Compound interest – Meaning, Formula and Solved Examples

Compound interest is a way of calculating interest where the interest earned over time is added back to the principal, so that the interest in the next period is calculated on the new total (which includes the previously earned interest). This means we earn interest on the original amount we invested or borrowed and also on the interest that has been added to it.

How Compound Interest Works

Imagine we put some money in a bank account that pays interest. With compound interest, after a certain period, the interest we earn is added to the original amount. Then, the next time interest is calculated, it’s based on this new, larger amount. This process continues, so over time, our money can grow faster than with simple interest.

The Formula for Compound Interest

The compound interest formula is:A=P(1+R/100)^T

Where:

• A = The amount of money accumulated after T years, including interest.
• P = The principal amount (the initial money).
• R = The annual interest rate (in percentage).
• T = The time the money is invested or borrowed for, in years.

To find just the interest earned, we subtract the principal from the total amount:

Compound Interest=A−P

Example 1:

Problem:
Emily invests $1000 in a savings account that offers 5% compound interest per year. How much will she have in her account after 3 years?

Solution:

1. Identify the values:
   • Principal (P) = $1000
   • Rate (R) = 5%
   • Time (T) = 3 years
2. Use the compound interest formula:
3. A=1000(1+5/100)³
4. Calculate:
   • First, calculate (1+5/100)=1.05.
   • Then, calculate (1.05)³≈1.157625.
   • Finally, multiply by the principal: 1000×1.157625=1157.63.
5. Answer: After 3 years, Emily will have $1157.63 in her account.
6. The interest earned is:
7. Compound Interest=1157.63−1000=157.63

Example 2:

Problem:
Liam saves $1500 in a bank account with a compound interest rate of 4% per year. How much will his savings grow to after 2 years?

Solution:

1. Identify the values:
   • Principal (P) = $1500
   • Rate (R) = 4%
   • Time (T) = 2 years
2. Use the compound interest formula:A=1500(1+4/100)²
3. Calculate:
   • First, calculate (1+4/100)=1.04.
   • Then, calculate (1.04)²≈1.0816.
   • Finally, multiply by the principal: 1500×1.0816=1622.40.
4. Answer: After 2 years, Liam will have $1622.40 in his account.The interest earned is:
5. Compound Interest=1622.40−1500=122.40

Why Compound Interest is Powerful

Compound interest is powerful because it helpsour money grow faster over time. The more frequently the interest is compounded, the more interest we earn. For example, interest compounded yearly earns less than interest compounded monthly or daily, because in the latter cases, the interest gets added to the principal more often.

Example 3:

Problem:
Olivia invests $2000 at an interest rate of 6% per year, compounded annually. How much will she have after 5 years?

Solution:

1. Identify the values:
   • Principal (P) = $2000
   • Rate (R) = 6%
   • Time (T) = 5 years
2. Use the compound interest formula: A=2000(1+6/100)⁵
3. Calculate:
   • First, calculate (1+6/100)=1.06.
   • Then, calculate 1.065≈1.338225.
   • Finally, multiply by the principal: 2000×1.338225=2676.45.
4. Answer: After 5 years, Olivia will have $2676.45 in her account.The interest earned is:
5. Compound Interest=2676.45−2000=676.45

Summary

• Simple Interest: Interest is calculated only on the principal amount.
• Compound Interest: Interest is calculated on the principal and the accumulated interest.

Compound interest can help our savings grow more quickly because we earn interest on the interest as well as on the principal. The longer we leave our money invested, the more we benefit from compound interest.