#### Topics

##### Commercial Mathematics

##### 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)

##### 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

##### Loci

##### Circles

- Concept of Circle - Centre, Radius, Diameter, Arc, Sector, Chord, Segment, Semicircle, Circumference, Interior and Exterior, Concentric Circles
- 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

##### Shares and Dividends

##### Algebra

##### Quadratic Equations

##### Banking

##### Geometry

##### Factorization

- Factor Theorem
- Remainder Theorem
- Factorising a Polynomial Completely After Obtaining One Factor by Factor Theorem

##### 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

##### Ratio and Proportion

- Concept of Ratio
- Concept of Proportion
- Componendo and Dividendo Properties
- Alternendo and Invertendo Properties
- Direct Applications
- Concept of Ratio

##### 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

##### Gst (Goods and Services Tax)

- Sales Tax, Value Added Tax, and Good and Services Tax
- Computation of Tax
- Concept of Discount
- List Price
- Concepts of Cost Price, Selling Price, Total Cost Price, and Profit and Loss, Discount, Overhead Expenses and GST
- Basic/Cost Price Including Inverse Cases.
- Selling Price
- Dealer
- Goods and Service Tax (Gst)
- Gst Tax Calculation
- Gst Tax Calculation
- Input Tax Credit (Itc)

##### Symmetry

##### 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

##### Similarity

- Similarity of Triangles
- Similarity of Triangles
- Axioms of Similarity of Triangles
- Areas of Similar Triangles Are Proportional to the Squares on Corresponding Sides

##### Linear Inequations

- Linear Inequations in One Variable
- Solving Algebraically and Writing the Solution in Set Notation Form
- Representation of Solution on the Number Line

##### Arithmetic Progression

- Arithmetic Progression - Finding Their General Term
- Sum of First ‘n’ Terms of an Arithmetic Progressions
- Simple Applications of Arithmetic Progression

##### 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

##### Probability

##### Geometric Progression

- Geometric Progression - Finding Their General Term.
- Geometric Progression - Finding Sum of Their First ‘N’ Terms
- Simple Applications - Geometric Progression

##### Matrices

##### Reflection

- Reflection Examples
- Reflection Concept
- Reflection of a Point in a Line
- Reflection of a Point in the Origin.
- Invariant Points.

##### 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.

- Proportion
- Continued or mean proportion
- Proportionality law

## Definition

**Proportion:** If two ratios are equal, then they are said to be in proportion. We use the symbol ‘::’ or ‘=’ to equate the two ratios.

**Continued proportion: **Four quantities are said to be in continued proportion if the ratio of the first term and second term be equal to the ratio of the second term and the third term be equal to the ratio of the third term and fourth term. If w, x, y, and z are four quantities such that w: x = x: y = y: z i.e., `w/x = x/y = y/z`, they are said to be continued proportion.

## Notes

**Proportion:**

- Bhavika has 28 marbles and Vini has 180 flowers. They want to share these among themselves. Bhavika gave 14 marbles to Vini and Vini gave 90 flowers to Bhavika. But Vini was not satisfied. She felt that she had given more flowers to Bhavika than the marbles given by Bhavika to her. What do you think? Is Vini correct?
- To solve this problem both went to Vini’s mother Pooja. Pooja explained that out of 28 marbles, Bhavika gave 14 marbles to Vini. Therefore, the ratio is 14: 28 = 1: 2. And out of 180 flowers, Vini had given 90 flowers to Bhavika. Therefore, the ratio is 90: 180 = 1: 2. Since both the ratios are the same, so the distribution is fair.
- If two ratios are equal, then they are said to be in proportion.
- If two ratios are equal, we say that they are in proportion and use the symbol ‘::’ or ‘=’ to equate the two ratios.

**Continued or Mean Proportion:**

Three quantities are said to be in continued proportion if the ratio of the first term and second term be equal to the ratio of the second term and third term.

**Continued proportion:** Three numbers ‘a’, ‘b’, and ‘c’ are said to be continued proportion if a, b and c are in proportion.

Thus, if a, b and c are in continued-proportion, then a, b, b, c are in proportion, that means; a: b: : b: c

⇒ Product of extremes = Product of means

⇒ a x c = b x b

⇒ a x c = b²

Continued proportion is also known as mean proportional.

If ‘b’ is a mean proportional between a and c then b^{2} = ac.

Suppose, the three quantities x, y and z are said to be in continued proportion if x: y = y: z, i.e., `x/y = y/z`

Similarly, four quantities are said to be in continued proportion if the ratio of the first term and second term be equal to the ratio of the second term and the third term be equal to the ratio of the third term and fourth term.

If w, x, y, and z are four quantities such that w: x = x: y = y: z i.e., `w/x = x/y = y/z`, they are said to be continued proportion.

- For example, we can say 2, 4, 180, and 360 are in the proportion which is written as 2: 4:: 180: 360 and is read as 2 is to 4 as 180 is to 360.
- If two ratios are not equal, then we say that they are not in proportion.

For example, the two ratios 2: 5 and 180: 45 are not equal, i.e. 2: 5 ≠ 180: 45. Therefore, the four quantities 2, 5, 180 and 45 are not in proportion - In a statement of proportion, the four quantities involved when taken in order are known as respective terms. The first and fourth terms are known as extreme terms. Second and third terms are known as middle terms.

## Example

Are the ratios 25g: 30g and 40 kg: 48 kg in proportion?

25 g: 30 g = `25/30 `= 5: 6

40 kg: 48 kg = `40/48` = 5: 6

So, 25: 30 = 40: 48.

Therefore, the ratios 25 g: 30 g and 40 kg: 48 kg are in proportion, i.e. 25: 30:: 40: 48

The middle terms in this are 30, 40 and the extreme terms are 25, 48.

## Example

Are 30, 40, 45, and 60 in proportion?

Ratio of 30 to 40 = `30/40` = 3: 4.

Ratio of 45 to 60 = `45/60` = 3: 4.

Since, 30: 40 = 45: 60

Therefore, 30, 40, 45, and 60 are in proportion.

## Example

Do the ratios 15 cm to 2 m and 10 sec to 3 minutes form a proportion?

15 cm : 2 m :: 10 sec : 3 min

15 cm : 2 × 100 cm :: 10 sec : 30 × 60 sec

15 : 200 :: 10 : 1800

3 : 40 :: 1 : 180

No, they donot form a proportion

Ratio of 15 cm to 2 m = 15: 2 × 100 (1 m = 100 cm)

= 3: 40

Ratio of 10 sec to 3 min = 10: 3 × 60 (1 min = 60 sec)

=1: 18

Since, 3: 40 ≠ 1: 18, therefore, the given ratios do not form a proportion.

## Example

A hostel is to be built for schoolgoing girls. Two toilets are to be built for every 15 girls. If 75 girls will be living in the hostel, how many toilets will be required in this proportion?

Let us suppose x toilets will be needed for 75 girls.

The ratio of the number of toilets to the number of girls is `2/15`.

∴ `x/75 = 2/15`

∴ `x/75 xx 75 = 2/15 xx 75`................(Multiplying both sides by 75)

∴ x = 2 × 5

∴ x = 10

∴ 10 toilets will be required for 75 girls.