remain the same in different situations.
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The length of a room is 30 m and its breadth is 20 m.
So, the ratio of the length of the room to the breadth of the room =`30/20=3/2`=3: 2.
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There are 24 girls and 16 boys going for a picnic. The ratio of the number of girls to the number of boys = `24/16=3/2`= 3: 2
The ratio in both examples is 3: 2.
Example
Ratio of distance of the school from Mary’s home to the distance of the school from John’s home is 2: 1.
(a) Who lives nearer to the school?
(b) Complete the following table which shows some possible distances that
Mary and John could live from the school.
|
Distance from Mary’s home to school (in km.)
|
10 | 4 | |||
|
Distance from John’s home to school (in km.)
|
5 | 4 | 3 | 1 |
(a) John lives nearer to the school (As the ratio is 2: 1).
(b)
|
Distance from Mary’s home to school (in km.)
|
10 | 8 | 4 | 6 | 2 |
|
Distance from John’s home to school (in km.)
|
5 | 4 | 2 | 3 | 1 |
Example
Find the ratio of 90 cm to 1.5 m.
The two quantities are not in the same units. Therefore, we have to convert them into the same units.
1.5 m = 1.5 × 100 cm = 150 cm.
Therefore, the required ratio is 90: 150.
= `90/150 = (90 xx 30)/(150 xx 30) = 3/5`.
Required ratio is 3: 5.
Example
There are 45 persons working in an office. If the number of females is 25 and the remaining are males, find the ratio of:
(a) The number of females to the number of males.
(b) The number of males to the number of females.
Number of females = 25
Total number of workers = 45
Number of males = 45 – 25 = 20
Therefore, the ratio of number of females to the number of males = 25: 20 = 5: 4
And the ratio of number of males to the number of females = 20: 25 = 4: 5.
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