Compound Interest

Key Formulas

1. \(I = \frac{P \times T \times R}{100}\)
2. \(P = \frac{I \times 100}{T \times R}\)
3. \(R = \frac{I \times 100}{P \times T}\)
4. \(T = \frac{I \times 100}{P \times R}\)
5. \(A = P + I\)
6. \(P = \frac{A \times 100}{100 + TR}\)

Where:
\(A\) = Amount, \(P\) = Principal, \(r\) = Rate, \(t\) = Time

Key Formulas

1. \(A = P\left(1 + \frac{r}{100}\right)^t\) (Annual compound interest)
   
2. \(A = P\left(1 + \frac{r}{200}\right)^{2t}\) (Semi-annual compound interest)
   
3. \(A = P\left(1 + \frac{r}{400}\right)^{4t}\) (Quarterly compound interest)
   

Where:
\(A\) = Amount, \(P\) = Principal, \(r\) = Rate, \(t\) = Time

Review

1. Compound Interest:
   Compound Interest is the interest which is calculated by adding the interest to the principal and calculating the interest at the end of certain time. Compound interest is also known as interest on interest.
2. Compound amount:
   The sum of compound interest and principal is called the compound amount.
3. Yearly Compound Interest:
   Yearly compound interest is the interest which is calculated by adding the interest to the principal and calculating the interest at the end of every year.
4. Half-yearly Compound Interest:
   Half-yearly compound interest is the interest which is calculated by adding the interest to the principal and calculating the interest at the end of every six months.
5. Quarterly Compound Interest:
   Quarterly compound interest is the interest which is calculated by adding the interest to the principal and calculating the interest at the end of every 3 months.

Comparison between Simple and Compound Interest

Year	Details	Year	Details
Present	P = 100 R = 10% p.a	Present	P = 100 R= 10%
First Year	\(P = 100\) \(I = 10\)	First Year	\(P = 100\) \(I = 10\)
Second Year	\(P = 100\) \(I = 10\)	Second Year	\(P = (100 + 10)=110\) \(I = 11\)
Third Year	\(P = 100\) \(I = 10\)	Third Year	\(P = 110+11=121\) \(I = 12.1\)

Comparison between Simple and Compound Amount

Time	Descriptions	Simple Interest	Compound Interest
First Year	Principal Interest (10%) Year-end amount	\(\text{Rs } 100\) \(\text{Rs } 10\) \(\text{Rs } 110\)	\(\text{Rs } 100\) \(\text{Rs } 10\) \(\text{Rs } 110\)
Second Year	Principal Interest (10%) Year-end amount	\(\text{Rs } 100\) \(\text{Rs } 10\) \(\text{Rs } 110\)	\(\text{Rs } 110\) \(\text{Rs } 11\) \(\text{Rs } 121\)
Third Year	Principal Interest (10%) Year-end amount	\(\text{Rs } 100\) \(\text{Rs } 10\) \(\text{Rs } 130\)	\(\text{Rs } 121\) \(\text{Rs } 12.1\) \(\text{Rs } 133.10\)

When the interest is compounded annually

The value of T must be in whole number.
Note: The annual compound interest for 1 year or less will be same as the simple interest.

When the interest is compounded semi-annually

2T must be a whole number.
Note: The annual compound interest for 6 months or less will be the same as the simple interest.

When the interest is compounded quarterly

4T must be in a whole number.

When the rate of interest are different for consecutive years

If \(R_1\) = Rate of interest of 1st year, \(R_2\) = Rate of interest of 2nd year, \(R_3\) = Rate of interest of 3rd year, then

1. Compound Amount: \(CA = P\left(1 + \frac{R_1}{100}\right)\left(1 + \frac{R_2}{100}\right)\left(1 + \frac{R_3}{100}\right)\)
2. Compound Interest: \(CI = P\left[\left(1 + \frac{R_1}{100}\right)\left(1 + \frac{R_2}{100}\right)\left(1 + \frac{R_3}{100}\right) - 1\right]\)

If compound amount T years is CA₁ and the compound amount for (T+1) years is CA₂

If compound amount for 1 years is \(CA_1\) and the compound amount for (T + 1) years is \(CA_2\)

1. Rate of Compound = \(\left(\frac{CA_2}{CA_1}\right) \times 100\%\)
2. principal = \(\frac{CA_1}{\left(1 + \frac{R}{100}\right)^T}\)

If compound interest for 1 year is CI₁ and the compound interest for 2 years is CI₂

If compound interest for 1 year is \(CI_1\) and compound interest for 2 years is \(CI_2\).

1. Rate of interest= \(\left(\frac{CI_2}{CI_1} - 2\right) \times 100\%\)
2. Principal = \(\frac{CI_1 \times 100}{R}\)

Difference between compound interest and simple interest

Type of Compound	Compound Amount	Compound Interest
Annually	\(CA = P\left(1 + \frac{R}{100}\right)^T\)	\(CI = P\left[\left(1 + \frac{R}{100}\right)^T - 1\right]\)
Semi-anually	\(CA = P\left(1 + \frac{R}{200}\right)^{2T}\)	\(CI = P\left[\left(1 + \frac{R}{200}\right)^{2T} - 1\right]\)
Quarterly	\(CA = P\left(1 + \frac{R}{400}\right)^{4T}\)	\(CI = P\left[\left(1 + \frac{R}{400}\right)^{4T} - 1\right]\)
Different rates of interest	\(CA = P\left(1 + \frac{R_1}{100}\right)\left(1 + \frac{R_2}{100}\right)\left(1 + \frac{R_3}{100}\right)\)	\(CI = P\left[\left(1 + \frac{R_1}{100}\right)\left(1 + \frac{R_2}{100}\right)\left(1 + \frac{R_3}{100}\right) - 1\right]\)

Rate of Interest

According to the system of annual compound interest, if the compound amount of 'p' years is Rs \(CA_p\) and that of 'q' years is Rs \(CA_q\), the formulae of finding the rate of interest is:

Conditions	Rate of Interest
If the difference between the years, q − p = 1	\(R = \left(\frac{CA_q}{CA_p} - 1\right)\) % p.a.
If the difference between the years, q − p = 2	\(R = \left(\sqrt{\frac{CA_q}{CA_p}} - 1\right)\) % p.a.
If the difference between the years, q − p = 3	\(R = \left(\sqrt[3]{\frac{CA_q}{CA_p}} - 1\right) \) % p.a.

You want to deposit Rs 60,000 in a bank for 2 years. The information in the notice board of the bank is given below.

Fixed deposit account (P)
Rate of half yearly compound interest 10% p.a.

Fixed deposit account (Q)
Rate of yearly compound interest 12% p.a.

The notice board shows that the accounts are going to be effective from 2079/01/01. Grab the opportunity of fixed deposit in time.

Now

1. Write the formula to calculate annual compound interest.
2. How much interest will be collected in account (P) after 2 years?
3. How much interest will be collected in account (Q) after 2 years?
4. After knowing the interest obtained from the account 'P' and interest from account 'Q', which account provides the maximum interest? And why?

A person deposited Rs 5,00,000 in a commercial bank for 2 years to get the half-yearly compound interest at the rate of 10% per annum. 5% tax on the interest will be levied. But right after a year, the bank has changed the policy and decided to accomplish the interest quarterly at the same rate of interest.

1. Find the interest of the first year by deducting the tax.
2. What would be the interest of the second year be after deducting the tax?
3. What is the difference between the interest of the first year and second year after deducting the tax?
4. After deducting the tax, by what percentage the interest of the second year differ from the interest of the first year?

Exercise

1. Definitions or Formula Recall
   1. Compound interest
   2. Compound amount
   3. Yearly compound interest
   4. Quarterly compound interest
   5. Semi-annual compound interest
2. According to annual compound interest, if A is the amount, R is the rate percent per annum and T is the time, what will be the principal (P)?
3. The annual compound interest on a sum P in T years at R% per annum is CI. Write down the relation among P, T, R and CI.
4. The semi-annual compound amount on a sum P in T years at R% per annum is CA. Write down the relation among P, T, R and CA.
5. In usual notations, what does \( P\left(1 + \frac{R}{200}\right)^{2T} \) stand for?
6. The quarterly compound amount on a sum P in T years at R% per annum is CA. Write down the relation among P, T, R and CA.
7. In usual notations, what does \( P\left(1 + \frac{R}{400}\right)^{4T} \) stand for?
8. The semi-annual compound interest on a sum P in T years at R% per annum is CI. Write down the relation among P, T, R and CI.
9. The quarterly compound interest on a sum P in T years at R% p.a. is CI. Write down the relation among P, T, R and CI.
10. According to annual compound interest, CA is the amount, R is the rate percent per annum, T years and M months is the time. What will be the principal (P)?
11. The annual compound interest on a sum P in T years and M months at R% p.a. is CI. Write down the relation among P, T, R and CI.
12. If CA is the compound amount of a sum P at different rates R₁%, R₂% and R₃% in the first, second and third years respectively, write the formula to calculate CA.
13. If CI is the compound interest of a sum P at different rates R₁%, R₂% and R₃% in the first, second and third years respectively, write the formula to calculate CI.

Exercise

1. The compound amount of Rs 5000 for three years is Rs 8500. Find the compound interest.
2. If the principal and compound interest are Rs 8000 and Rs 3000 respectively, find the compound amount.
3. If the compound interest of Rs 100 for 1 year is Rs 15, find the annual rate of interest.
4. What sum of money has the compound interest Rs 5000 and compound amount Rs 8000 in 3 years?
5. If the principal and compound interest are equal, how many times of principal is the compound amount?
6. If the principal is 4 times the compound interest, how many times of compound interest is the compound amount?
7. If the principal is one third the compound interest, how many times of compound interest is the compound amount?
8. If the principal is five times the compound interest and the compound amount is Rs 6000, find the compound interest.

Exercise

1. Using formula, find the annual compound amount:
   1. Rs 10,000 at 10% for 2 years
   2. Rs 8,000 at 8% for 3 years
   3. Rs 12,000 at 5% for 3 years
   4. Rs 15,000 at 4% for 2 years
2. Using formula, find the annual compound interest:
   1. Rs 3,000 at 10% for 2 years
   2. 4,000 at 10% for 3 years
   3. Rs 5,000 at 10% for 3 years
   4. Rs 6,000 at 10% for 2 years
3. Using formula, find the semi-annual compound amount:
   1. Rs 10,000 at 10% for 2 years
   2. Rs 40,000 at 12% for 2 years
   3. Rs 50,000 at 15% for 1½ years
   4. Rs 20,000 at 8% for 1½ years
4. Using formula, find the semi-annual compound interest:
   1. Rs 80,000 at 10% for 2 years
   2. Rs 10,000 at 14% for 1 year
   3. Rs 75,000 at 15% for 1½ years
   4. Rs 90,000 at 18% for 1½ years
5. Using formula, find the quarterly compound amount.
   1. Rs 25,000 at 10% for 1 year
   2. Rs 30,000 at 10% for 9 months
   3. Rs 40,000 at 8% for 6 months
   4. Rs 60,000 at 12% for 12 months
6. Using formula, find the quarterly compound interest.
   1. Rs 16,000 at 10% for 6 months
   2. Rs 24,000 at 10% for 9 months
   3. Rs 40,000

Exercise

1. Pemba borrowed Rs 48,000 from a cooperative at 10% rate of interest p.a. How much compound amount should be paid after 3 years?
2. Dorje borrowed Rs 2,500 from a cooperative at 8% rate of interest p.a. How much compound amount should be paid after 2 years?
3. Rupa deposited Rs 50,000 in a bank at the rate of 8% p.a. If the bank provides semi-annual compound interest, find the compound amount after 2 years.
4. A bank provides semi-annual compound interest in which Manju deposited Rs 1,00,000 at the rate of 10% p.a. Find the compound amount she gets after 2 years.
5. Semi-annual compound interest at 8% p.a. is available in Nepal Bank Limited for money deposit. If the bank uses semi-annual compound interest method to calculate the interest, calculate the interest for Rs 20,000 for 1 year.
6. Manisha deposited Rs 50,000 in a bank at the rate of compound interest 8% p.a. If the bank provides half-yearly compound interest, find the amount and compound interest she receives after 2 years.
7. Ramila deposited Rs 50,000 in a bank at the rate of 8% p.a. If the bank provides quarterly compound interest, find the compound amount she gets after 2 years.
8. A bank provides quarterly compound interest in which Sarmila deposited Rs 1,00,000 at the rate of 10% p.a. Find the compound amount she gets after 2 years.
9. A bank provides quarterly compound interest. If Sunil deposited Rs 50,00,000 for 1 year at the rate of 12% p.a., find the amount and compound interest.
10. Sunita deposited Rs 50,000 in a bank at the rate of compound interest 8% p.a. If the bank provides quarterly compound interest, find the amount and compound interest she receives after 1 year.
11. Find the amount compounded annually on Rs 25,000 for 2 years if the rates of interest for two years are 10% and 12% respectively.
12. Find the amount compounded annually on Rs 40,000 for 2 years if the rates of interest for two years are 10% and 15% respectively.

Exercise

1. Bijaya deposited Rs 10,000 in a bank. If the bank gives 8% p.a. compound interest, find the compound amount after 1 year 6 months.
2. Sarthak deposited Rs 20,000 in a bank. If the bank gives 8% p.a. compound interest, find the compound amount after 2 years 6 months.
3. A person deposited Rs 50,000 in a bank for 20 months at the rate of 10% p.a. compound interest compounded semi-annually. Find the amount and interest.
4. A person deposited Rs 60,000 in a bank for 15 months at the rate of 8% p.a. compound interest compounded semi-annually. Find the amount and interest.
5. A person deposited Rs 40,000 in a bank for 9 months at the rate of 10% p.a. compound interest compounded quarterly. Find the amount and interest.
6. A person deposited Rs 55,000 in a bank for 10 months at the rate of 10% p.a. compound interest compounded quarterly. Find the amount and interest.

Exercise

1. Jharana got Rs 1881 interest of a certain sum for 2 years at 9% compounded yearly. What was the sum deposited?
2. Rina got Rs 2500 interest of a certain sum for 2 years at 25% compounded yearly. What was the sum deposited?
3. At what percent rate of compound interest p.a. will the compound interest on Rs 34,300 be Rs 16,900 in 3 years?
4. At what rate percent per annum, the compound amount on Rs 576 amounts to Rs 625 in 2 years?
5. At what rate percent per annum, the semi-annual compound interest on Rs 40,000 will be Rs 3264 in one year?
6. At what rate percent per annum, the semi-annual compound interest on Rs 50,000 will be Rs 5125 in one year?
7. At what rate percent per annum, the quarterly compound interest on Rs 10,000 will be Rs 506.25 in 6 months?
8. At what rate percent per annum, the quarterly compound interest on Rs 20,000 will be Rs 808 in 6 months?

Exercise

1. In how many years will Rs 8000 amount to Rs 13,824 at 20% per annum interest compounded annually?
2. In how many years will Rs 7500 amount to Rs 9408 at 12% per annum interest compounded annually?
3. In how many years will compound interest payable yearly on Rs 10,000 at 10% per year be Rs 3310?
4. According to the yearly compound interest, in what time will the compound interest on Rs 4,00,000 at the rate of interest 6.5% per annum be Rs 53,690?
5. According to the half-yearly compound interest, in what time will the semi-annual compound interest on Rs 80,000 at the rate of interest 8% p.a. be Rs 6528?
6. According to the half-yearly compound interest, in what time will the semi-annual compound interest on Rs 50,000 at the rate of interest 10% p.a. be Rs 5125?
7. According to the quarterly compound interest, in what time will the quarterly compound interest on Rs 40,000 at the rate of interest 10% p.a. be Rs 2025?
8. According to the quarterly compound interest, in what time will the quarterly compound interest on Rs 10,000 at the rate of interest 8% p.a. be Rs 404?

Exercise

1. Rupak borrowed Rs 14,000 with Bidur at 12% annual rate of interest.
   1. What is the annual compound interest for 3 years?
   2. Find the semi-annual compound interest for 2 years.
   3. Calculate the quarterly compound interest for 1 year.
   4. By how much is the quarterly compound interest more than the annual compound interest of 1 year?
2. Rupa borrowed Rs 10,000 with Bindhya at 10% annual rate of interest.
   1. What is the annual compound interest for 3 years?
   2. Find the semi-annual compound interest for 2 years.
   3. Calculate the quarterly compound interest for 1 year.
   4. By how much is the quarterly compound interest more than the annual compound interest of 1 year?

Exercise

1. Rs 64,000 is invested for 3 years at 4% per annum rate of interest for first year, 5% per annum rate of interest for second year and 6% per annum rate of interest for third year.
   1. Find the annual compound amount at the end of second year.
   2. What is the annual compound amount at the end of third year?
   3. Calculate the total interest of 3 years.
   4. What will be the difference in interest for 3 years if the single interest rate 5% is used instead of different rates?
2. Rs 80,000 is invested for 3 years at 8% per annum rate of interest for first year, 10% per annum rate of interest for second year and 12% per annum rate of interest for third year.
   1. Find the annual compound amount at the end of second year.
   2. What is the annual compound amount at the end of third year?
   3. Calculate the total interest of 3 years.
   4. What will be the difference in interest for 3 years if the single interest rate 10% is used instead of different rates?

Exercise

1. Santosh borrowed Rs 130,000 from Suresh at the rate of 21% per annum.
   1. Find the simple interest at the end of 2 years.
   2. What is the annual compound interest at the end of 2 years?
   3. By how much is the semi-annual compound interest at the end of 2 years more than that of simple interest?
2. Sarmila borrowed Rs 150,000 from Ramila at the rate of 21% per annum.
   1. Find the simple interest at the end of 2 years.
   2. What is the annual compound interest at the end of 2 years?
   3. By how much is the semi-annual compound interest at the end of 2 years more than that of simple interest?
3. Hari borrowed Rs 130,000 from Krishna at the rate of 21% per annum. At the end of 1 year and 6 months,
   1. How much simple interest will he have to pay?
   2. How much compound interest will he have to pay?
   3. What should be the new rate of simple interest to make simple interest and compound interest equal?
4. Sita borrowed Rs 170,000 from Radha at the rate of 21% p.a. At the end of 1 year and 6 months,
   1. How much simple interest will she have to pay?
   2. How much compound interest will she have to pay?
   3. What should be the new rate of simple interest to make simple interest and compound interest equal?
5. Hari Bahadur borrowed Rs 40,000 from Madan Bahadur at the rate of 10% p.a. At the end of 1 year and 6 months,
   1. What will be the annual compound interest?
   2. How much will be the semi-annual compound interest?
   3. If the annual compound interest is used instead of semi-annual compound interest then who will receive how much more benefit from interest?
6. Bir Bahadur borrowed Rs 1,60,000 from Bal Bahadur at the rate of 8% p.a. At the end of 1 year and 6 months,
   1. What will be the annual compound interest?
   2. How much will be the semi-annual compound interest?
   3. If the annual compound interest is used instead of semi-annual compound interest then who will receive how much more benefit from interest?
7. Manisha borrowed Rs 2,40,000 from Nisha at the rate of 10% p.a. for 9 months.
   1. What will be the semi-annual compound interest?
   2. How much will be the quarterly compound interest?
   3. If the quarterly compound interest is used instead of semi-annual compound interest, then who will receive how much more benefit from interest?
8. Ranjita borrowed Rs 3,20,000 from Sanchita at the rate of 8% p.a. for 9 months.
   1. What will be the semi-annual compound interest?
   2. How much will be the quarterly compound interest?
   3. If the quarterly compound interest is used instead of semi-annual compound interest, then who will receive how much more benefit from interest?

Exercise

1. A farmer visits different local banks to deposit Rs 4,00,000 for 2 years. The interest rates provided by the banks are given below.

   Bank A	Bank B	Bank C
   Annual compound interest 10% p.a.	Semi-annual compound interest 8% p.a.	Simple interest 11% p.a.

   1. Find the interest that Bank A provides.
   2. What is the interest that Bank B provides?
   3. In which bank do you suggest the farmer to deposit his money and why?
2. A teacher visits different local banks to deposit Rs 5,00,000 for 2 years. The interest rates provided by the banks are given below.

   Bank A	Bank B	Bank C
   Annual compound interest 20% p.a.	Semi-annual compound interest 18% p.a.	Simple interest 22% p.a.

   1. Find the interest that Bank A provides.
   2. What is the interest that Bank B provides?
   3. In which bank do you suggest the teacher to deposit his money and why?
3. You want to deposit Rs 1,20,000 in a bank for 2 years. The information shown on the notice board of the bank is given below.

   It is notified that two types of fixed deposit accounts are going to be effective from 2080/01/01. Grab the opportunity of fixed deposit on time.
   Fixed Deposit Account (A)	Fixed Deposit Account (B)
   Rate of half-yearly compound interest 10% p.a.	Rate of yearly compound interest 12% p.a.

   1. Write the formula to calculate annual compound interest.
   2. How much interest will be collected in account (A) after 2 years?
   3. How much interest will be collected in account (B) after 2 years?
   4. After knowing the interest rates of both options, by which option will you deposit the money? And why?
4. You went to deposit Rs 80,000 in a bank for 2 years. The information in the notice board of the bank is given below.

   It is notified that two types of fixed deposit account are going to be effective from 2081/01/01. Grab the opportunity of fixed deposit on time.
   Fixed deposit account (M)	Fixed deposit account (N)
   Rate of half yearly compound interest 10% p.a.	Rate of yearly compound interest 12% p.a.

   1. Write the formula to calculate annual compound interest.
   2. How much interest will be collected in account (M) after 2 years?
   3. How much interest will be collected in account (N) after 2 years?
   4. After knowing the interest rates of both options, by which option will you deposit the money? And why?