Financial Maths Class 11 Applied Maths Full Chapter Notes

Financial Mathematics is an important part of Class 11 Applied Maths that connects mathematical concepts to everyday financial decisions. This chapter covers topics like Simple and Compound Interest, Annuities, GST calculations, and utility bills, helping students understand how money grows over time, how payments are structured, and how taxes and bills are calculated. Learning these concepts equips students with practical skills to manage personal finance, evaluate investments, and solve real-life financial problems effectively.

Financial Maths Class 11 Applied Maths Chapter

Financial Mathematics is a crucial part of the Class 11 Applied Mathematics syllabus, providing essential tools for understanding personal finance, investment, and business operations. This chapter lays the groundwork for complex financial concepts, enabling students to calculate interest, analyze annuities, and interpret various tax and utility bills.

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Simple Interest (SI)

Simple Interest, or 'साधारण ब्याज', is calculated only on the original sum of money (the initial principal). The interest earned remains constant for each period.

(Memory Tip: Interest and cutting an onion can both bring tears to your eyes – onion physically, interest financially when paying.)

Key Principle of Simple Interest

The interest is always charged on the original sum of money (the initial principal), regardless of the year of calculation.

Simple Interest Formulas

1. Simple Interest (SI) Formula:
   SI = (P × R × T) / 100

• P (Principal): Initial sum.

• R (Rate of Interest): Annual interest rate (numerical value, e.g., 8 for 8%).

• T (Time Period): Duration in years.

2. Amount / Future Value (A) Formula:
   A = Principal (P) + Simple Interest (SI)
   Alternatively: A = P (1 + (RT / 100))

Worked Examples: Simple Interest

Example 1: Basic Calculation

Calculate the amount after 4 years if ₹10,000 is invested at 8% p.a. simple interest.

• SI: (10000 × 8 × 4) / 100 = ₹3200

• Amount: ₹10,000 + ₹3200 = ₹13,200

Example 2: Time in Months

Calculate SI on ₹10,000 at 8% p.a. for 5 months.

• T: 5 / 12 years

• SI: (10000 × 8 × (5/12)) / 100 ≈ ₹333.33

Example 3: Finding the Rate of Interest

At what rate will ₹75,000 yield ₹3,375 as simple interest in 6 months?

• P: ₹75,000, SI: ₹3,375, T: 0.5 years

• 3375 = (75000 × R × 0.5) / 100

• R = 3375 / 375 = 9% p.a.

Example 4: Doubling Time

How much time will a sum double itself at 5% p.a. simple interest?

• Let P = x, then A = 2x. R = 5%.

• 2x = x(1 + (5 × T) / 100)

• 2 = 1 + 5T / 100 => 1 = 5T / 100 => T = 20 years.

Compound Interest (CI)

Compound Interest, or 'चक्रवर्ती ब्याज', is fundamentally different from simple interest. Interest is calculated not just on the principal, but also on the accumulated interest from previous periods. This is often referred to as "interest on interest".

(Memory Tip: When you receive compound interest, it's "Balle Balle" (celebratory); when you pay, it's "Thalle Thalle" (downfall). )

Key Principle of Compound Interest

The interest charged in each time interval is calculated on the total amount (principal + accumulated interest) from the previous time interval.

Compound Interest Formulas

The formula calculates the final amount directly.

A = P (1 + i)ⁿ

• A: Final Amount (Future Value).

• P: Principal (Original Sum of Money).

• i: Compounded Interest Rate per interval.

• n: Number of Compounding Intervals.

Defining i and n:

These variables depend on the compounding factor (m), which is the number of times interest is compounded in one year.

Common Compounding Frequencies

• Compounded Interest Rate per Interval (i): i = (r / m) / 100 (where 'r' is annual rate).

• Total Number of Intervals (n): n = m * t (where 't' is total time in years).

The Compound Interest (CI) is calculated as: CI = A - P or CI = P * [(1 + i)ⁿ - 1]

Understanding the Compounding Effect (Example)

Principal = ₹1000, Rate = 10% p.a., compounded annually.

• Year 1: Amount = ₹1000 + (10% of ₹1000) = ₹1100. (Same as SI for 1 year)

• Year 2: Amount = ₹1100 + (10% of ₹1100) = ₹1210. (Interest on previous interest)

• Compare with SI: For 2 years, SI would give ₹1200. CI gives ₹1210, demonstrating "interest on interest".

Worked Example: Compound Interest Calculation

Find the CI on ₹16,000 for 1.5 years at 10% p.a., with interest payable half-yearly.

• P = ₹16,000, t = 1.5 years, r = 10%, m = 2

• i = (10 / 2) / 100 = 0.05

• n = 2 * 1.5 = 3

• CI = 16000 * [(1 + 0.05)³ - 1] = 16000 * [(1.05)³ - 1]

• CI = 16000 * [1.157625 - 1] = 16000 * 0.157625 = ₹2,522

Fundamental Difference: Simple vs. Compound Interest

Effective Rate of Interest

The Effective Rate of Interest is the actual annual rate earned on an investment when compounding occurs more than once a year. It reveals the true annual return.

• The effective rate is for one year and does not depend on the principal amount.

• Formula: E = [ (1 + i)ⁿ - 1 ] * 100 (where i is rate per period, n is compounding periods in one year, i.e., m).

Example: Effective rate for 10% p.a. compounded quarterly.

• m = 4, i = 10 / (4 * 100) = 0.025

• E = [ (1 + 0.025)⁴ - 1 ] * 100 = [ (1.025)⁴ - 1 ] * 100

• E = [ 1.10381 - 1 ] * 100 = 10.38%

Annuities

Definition of an Annuity

An annuity is a sequence of payments that meets two strict conditions:

1. Regular Amount: The value of each payment is always the same.

2. Regular Interval of Time: The time gap between any two consecutive payments is always the same.
   (Memory Tip: A Fixed Deposit (FD) is not an annuity. A Systematic Investment Plan (SIP) is an annuity.)

Examples of Annuities

Types of Annuities

Based on Time Duration:

1. Annuity Certain: Fixed time period (known start and end).

2. Contingent Annuity: Known start, but end depends on an event.

3. Perpetuity: No end date; payments continue forever.

Based on Time of Payment:

1. Ordinary Annuity: Payments at the end of each interval (Analogy: Receiving a salary).

2. Annuity Due: Payments at the beginning of each interval (Analogy: Paying rent).

3. Deferred Annuity: First payment is delayed for specified periods.

Core Concepts in Annuity Calculations

• Future Value (FV): The total accumulated value of a series of payments at a specific point in the future. (e.g., total savings from monthly deposits).

• Present Value (PV): The current worth of a series of future payments. (e.g., lump sum needed today to receive future income stream).

Formulas for Future Value (FV)

1. Ordinary Annuity (payments at end): FV = R * [((1 + i)ⁿ - 1) / i]

2. Annuity Due (payments at beginning): FV = { R * [((1 + i)ⁿ - 1) / i] } * (1 + i)

• R: Regular payment amount.

• i: Interest rate per compounding period.

• n: Total number of payment intervals.

Formulas for Present Value (PV)

1. Ordinary Annuity (payments at end): PV = R * [(1 - (1 + i)⁻ⁿ) / i]

2. Immediate Annuity (Annuity Due, payments at beginning): PV = { R * [(1 - (1 + i)⁻ⁿ) / i] } * (1 + i)

Application: Loan Repayments (Amortization): The loan amount is the Present Value (PV), and equal installments are the Regular Payments (R).

Worked Example: Sinking Fund (FV Application)

A machine costs ₹98,000, scrap value ₹3,000 after 12 years. What amount (R) should be retained from profits annually (end of year) at 5% p.a. to accumulate funds for a new machine?

• Target FV (A) = ₹98,000 - ₹3,000 = ₹95,000

• n = 12, i = 0.05

• A = R * [((1 + i)ⁿ - 1) / i]

• 95,000 = R * [((1.05)¹² - 1) / 0.05]

• Assuming (1.05)¹² ≈ 1.797

• 95,000 = R * [(1.797 - 1) / 0.05] = R * (0.797 / 0.05)

• R = (95,000 * 0.05) / 0.797 ≈ ₹5,960.00

Net Present Value (NPV)

Net Present Value (NPV) evaluates investment profitability. It is the difference between the present value of all future cash inflows and the present value of all cash outflows.

• Formula: NPV = (PV of all Cash Inflows) - (PV of all Cash Outflows)

• Decision Rule:

• NPV > 0: Investment is good (profitable).

• NPV < 0: Investment is bad.

GST (Goods and Services Tax)

Problem 1: GST Chain (Intra-State)

A Wholesaler gives 10% discount on ₹45,000 AC to a Dealer. Dealer sells to Consumer at 4% markup on printed price. Intra-state, GST 18%.

• Wholesaler to Dealer (Taxable Value): ₹45,000 - (10% of ₹45,000) = ₹40,500.

• Input GST (for Dealer): 18% of ₹40,500 = ₹7,290.

• Dealer to Consumer (Taxable Value): ₹45,000 + (4% of ₹45,000) = ₹46,800.

• Output GST (for Dealer): 18% of ₹46,800 = ₹8,424.

• Tax Paid by Dealer to Government: Output GST - Input GST = ₹8,424 - ₹7,290 = ₹1,134 (₹567 CGST, ₹567 SGST).

• Total Tax Received by Government: Final Output GST = ₹8,424 (₹4,212 CGST, ₹4,212 SGST).

• Total Amount Paid by Consumer: Taxable Value + Output GST = ₹46,800 + ₹8,424 = ₹55,224.

Problem 2: GST Chain (Inter-State / IGST)

Manufacturer (Maharashtra) sells a ₹2,500 carpet to Wholesaler (West Bengal) at 12% discount. Wholesaler sells to Retailer (Bihar) at 32% above marked price. GST 5%.

• Transaction 1 (Inter-state): Manufacturer to Wholesaler

• Taxable Value: ₹2,500 - (12% of ₹2,500) = ₹2,200.

• IGST: 5% of ₹2,200 = ₹110.

• Price for Wholesaler (incl. tax): ₹2,200 + ₹110 = ₹2,310.

• Transaction 2 (Inter-state): Wholesaler to Retailer

• Taxable Value: ₹2,500 + (32% of ₹2,500) = ₹3,300.

• IGST: 5% of ₹3,300 = ₹165.

• Price for Retailer (incl. tax): ₹3,300 + ₹165 = ₹3,465.

• Tax Received by Central Government: In inter-state transactions, the entire IGST goes to the Central Government. The final IGST in the chain is ₹165.

Utility Bills

Conceptual Framework of a Utility Bill

Utility bills (electricity, water, gas) typically comprise:

1. Consumption-Based Charges: Based on usage, often with a slab system (price per unit increases with consumption).

2. Fixed Charges: Constant fees regardless of usage (e.g., meter rent, load charges).

3. Taxes and Surcharges: Additional fees (e.g., Energy Tax, specific surcharges).

Example: Calculating an Electricity Bill

Consumer uses 587 units; load 4 kW.

• Slabs: 0-200 units @ ₹3.00; 201-400 units @ ₹4.50; 401-800 units @ ₹6.50.

• Fixed Charge: ₹50/kW.

• Surcharge: ₹0.40/unit.

• Energy Tax: 5% on (consumption + fixed charges).

1. Consumption Charges:

• (200 units * ₹3.00) = ₹600.00

• (200 units * ₹4.50) = ₹900.00

• (187 units * ₹6.50) = ₹1,215.50 (Remaining units: 587 - 400 = 187)

• Total Consumption Charge: ₹600 + ₹900 + ₹1,215.50 = ₹2,715.50

2. Fixed Charge: 4 kW * ₹50/kW = ₹200.00

3. Surcharge: 587 units * ₹0.40/unit = ₹234.80

4. Energy Tax: 5% of (₹2,715.50 + ₹200.00) = 5% of ₹2,915.50 = ₹145.78

5. Total Electricity Bill: ₹2,715.50 + ₹200.00 + ₹234.80 + ₹145.78 = ₹3,296.08