Compound Interest Solutions
7 Cards
(ii) Principal (P) = Rs 75,000, time (T) = 9 months, rate (R) = 10 % p.a.
Solution
Here, principal (P) = Rs 75,000
Rate of interest (R) = 10% p.a.
Time (T) = 9 months = \(\frac{9}{12} \text{ years} = \frac{3}{4} \text{ years}\)
\[
\begin{aligned}
\text{Now, C.I. quarterly} &= P \left[ \left( 1 + \frac{R}{400} \right)^{4T} - 1 \right] \\
&= \text{Rs } 75,000 \left[ \left( 1 + \frac{10}{400} \right)^{4 \times \frac{3}{4}} - 1 \right]
\end{aligned}
\]
\[ = \text{Rs } 5,766.80 \]
4. a) Mrs. Tharu deposited Rs 65,000 in Lumbini Bank at the rate of 10% p.a. for 1 year. How much amount will she get under the following cases?
(i) The bank pays the interest compounding yearly.
(ii) The bank pays the interest compounding half-yearly.
(iii) The bank pays the interest compounding quarterly.
Solution
Here, principal (P) = Rs 65,000
Rate of interest (R) = 10% p.a.
Time (T) = 1 year
\[
\begin{aligned}
\text{C.A. compounded yearly} &= P \left( 1 + \frac{R}{100} \right)^T \\
&= \text{Rs } 65,000 \left( 1 + \frac{10}{100} \right)^1
\end{aligned}
\]
\[ = \text{Rs } 71,500 \]
\[
\begin{aligned}
\text{C.A. compounded half-yearly} &= P \left( 1 + \frac{R}{200} \right)^{2T} \\
&= \text{Rs } 65,000 \left( 1 + \frac{10}{200} \right)^{2 \times 1}
\end{aligned}
\]
\[ = \text{Rs } 71,662.50 \]
\[
\begin{aligned}
\text{C.A. compounded quarterly} &= P \left( 1 + \frac{R}{400} \right)^{4T} \\
&= \text{Rs } 65,000 \left( 1 + \frac{10}{400} \right)^{4 \times 1}
\end{aligned}
\]
\[ = \text{Rs } 71,747.84 \]
b) Minakshi invested Rs 85,000 for 1 year in Goodwill Finance at the rate of 8% per annum.
(i) How much interest will she receive if it is compounded in each year?
(ii) How much interest will she receive if it is compounded in each 6 month?
(iii) How much interest will she receive if it is compounded in each 3 month?
Solution
Here, principal (P) = Rs 85,000
Rate of interest (R) = 8% p.a.
Time (T) = 1 year
\[
\begin{aligned}
\text{C.I. compounded yearly} &= \left[ \left( 1 + \frac{R}{100} \right)^T - 1 \right] \\
&= \text{Rs } 85,000 \left[ \left( 1 + \frac{8}{100} \right)^1 - 1 \right]
\end{aligned}
\]
\[ = \text{Rs } 6,800 \]
\[
\begin{aligned}
\text{C.I. compounded half-yearly} &= P \left[ \left( 1 + \frac{R}{200} \right)^{2T} - 1 \right] \\
&= \text{Rs } 85,000 \left[ \left( 1 + \frac{8}{200} \right)^{2 \times 1} - 1 \right]
\end{aligned}
\]
\[ = \text{Rs } 6,936 \]
\[
\begin{aligned}
\text{C.I. compounded quarterly} &= P \left[ \left( 1 + \frac{R}{400} \right)^{4T} - 1 \right] \\
&= \text{Rs } 85,000 \left[ \left( 1 + \frac{8}{400} \right)^{4 \times 1} - 1 \right]
\end{aligned}
\]
\[ = \text{Rs } 7,006.73 \]
5. a) Samriddhi has deposited Rs 40,000 in Lumbini Bank at 10% p.a. for 1 year.
(i) Write the formula to find the yearly compound amount on a sum P for T years at R% per annum.
(ii) Calculate the yearly compound amount.
(iii) Find the amount if the bank pays the interest compounding half-yearly.
(iv) Find the difference between the amounts compounded yearly and quarterly.
Solution
Here, principal (P) = Rs 40,000
Rate of interest (R) = 10% p.a.
Time (T) = 1 year
\[ \text{C.A. compounded yearly} = P \left( 1 + \frac{R}{100} \right)^T \]
\[
\begin{aligned}
\text{C.A. compounded yearly} &= P \left( 1 + \frac{R}{100} \right)^T \\
&= \text{Rs } 40,000 \left( 1 + \frac{10}{100} \right)^1
\end{aligned}
\]
\[ = \text{Rs } 44,000 \]
\[
\begin{aligned}
\text{C.A. compounded half-yearly} &= P \left( 1 + \frac{R}{200} \right)^{2T} \\
&= \text{Rs } 40,000 \left( 1 + \frac{10}{200} \right)^{2 \times 1}
\end{aligned}
\]
\[ = \text{Rs } 44,100 \]
\[
\begin{aligned}
\text{C.A. compounded quarterly} &= P \left( 1 + \frac{R}{400} \right)^{4T} \\
&= \text{Rs } 40,000 \left( 1 + \frac{10}{400} \right)^{4 \times 1} \\
&= \text{Rs } 44,152.52
\end{aligned}
\]
So, difference between C.A. yearly and quarterly = Rs 44,152.52 – Rs 44,000
\[ = \text{Rs } 152.52 \]
b) Bishwant took a loan of Rs 50,000 at the rate of 6% p.a. for 1 year from Agriculture Development Bank to promote his vegetable farming.
(i) What does P represent in the usual notation, C.A. = \( P \left( 1 + \frac{R}{100} \right)^T \)?
(ii) Find the annual compound amount that he has to pay to the bank.
(iii) How much amount should he pay to the bank according to the semi-annual compound interest system?
(iv) Find the difference between the amounts compounded semi-annually and quarterly.
Solution
Here, principal (P) = Rs 50,000
Rate of interest (R) = 6% p.a.
Time (T) = 1 year
P represents principal in the formula \( \text{C.A.} = P \left( 1 + \frac{R}{100} \right)^T \).
\[
\begin{aligned}
\text{C.A. compounded yearly} &= P \left( 1 + \frac{R}{100} \right)^T \\
&= \text{Rs } 50,000 \left( 1 + \frac{6}{100} \right)^1 \\
&= \text{Rs } 53,000
\end{aligned}
\]
Thus, he has to pay Rs 53,000 yearly compound amount to the bank.
\[
\begin{aligned}
\text{C.A. compounded half-yearly} &= P \left( 1 + \frac{R}{200} \right)^{2T} \\
&= \text{Rs } 50,000 \left( 1 + \frac{6}{200} \right)^{2 \times 1} \\
&= \text{Rs } 53,045
\end{aligned}
\]
So, he should pay Rs 53,045 to the bank according to the semi-annual compound interest system.
\[
\begin{aligned}
\text{C.A. compounded quarterly} &= P \left( 1 + \frac{R}{400} \right)^{4T} \\
&= \text{Rs } 50,000 \left( 1 + \frac{6}{400} \right)^{4 \times 1} \\
&= \text{Rs } 53,068.18
\end{aligned}
\]
So, the difference between the amounts compounded semi-annually and quarterly
\[
\begin{aligned}
&= \text{Rs } 53,068.18 - \text{Rs } 53,045
\end{aligned}
\]
\[ = \text{Rs } 23.18 \]
6. a) You are going to deposit Rs 1,00,000 for 2 years in a bank. You have got the following information about the rate of interest given by two banks.
| Bank- A | Yearly compound interest rate: 9% p.a. |
| Bank- B | Half-yearly compound interest rate: 8% p.a. |
(i) Which formula should be used to find the interest on a sum P for T years at R% per annum compounded annually?
(ii) Calculate the interest earned by saving the money in bank-A.
(iii) In which bank would you deposit the money and why? Write with reason.
Solution
Here, principal (P) = Rs 1,00,000
Time (T) = 2 years
\[ \text{C.I. compounded annually} = P \left[ \left( 1 + \frac{R}{100} \right)^T - 1 \right] \]
For bank-A
Rate of interest (R) = 9% p.a.
\[
\begin{aligned}
\text{C.I. compounded yearly} &= P \left[ \left( 1 + \frac{R}{100} \right)^T - 1 \right] \\
&= \text{Rs } 1,00,000 \left[ \left( 1 + \frac{9}{100} \right)^2 - 1 \right]
\end{aligned}
\]
\[ = \text{Rs } 18,810 \]
For bank-B
Rate of interest (R) = 8% p.a.
\[
\begin{aligned}
\text{C.I. compounded half-yearly} &= P \left[ \left( 1 + \frac{R}{200} \right)^{2T} - 1 \right] \\
&= \text{Rs } 1,00,000 \left[ \left( 1 + \frac{8}{200} \right)^{2 \times 2} - 1 \right] \\
&= \text{Rs } 16,985.86
\end{aligned}
\]
I would deposit the money in bank-A because I could earn Rs. 1,824.14 more interest from bank-A than from bank-B.
b) Your sister is going to deposit Rs 2,00,000 for 2 years in a bank. She has got the following information about the rate of interest given by two banks.
| Bank- P | Yearly compound interest rate: 5% p.a. |
| Bank- Q | Half-yearly compound interest rate: 4% p.a. |
(i) Write the formula which is used to find the interest compounded half-yearly.
(ii) Find the interest that she earned by saving the money in bank-P.
(iii) Which bank would you suggest her to deposit the money and why? Write with reason.
Solution
Here, principal (P) = Rs 2,00,000
Time (T) = 2 years
\[ \text{C.I. compounded half-yearly} = P \left[ \left( 1 + \frac{R}{200} \right)^{2T} - 1 \right] \]
For bank-P
Rate of interest (R) = 5% p.a.
\[
\begin{aligned}
\text{C.I. compounded yearly} &= P \left[ \left( 1 + \frac{R}{100} \right)^T - 1 \right] \\
&= \text{Rs } 2,00,000 \left[ \left( 1 + \frac{5}{100} \right)^2 - 1 \right]
\end{aligned}
\]
\[ = \text{Rs } 20,500 \]
For bank-Q
Rate of interest (R) = 4% p.a.
\[
\begin{aligned}
\text{C.I. compounded half-yearly} &= P \left[ \left( 1 + \frac{R}{200} \right)^{2T} - 1 \right] \\
&= \text{Rs } 2,00,000 \left[ \left( 1 + \frac{4}{200} \right)^{2 \times 2} - 1 \right] \\
&= \text{Rs } 16,486.43
\end{aligned}
\]
I would suggest my sister to deposit the money in bank-P because she could earn