Exercise: 8-D
Q1: Find the amount and the compound interest on ₹120000 at 8% per annum for 1 year, compounded half-yearly.
Step 1: Use the compound interest formula for half-yearly compounding:
\[
A = P \left(1 + \frac{R}{2 \times 100}\right)^{2T}
\]
Where:
P = ₹120000, R = 8%, T = 1 year
Step 2: Substitute the values:
\[
A = 120000 \left(1 + \frac{8}{200}\right)^2 = 120000 \times (1.04)^2 = 120000 \times 1.0816 = ₹129792
\]Step 3: Calculate compound interest:
\[
CI = A – P = 129792 – 120000 = ₹9792
\]Answer: Amount = ₹129792, Compound Interest = ₹9792
Q2: Find the amount and the compound interest on ₹32500 for 1 year; at 12% per annum, compounded half yearly.
Step 1: Use the compound interest formula for half-yearly compounding:
\[
A = P \left(1 + \frac{R}{2 \times 100} \right)^{2T}
\]
Where:
P = ₹32500, R = 12%, T = 1 year
Step 2: Substitute the values:
\[
A = 32500 \left(1 + \frac{12}{200} \right)^2 = 32500 \times (1.06)^2 = 32500 \times 1.1236 = ₹36517
\]Step 3: Calculate compound interest:
\[
CI = A – P = 36517 – 32500 = ₹4017
\]Answer: Amount = ₹36517, Compound Interest = ₹4017
Q3: Calculate the amount and the compound interest on ₹24000 for \(1\frac{1}{2}\) years at 10% per annum, compounded half-yearly.
Step 1: Convert time to months or half-years:
\[
T = 1\frac{1}{2} = 1.5 \text{ years} = 3 \text{ half-years}
\]Step 2: Use compound interest formula for half-yearly compounding:
\[
A = P \left(1 + \frac{R}{2 \times 100}\right)^{2T} = P \left(1 + \frac{R}{200}\right)^n
\]
Where:
\(P = 24000, R = 10\%, n = 3 \text{ (half-years)}\)
Step 3: Substitute values:
\[
A = 24000 \times \left(1 + \frac{10}{200}\right)^3 = 24000 \times (1.05)^3 = 24000 \times 1.157625 = ₹27783
\]Step 4: Calculate Compound Interest:
\[
CI = A – P = 27783 – 24000 = ₹3783
\]Answer: Amount = ₹27783, Compound Interest = ₹3783
Q4: Calculate the amount and the compound interest on ₹10000 for 6 months at 12% per annum, compounded quarterly.
Step 1: Convert time to quarters:
\[
T = 6 \text{ months} = \frac{6}{12} = 0.5 \text{ years}
\]
Number of quarters \(n = 0.5 \times 4 = 2\)
Step 2: Use the compound interest formula for quarterly compounding:
\[
A = P \left(1 + \frac{R}{4 \times 100}\right)^n
\]
Where:
\(P = 10000, R = 12\%, n = 2\)
Step 3: Substitute values:
\[
A = 10000 \times \left(1 + \frac{12}{400}\right)^2 = 10000 \times (1.03)^2 = 10000 \times 1.0609 = ₹10609
\]Step 4: Calculate Compound Interest:
\[
CI = A – P = 10609 – 10000 = ₹609
\]Answer: Amount = ₹10609, Compound Interest = ₹609
Q5: Calculate the amount and the compound interest on ₹15625 for 9 months at 16% per annum, compounded quarterly.
Step 1: Convert time to quarters:
\[
T = 9 \text{ months} = \frac{9}{12} = 0.75 \text{ years}
\]
Number of quarters \(n = 0.75 \times 4 = 3\)
Step 2: Use the compound interest formula for quarterly compounding:
\[
A = P \left(1 + \frac{R}{4 \times 100}\right)^n
\]
Where:
\(P = 15625, R = 16\%, n = 3\)
Step 3: Substitute the values:
\[
A = 15625 \times \left(1 + \frac{16}{400}\right)^3 = 15625 \times (1.04)^3 = 15625 \times 1.124864 = ₹17576
\]Step 4: Calculate Compound Interest:
\[
CI = A – P = 17576 – 15625 = ₹1951
\]Answer: Amount = ₹17576, Compound Interest = ₹1951
Q6: Calculate the amount and the compound interest on ₹2560000 for 1 year at 10% per annum, compounded quarterly.
Step 1: Use the compound interest formula for quarterly compounding:
\[
A = P \left(1 + \frac{R}{4 \times 100} \right)^{4T}
\]
Where:
P = ₹2560000, R = 10%, T = 1 year
Step 2: Substitute the values:
\[
A = 2560000 \times \left(1 + \frac{10}{400} \right)^4 = 2560000 \times \left(\frac{41}{40} \right)^4 \\
A = 2560000 \times \frac{41}{40} \times \frac{41}{40} \times \frac{41}{40} \times \frac{41}{40} \\
A = 2560000 \times \frac{2825761}{2560000}
\]
Therefore:
\[
A = ₹2825761
\]Step 3: Calculate Compound Interest:
\[
CI = A – P = 2825761 – 2560000 = ₹265761
\]Answer: Amount = ₹2825761, Compound Interest = ₹265761
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