Question 1

₹5,000 is invested at 8% per annum compound interest, compounded quarterly. Find the compound interest earned in 6 months.

Accepted Answer

Step 1: For quarterly compounding, rate per quarter = 8% ÷ 4 = 2% per quarter. Step 2: Number of quarters in 6 months = 2. Step 3: Amount = 5000(1 + 2/100)^2 = 5000(1.02)^2 = 5000 × 1.0404 = ₹5,202. Step 4: Compound Interest = 5202 - 5000 = ₹202.

Question 2

A sum of ₹8,000 is invested at 12% per annum compound interest, compounded half-yearly. What will be the amount after 1 year?

Accepted Answer

Step 1: For half-yearly compounding, rate per half-year = 12% ÷ 2 = 6% per half-year. Step 2: Number of half-years in 1 year = 2. Step 3: Amount = P(1 + r/100)^n = 8000(1 + 6/100)^2 = 8000(1.06)^2 = 8000 × 1.1236 = ₹8,988.80.

Question 3

A principal amount becomes ₹10,648 in 2 years at 8% per annum compound interest, compounded half-yearly. What is the principal?

Accepted Answer

Step 1: For half-yearly compounding, rate per half-year = 8% ÷ 2 = 4% per half-year. Step 2: Number of half-years in 2 years = 4. Step 3: Amount = P(1 + r/100)^n, so 10648 = P(1.04)^4. Step 4: (1.04)^4 = 1.16985856 ≈ 1.1699. Step 5: P = 10648 ÷ 1.1699 ≈ ₹9,100 (exact: 10648 ÷ 1.16985856 = 9100).

Question 4

₹12,000 is invested at 10% per annum compound interest, compounded half-yearly. After how many half-years will the amount become ₹13,230?

Accepted Answer

Step 1: For half-yearly compounding, rate per half-year = 10% ÷ 2 = 5% per half-year. Step 2: Amount = P(1 + r/100)^n, so 13230 = 12000(1.05)^n. Step 3: (1.05)^n = 13230 ÷ 12000 = 1.1025. Step 4: (1.05)^2 = 1.1025. Step 5: n = 2 half-years.

Question 5

The compound interest on ₹6,400 at 12.5% per annum for 1 year, compounded quarterly, is:

Accepted Answer

Step 1: For quarterly compounding, rate per quarter = 12.5% ÷ 4 = 3.125% per quarter. Step 2: Number of quarters in 1 year = 4. Step 3: Amount = 6400(1 + 3.125/100)^4 = 6400(1.03125)^4. Step 4: (1.03125)^4 = 1.13141... ≈ 1.1314. Step 5: Amount = 6400 × 1.1314 = 7,241 (exact: 7240.96). Step 6: CI = 7241 - 6400 = ₹841.

Question 6

The difference between compound interest and simple interest on a sum for 1 year at 8% per annum, compounded half-yearly, is ₹16. What is the principal?

Accepted Answer

Step 1: Let Principal = P. Rate = 8% p.a., Time = 1 year. Step 2: For SI: SI = (P × 8 × 1)/100 = 0.08P. Step 3: For half-yearly CI: A = P(1 + 4/100)^2 = P(1.04)^2 = 1.0816P. CI = 1.0816P - P = 0.0816P. Step 4: Difference = CI - SI = 0.0816P - 0.08P = 0.0016P. Step 5: 0.0016P = 16 → P = 16 ÷ 0.0016 = ₹10,000.

Question 7

₹12,000 is invested at 8% per annum compound interest, compounded quarterly. Find the compound interest earned in 6 months.

Accepted Answer

Step 1: Principal (P) = ₹12,000, Rate (R) = 8% p.a., Time (T) = 6 months = 0.5 years, Compounded quarterly. Step 2: For quarterly compounding, rate per quarter = 8/4 = 2%, and number of quarters in 6 months = 2. Step 3: Amount = P(1 + r/100)^n = 12000(1 + 2/100)^2 = 12000(1.02)^2 = 12000 × 1.0404 = ₹12,484.80. Step 4: Compound Interest = ₹12,484.80 - ₹12,000 = ₹484.80.

Question 8

A sum of money triples itself in 3 years at compound interest, compounded half-yearly. What is the rate of interest per annum (approximately)?

Accepted Answer

Step 1: Let Principal = P, Amount = 3P (triples). Time = 3 years, Compounded half-yearly. Step 2: Number of half-years = 3 × 2 = 6. Step 3: A = P(1 + r/100)^n → 3P = P(1 + r/100)^6. Step 4: 3 = (1 + r/100)^6 → 1 + r/100 = 3^(1/6) ≈ 1.1956. Step 5: r/100 ≈ 0.1956, so r ≈ 19.56% ≈ 20% (approximately).

Question 9

A principal amount becomes ₹14,641 in 2 years at 20% per annum compound interest, compounded half-yearly. What is the principal?

Accepted Answer

Step 1: Amount (A) = ₹14,641, Rate (R) = 20% p.a., Time (T) = 2 years, Compounded half-yearly. Step 2: For half-yearly compounding, rate per half-year = 20/2 = 10%, and number of half-years = 2 × 2 = 4. Step 3: A = P(1 + r/100)^n → 14641 = P(1.10)^4 = P × 1.4641. Step 4: P = 14641 ÷ 1.4641 = ₹10,000.

Question 10

A sum of ₹8,000 is invested at 10% per annum compounded half-yearly for 1 year. What is the compound interest earned?

Accepted Answer

Step 1: For half-yearly compounding, Rate per period = 10/2 = 5%, Number of periods = 1×2 = 2. Step 2: Amount = P × (1 + r/100)^n = 8000 × (1.05)^2 = 8000 × 1.1025 = ₹8,820. Step 3: CI = Amount − Principal = 8820 − 8000 = ₹820. Common mistake: Using annual rate directly gives 8000×0.10 = ₹800 (simple interest trap). Therefore, answer is ₹820.

Question 11

The difference between compound interest and simple interest on a sum of ₹10,000 for 2 years, compounded half-yearly at 8% per annum, is:

Accepted Answer

Step 1: Principal P = ₹10,000, Rate R = 8% p.a., Time = 2 years. Step 2: For half-yearly CI: n = 4 half-years, r = 4% per half-year. Amount = 10,000(1.04)^4 = 10,000 × 1.16985856 = 11,698.59. CI = 11,698.59 - 10,000 = 1,698.59. Step 3: For SI: SI = (10,000 × 8 × 2)/100 = 1,600. Step 4: Difference = 1,698.59 - 1,600 = 98.59 ≈ ₹98.60 or ₹99.

Question 12

A sum of ₹8,000 is invested at 12% per annum compound interest, compounded half-yearly. What will be the amount after 18 months?

Accepted Answer

Step 1: Principal P = ₹8,000, Rate R = 12% p.a., Time = 18 months = 3 half-years. Step 2: For half-yearly compounding, rate per half-year = 12/2 = 6% per half-year. Step 3: Amount = P(1 + r/100)^n = 8000(1 + 6/100)^3 = 8000(1.06)^3. Step 4: (1.06)^3 = 1.191016. Step 5: Amount = 8000 × 1.191016 = ₹9,528.13 ≈ ₹9,528.

Question 13

A person invests ₹5,000 at 12% per annum compound interest, compounded half-yearly. Another person invests ₹5,000 at 12% per annum compound interest, compounded quarterly. What is the difference in their amounts after 1 year?

Accepted Answer

Step 1: Both invest ₹5,000 at 12% p.a. for 1 year. Step 2: Person A (half-yearly): n = 2 half-years, r = 6% per half-year. Amount_A = 5,000(1.06)^2 = 5,000 × 1.1236 = 5,618. Step 3: Person B (quarterly): n = 4 quarters, r = 3% per quarter. Amount_B = 5,000(1.03)^4 = 5,000 × 1.12550881 = 5,627.54. Step 4: Difference = 5,627.54 - 5,618 = 9.54 ≈ ₹9.50 or ₹10.

Question 14

The difference between compound interest and simple interest on a sum of ₹10,000 at 10% per annum for 1 year, compounded half-yearly, is:

Accepted Answer

Step 1: Principal (P) = ₹10,000, Rate (R) = 10% p.a., Time (T) = 1 year. Step 2: For Compound Interest (half-yearly): A = P(1 + R/200)^(2T) = 10,000(1 + 10/200)^2 = 10,000(1.05)^2 = 10,000 × 1.1025 = ₹11,025. CI = 11,025 - 10,000 = ₹1,025. Step 3: For Simple Interest: SI = (P × R × T)/100 = (10,000 × 10 × 1)/100 = ₹1,000. Step 4: Difference = CI - SI = 1,025 - 1,000 = ₹25.

Question 15

₹5,000 is invested at 8% per annum compound interest, compounded quarterly. What is the compound interest earned after 6 months?

Accepted Answer

Step 1: Principal (P) = ₹5,000, Rate (R) = 8% p.a., Time (T) = 6 months = 0.5 years, Compounded quarterly. Step 2: For quarterly compounding, we use A = P(1 + R/400)^(4T). Step 3: Number of quarters in 6 months = 2. Step 4: A = 5000(1 + 8/400)^2 = 5000(1 + 0.02)^2 = 5000(1.02)^2 = 5000 × 1.0404 = ₹5,202. Step 5: Compound Interest = A - P = 5,202 - 5,000 = ₹202.

Question 16

₹5,000 is invested at 8% per annum compound interest, compounded quarterly. Find the compound interest earned in 1 year.

Accepted Answer

Step 1: For quarterly compounding, rate per quarter = 8% ÷ 4 = 2% per quarter. Step 2: Number of quarters in 1 year = 4. Step 3: Amount = 5000(1 + 2/100)^4 = 5000(1.02)^4 = 5000 × 1.08243216 = ₹5,412.16. Step 4: Compound Interest = 5412.16 - 5000 = ₹412.16. Therefore, the answer is ₹412.16.

Question 17

₹12,000 is invested at 10% per annum compound interest, compounded quarterly. What is the amount after 6 months?

Accepted Answer

Step 1: For quarterly compounding, rate per quarter = 10% ÷ 4 = 2.5% per quarter. Step 2: 6 months = 2 quarters. Step 3: Amount = 12000(1 + 2.5/100)^2 = 12000(1.025)^2 = 12000 × 1.050625 = ₹12,607.50. Therefore, the answer is ₹12,607.50.

Question 18

A sum of ₹8,000 is invested at 12% per annum compound interest, compounded half-yearly. What will be the amount after 1 year?

Accepted Answer

Step 1: For half-yearly compounding, rate per half-year = 12% ÷ 2 = 6% per half-year. Step 2: Number of half-years in 1 year = 2. Step 3: Amount = P(1 + r/100)^n = 8000(1 + 6/100)^2 = 8000(1.06)^2 = 8000 × 1.1236 = ₹8,988.80. Therefore, the answer is ₹8,988.80.

Question 19

A principal amount becomes ₹10,648 in 2 years at 8% per annum compound interest, compounded half-yearly. What is the principal?

Accepted Answer

Step 1: For half-yearly compounding, rate per half-year = 8% ÷ 2 = 4% per half-year. Step 2: Number of half-years in 2 years = 4. Step 3: Amount = P(1 + r/100)^n, so 10648 = P(1.04)^4. Step 4: (1.04)^4 = 1.16985856 ≈ 1.1699. Step 5: P = 10648 ÷ 1.1699 = ₹9,100. Therefore, the principal is ₹9,100.

Question 20

₹20,000 is invested at 20% per annum compound interest, compounded quarterly. What will be the amount after 6 months?

Accepted Answer

Step 1: For quarterly compounding, rate per quarter = 20% ÷ 4 = 5% per quarter. Step 2: 6 months = 2 quarters. Step 3: Amount = 20000(1 + 5/100)^2 = 20000(1.05)^2 = 20000 × 1.1025 = ₹22,050. Therefore, the answer is ₹22,050.

Question 21

The compound interest on ₹6,400 at 12.5% per annum, compounded half-yearly, for 1 year is:

Accepted Answer

Step 1: For half-yearly compounding, rate per half-year = 12.5% ÷ 2 = 6.25% per half-year. Step 2: Number of half-years in 1 year = 2. Step 3: Amount = 6400(1 + 6.25/100)^2 = 6400(1.0625)^2 = 6400 × 1.12890625 = ₹7,224.90. Step 4: Compound Interest = 7224.90 - 6400 = ₹824.90. Therefore, the answer is ₹824.90.

Question 22

₹5,000 is invested at 16% per annum compound interest, compounded quarterly. What is the difference between the amount after 6 months and after 3 months?

Accepted Answer

Step 1: Principal (P) = ₹5,000, Rate (R) = 16% p.a., Compounded quarterly. Step 2: Quarterly rate = 16/4 = 4%. Step 3: Amount after 3 months (1 quarter) = 5000(1.04)^1 = ₹5,200. Step 4: Amount after 6 months (2 quarters) = 5000(1.04)^2 = 5000 × 1.0816 = ₹5,408. Step 5: Difference = 5408 - 5200 = ₹208.

Question 23

A sum of ₹5,000 is invested at 16% per annum compound interest, compounded quarterly. What is the difference between the compound interest for the 1st quarter and the 2nd quarter?

Accepted Answer

Step 1: Principal (P) = ₹5,000, Rate (R) = 16% p.a., Compounded quarterly. Step 2: Quarterly rate = 16/4 = 4%. Step 3: Interest for 1st quarter = 5,000 × 4/100 = ₹200. Step 4: Amount after 1st quarter = 5,000 + 200 = ₹5,200. Step 5: Interest for 2nd quarter = 5,200 × 4/100 = ₹208. Step 6: Difference = 208 - 200 = ₹8.

Question 24

A principal amount becomes ₹13,310 in 1 year at 10% per annum compound interest, compounded half-yearly. What is the principal?

Accepted Answer

Step 1: Amount (A) = ₹13,310, Rate (R) = 10% p.a., Time (T) = 1 year, Compounded half-yearly. Step 2: For half-yearly compounding, rate per half-year = 10/2 = 5%, and number of periods = 1 × 2 = 2. Step 3: A = P(1 + r/100)^n ⟹ 13310 = P(1.05)^2 ⟹ 13310 = P × 1.1025 ⟹ P = 13310/1.1025 = ₹12,000.

Question 25

₹12,000 is lent at 8% per annum compound interest, compounded quarterly. Find the compound interest earned in 6 months.

Accepted Answer

Step 1: Principal (P) = ₹12,000, Rate (R) = 8% p.a., Time (T) = 6 months = 0.5 years, Compounded quarterly. Step 2: For quarterly compounding, rate per quarter = 8/4 = 2%, and number of quarters in 6 months = 2. Step 3: Amount = P(1 + r/100)^n = 12000(1 + 2/100)^2 = 12000(1.02)^2 = 12000 × 1.0404 = ₹12,484.80. Step 4: Compound Interest = Amount - Principal = 12,484.80 - 12,000 = ₹484.80.

Question 26

₹12,000 is invested at 8% per annum compound interest, compounded quarterly. Find the compound interest earned in 6 months.

Accepted Answer

Step 1: Principal (P) = ₹12,000, Rate (R) = 8% p.a., Time (T) = 6 months = 0.5 years, Compounded quarterly. Step 2: For quarterly compounding, rate per quarter = 8/4 = 2%, and number of quarters in 6 months = 2. Step 3: Amount = 12000(1 + 2/100)^2 = 12000(1.02)^2 = 12000 × 1.0404 = ₹12,484.80. Step 4: Compound Interest = 12484.80 - 12000 = ₹484.80.

Question 27

₹10,000 is invested at 12% per annum compound interest, compounded quarterly for 9 months. What is the compound interest earned?

Accepted Answer

Step 1: Principal (P) = ₹10,000, Rate (R) = 12% p.a., Time (T) = 9 months = 3 quarters, Compounded quarterly. Step 2: Quarterly rate = 12/4 = 3%, and number of quarters = 3. Step 3: Amount = 10000(1 + 3/100)^3 = 10000(1.03)^3. Step 4: (1.03)^3 = 1.03 × 1.03 × 1.03 = 1.092727. Step 5: Amount = 10000 × 1.092727 = ₹10,927.27. Step 6: Compound Interest = 10927.27 - 10000 = ₹927.27.

Question 28

The difference between compound interest (half-yearly) and simple interest on ₹10,000 for 1.5 years at 16% per annum is:

Accepted Answer

Step 1: P = ₹10,000, R = 16% p.a., T = 1.5 years. Step 2: Simple Interest: SI = (P × R × T)/100 = (10,000 × 16 × 1.5)/100 = ₹2,400. Step 3: For half-yearly CI, rate per half-year = 16/2 = 8%, time = 1.5 years = 3 half-years. Step 4: Amount A = P(1.08)^3 = 10,000 × 1.259712 = ₹12,597.12. Step 5: CI = A - P = 12,597.12 - 10,000 = ₹2,597.12. Step 6: Difference = CI - SI = 2,597.12 - 2,400 = ₹197.12 ≈ ₹197.

Question 29

₹5,000 is invested at 20% per annum compound interest, compounded half-yearly. If ₹2,000 is withdrawn after 1 year, what is the amount after another 1 year (i.e., at the end of 2 years)?

Accepted Answer

Step 1: P = ₹5,000, R = 20% p.a., half-yearly rate = 10%, T = 1 year = 2 half-years. Step 2: Amount after 1 year: A₁ = 5,000(1.10)^2 = 5,000 × 1.21 = ₹6,050. Step 3: After withdrawal: 6,050 - 2,000 = ₹4,050 (new principal). Step 4: This amount compounds for another 1 year (2 half-years) at 10% per half-year. Step 5: A₂ = 4,050(1.10)^2 = 4,050 × 1.21 = ₹4,900.50 ≈ ₹4,900 (or ₹4,901 if rounding to nearest rupee).

Question 30

A sum of ₹8,000 is invested at 12% per annum compound interest, compounded half-yearly. What is the compound interest earned in 1.5 years?

Accepted Answer

Step 1: Principal P = ₹8,000, Rate R = 12% p.a., Time T = 1.5 years = 3 half-years. Step 2: For half-yearly compounding, rate per half-year = 12/2 = 6%. Step 3: Amount A = P(1 + r/100)^n = 8000(1 + 6/100)^3 = 8000(1.06)^3. Step 4: (1.06)^3 = 1.191016, so A = 8000 × 1.191016 = ₹9,528.13. Step 5: CI = A - P = 9,528.13 - 8,000 = ₹1,528.13 ≈ ₹1,528.

Question 31

A sum of money doubles itself in 2 years at compound interest, compounded quarterly. At what rate per annum is it invested?

Accepted Answer

Step 1: Let principal = P, amount = 2P (doubles), T = 2 years = 8 quarters. Step 2: For quarterly compounding, 2P = P(1 + r/100)^8, where r is the quarterly rate. Step 3: 2 = (1 + r/100)^8. Step 4: Taking 8th root: (1 + r/100) = 2^(1/8) = 1.09051. Step 5: r/100 = 0.09051, so r = 9.051% per quarter. Step 6: Annual rate = 9.051 × 4 = 36.2% ≈ 36% (or more precisely 36.2%). Verification: (1.09051)^8 ≈ 2.00. For cleaner calculation: if annual rate R, then quarterly rate = R/4. So 2 = (1 + R/400)^8. Testing R = 36: (1.09)^8 = 1.9926 ≈ 2 ✓

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