Concept of Interest Rates – Class 11 Applied Mathematics

Simple Interest Fundamentals

Simple interest is the interest calculated only on the principal amount borrowed or invested over a specific period. It remains constant for each time period because the principal does not change. The formula relates the principal, rate of interest, and time.

I = (P × R × T) / 100

Example: Solved Example: Calculating Simple Interest

Show Step-by-Step Solution

Step 1: Identify P = ₹5000, R = 10% p.a., T = 3 years.
Step 2: Apply formula I = (5000 × 10 × 3) / 100.
Step 3: Calculate I = 1500.
Answer: The simple interest is ₹1500.

Compound Interest Mechanics

Compound interest is calculated on the principal plus the accumulated interest from previous periods. This leads to exponential growth of the investment. The amount after n periods is determined by the compound interest formula.

A = P(1 + r/100)ⁿ

Example: Solved Example: Calculating Compound Amount

Show Step-by-Step Solution

Step 1: Identify P = ₹10000, r = 5%, n = 2 years.
Step 2: Apply A = 10000(1 + 5/100)².
Step 3: A = 10000(1.05)² = 10000 × 1.1025 = 11025.
Answer: The total amount is ₹11025.

Effective Rate of Interest

The effective annual rate (EAR) represents the actual interest earned due to compounding within a year. If interest is compounded k times per year, the effective rate is higher than the nominal rate. This allows for comparing different investment schemes.

EAR = (1 + r/k)ᵏ - 1

Example: Solved Example: Finding EAR

Show Step-by-Step Solution

Step 1: Nominal rate r = 12% (0.12), compounded quarterly (k=4).
Step 2: EAR = (1 + 0.12/4)⁴ - 1.
Step 3: EAR = (1.03)⁴ - 1 = 1.1255 - 1 = 0.1255.
Answer: The effective rate is 12.55%.

Section A: Multiple Choice Questions (MCQs)

easyFind the simple interest on ₹2000 at 5% per annum for 2 years.

A. ₹100 B. ₹200 C. ₹300 D. ₹400

Show Answer

✔ Correct: B

I = (2000 × 5 × 2) / 100 = 200.

easyIf the principal is ₹1000 and the amount after 1 year is ₹1080, what is the rate of interest?

A. 5% B. 6% C. 8% D. 10%

Show Answer

✔ Correct: C

Interest = 1080 - 1000 = 80. R = (80 × 100) / (1000 × 1) = 8%.

mediumCalculate the compound interest on ₹5000 for 2 years at 10% per annum compounded annually.

A. ₹1000 B. ₹1050 C. ₹1100 D. ₹1150

Show Answer

✔ Correct: B

A = 5000(1.1)² = 6050. CI = 6050 - 5000 = 1050.

mediumA sum of ₹4000 amounts to ₹4410 in 2 years at compound interest. Find the rate per annum.

A. 4% B. 5% C. 6% D. 10%

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✔ Correct: B

4410 = 4000(1 + r/100)². 1.1025 = (1 + r/100)². √1.1025 = 1.05. r = 5%.

hardFind the difference between simple and compound interest on ₹10000 for 2 years at 10% per annum.

A. ₹50 B. ₹100 C. ₹150 D. ₹200

Show Answer

✔ Correct: B

SI = 2000. CI = 10000(1.1)² - 10000 = 2100. Difference = 100.

hardIf the effective rate of interest is 10.25% for a nominal rate compounded semi-annually, find the nominal rate.

A. 8% B. 9% C. 10% D. 12%

Show Answer

✔ Correct: C

(1 + r/2)² - 1 = 0.1025. (1 + r/2)² = 1.1025. 1 + r/2 = 1.05. r = 0.10 or 10%.

very hardA sum of money doubles itself in 8 years at simple interest. In how many years will it triple itself?

A. 12 years B. 16 years C. 20 years D. 24 years

Show Answer

✔ Correct: B

P becomes 2P in 8 years, so interest = P. P = (P × R × 8)/100 → R = 12.5%. To triple, interest = 2P. 2P = (P × 12.5 × T)/100 → T = 200/12.5 = 16 years.

Section B: Assertion–Reason Questions

Assertion (A): The simple interest on ₹1000 at 10% for 2 years is ₹200.
Reason (R): Simple interest is calculated as (P × R × T) / 100.

Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

A. Option A B. Option B C. Option C D. Option D

Show Answer

✔ Correct: A

Both are true and the reason correctly explains the assertion calculation.

Assertion (A): Compound interest is always greater than or equal to simple interest for the same principal, rate, and time (T > 1).
Reason (R): Compound interest includes interest on interest, whereas simple interest does not.

Choose the correct option:
(A) Both A and R are true and R is the correct explanation of A.
(B) Both A and R are true but R is not the correct explanation of A.
(C) A is true but R is false.
(D) A is false but R is true.

A. Option A B. Option B C. Option C D. Option D

Show Answer

✔ Correct: A

Compound interest grows exponentially, making it greater than linear simple interest growth.

Section C: Short Answer Questions

mediumGiven P = ₹8000, R = 5%, T = 3 years. Find: (i) Simple Interest (ii) Amount at Simple Interest (iii) Difference if compounded annually.

Show Answer & Solution

(i) SI = (8000 × 5 × 3)/100 = ₹1200.
(ii) A = 8000 + 1200 = ₹9200.
(iii) CI = 8000(1.05)³ - 8000 = 9261 - 8000 = 1261. Diff = 1261 - 1200 = ₹61.

mediumGiven nominal rate 8% compounded quarterly. Find: (i) Periodic rate (ii) Number of periods in 2 years (iii) Effective annual rate.

Show Answer & Solution

(i) 8%/4 = 2% or 0.02.
(ii) 2 years × 4 = 8 periods.
(iii) EAR = (1 + 0.02)⁴ - 1 = 1.0824 - 1 = 8.24%.

hardA sum of ₹5000 is invested. Find: (i) Amount after 1 year at 10% simple interest (ii) Amount after 1 year at 10% compounded semi-annually (iii) The difference between (i) and (ii).

Show Answer & Solution

(i) A = 5000(1 + 0.1) = ₹5500.
(ii) A = 5000(1 + 0.05)² = 5000(1.1025) = ₹5512.50.
(iii) Diff = 5512.50 - 5500 = ₹12.50.

Section D: Long Answer Questions

hardAn investor deposits ₹20000 in a bank offering 12% interest compounded monthly. Calculate the total amount after 2 years and the total interest earned.

Show Answer & Solution

Step 1: Identify P = 20000, r = 0.12, k = 12, t = 2.
Step 2: Monthly rate = 0.12 / 12 = 0.01.
Step 3: Total periods n = 12 × 2 = 24.
Step 4: Use A = P(1 + i)ⁿ → A = 20000(1.01)²⁴.
Step 5: A = 20000 × 1.2697 = 25394. Interest = 25394 - 20000 = ₹5394.

very hardCompare two investment options for ₹10000: Option A: 10% simple interest for 3 years. Option B: 9% interest compounded annually for 3 years. Which is better?

Show Answer & Solution

Step 1: Calculate Option A: SI = (10000 × 10 × 3)/100 = ₹3000. Total = ₹13000.
Step 2: Calculate Option B: A = 10000(1.09)³.
Step 3: 1.09³ = 1.295029.
Step 4: A = 10000 × 1.295029 = ₹12950.29.
Step 5: Compare: Option A (13000) > Option B (12950.29). Option A is better.

Section E: Case-Based Question

★

Case Study: A student receives a scholarship of ₹50000. They decide to invest it for 2 years. Bank X offers 8% simple interest. Bank Y offers 7.5% interest compounded annually.

Sub-question 1: (i) Calculate the maturity value for Bank X. (ii) Calculate the maturity value for Bank Y.

Show Solution

(i) Bank X: A = 50000(1 + 0.08 × 2) = 50000(1.16) = ₹58000.
(ii) Bank Y: A = 50000(1.075)² = 50000(1.155625) = ₹57781.25.

Sub-question 2: Which bank provides a higher return and by how much?

Show Solution

Bank X provides ₹58000 and Bank Y provides ₹57781.25. Bank X is better by 58000 - 57781.25 = ₹218.75.

CBSE Board & Sample Paper Past Year Questions

📋 CBSE 2023 📝 2 Marks

If a sum of money becomes 4 times in 10 years at simple interest, find the rate of interest.

Show Answer & Solution

Let P = x, A = 4x. Interest = 3x. 3x = (x × R × 10)/100. 3 = R/10. R = 30%.

📋 CBSE 2022 📝 2 Marks

Calculate the compound interest on ₹10000 at 10% p.a. for 1 year compounded half-yearly.

Show Answer & Solution

P = 10000, r = 5% per half-year, n = 2. A = 10000(1.05)² = 11025. CI = 11025 - 10000 = ₹1025.

📋 CBSE Sample Paper 2024 📝 1 Mark

The effective rate of interest for a nominal rate of 12% compounded semi-annually is:

Show Answer & Solution

✔ Answer: C

EAR = (1 + 0.12/2)² - 1 = (1.06)² - 1 = 1.1236 - 1 = 12.36%.

📋 CBSE 2020 📝 1 Mark

Find the simple interest on ₹5000 at 6% p.a. for 9 months.

Show Answer & Solution

T = 9/12 = 0.75 years. SI = (5000 × 6 × 0.75)/100 = 50 × 4.5 = ₹225.

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