#### 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.

## Definition

**Cost Price(C.P.):**The buying price of any item is known as its cost price.**Selling Price(S.P):**The price at which you sell any item is known as the selling price.**Total cost price:**In trading, all expenses incurred on an article before it can be sold have to be added to the cost price of the article. That is called the total cost price of the article.**Profit:**Profit is the revenue remaining after all costs are paid from the selling price.**Loss:**When the item is sold and the selling price is less than the cost price, then it is said, the seller has incurred a loss.-
**Overhead expenses:**Additional expenses made after buying an article are included in the cost price and are known as overhead expenses.

## Formula

- Profit = Selling Price - Cost Price, which means CP < SP.
- Loss = Cost Price - Selling Price, which means CP > SP.
- CP = Buying price + Overhead expenses

## Notes

**Concepts of Cost Price, Selling Price, Profit, and Loss, Discount, Overhead Expenses, and GST:**

**Cost Price(C.P.): **The buying price of any item is known as its cost price. It is written in short as CP.

**Selling Price(S.P):** The price at which you sell any item is known as the selling price. It is written in short as SP.

**Total cost price:** In trading, all expenses incurred on an article before it can be sold have to be added to the cost price of the article. That is called the total cost price of the article.

**Profit: **Profit is the revenue remaining after all costs are paid from selling price.

**Loss: **When the item is sold and the selling price is less than the cost price, then it is said, the seller has incurred a loss.

You can decide whether the sale was profitable or not depending on the CP and SP.

- If CP < SP then you made a profit = SP – CP.
- If CP = SP then you are in a no-profit no loss situation.
- If CP > SP then you have a loss = CP – SP.

**Overhead expenses:**Additional expenses made after buying an article are included in the cost price and are known as overhead expenses.

## Example

Hamidbhai bought bananas worth 2000 rupees and sold them all for 1890 rupees. Did he make a profit or a loss? How much was it?

He bought bananas for Rs. 2000.

Hence, Cost price = Rs. 2000

Selling price = Rs. 1890

Cost price is greater than selling price. Therefore, Hamidbhai suffered a loss.

Loss = Cost price - Selling price

= 2000 - 1890

= Rs. 110

∴ Hamidbhai suffered a loss of Rs. 110 in this transaction.

## Example

Harbhajan Singh bought 500 kg of rice for 22000 rupees and sold it all at the rate of Rs. 48 per kg. How much profit did he make?

The cost price of 500 kg rice is Rs. 22000.

Selling price of 500 kg of rice is = 500 × 48 = Rs. 24000

Selling price is greater than cost price.

Therefore, there is a profit.

Profit = Selling price - Cost price

= 24000 - 22000

= Rs. 2000

∴ In this transaction, Harbhajan Singh made a profit of Rs. 2000.

## Example

Javedbhai bought 35 electric mixers for Rs. 4300 each. To transport them to the shop, he spent Rs. 2100. If he expects to make a profit of Rs. 21000, at what price should he sell each mixer?

Cost price of one mixer Rs. 4300.

Hence cost price of 35 mixers = 4300 × 35 = Rs. 150500

Total cost price = cost of mixers + cost of transport

= 150500 + 2100

= Rs. 152600

Javedbhai wants a profit of 21000 rupees.

∴ Hence, amount expected on selling = 152600 + 21000 = Rs. 173600

Selling price of 35 mixers = Rs. 173600

∴ Selling price of one mixer = 173600 ÷ 35 = Rs. 4960

Javedbhai should sell every mixer for Rs. 4960.

## Example

The cost of a flower vase is Rs. 120. If the shopkeeper sells it at a loss of 10%, find the price at which it is sold.

**Given:** CP = Rs. 120 and Loss percent = 10.

Loss of 10% means if CP is Rs. 100, Loss is Rs. 10

Therefore,

SP would be Rs.(100 - 10) = Rs. 90

When CP is Rs. 100, SP is Rs. 90.

Therefore, if CP were Rs. 120 then

SP = `90/100 xx 120` = Rs. 108.

**Given:** CP = Rs. 120 and Loss percent = 10.

Loss is 10% of the cost price

= 10% of Rs. 120

= `10/100 xx 120`

= Rs. 12

Therefore,

SP = CP - Loss = Rs. 120 - Rs. 12 = Rs. 108

## Example

Selling price of a toy car is Rs. 540. If the profit made by the shopkeeper is 20%, what is the cost price of this toy?

**Given:** SP = Rs. 540 and the Profit = 20%.

20% profit will mean if CP is Rs. 100, profit is Rs. 20.

Therefore, SP = 100 + 20 = 120

Now, when SP is Rs. 120, then CP is Rs. 100.

Therefore, when SP is Rs. 540,

Then CP = `100/120 xx 540` = Rs. 450.

**Given:** SP = Rs. 540 and the Profit = 20%.

Profit = 20% of CP and SP = CP + Profit

So, 540 = CP + 20% of CP

= CP + `20/100 xx "CP" = [1 + 1/5]"CP"`

= `6/5 "CP"`.

Therefore, `540 xx 5/6` = CP or Rs. 450 = CP

## Example

Meenu bought two fans for ₹ 1200 each. She sold one at a loss of 5% and the other at a profit of 10%. Find the selling price of each. Also, find out the total profit or loss.