# Compound interest. Examples of the calculation of compound interest.

Definition.

**Compound interest**is the effect often encountered in economics and finance, when the interest income at the end of each period are added to the principal amount and the obtained value in the future becomes a starting point for the calculation of new interest.

## A formula for calculating annual compound interest

P = A(1 + | I | )^{n} |

100% |

A - is the original principal;

I - nominal annual interest rate in percentage terms;

n - number of compounding periods.

## The derivation of formula for calculating compound interest

- To calculate the value for one period we use the formula to calculate the number that is greater than the original number of a given percentage
P _{1}= A(1 +I ) 100% - for the second period
P _{2}= P_{1}(1 +I ) = A(1 + I ) ^{2}100% 100% . . . - for the n-th period
P _{n}= P_{n-1}(1 +I ) = A(1 + I ) ^{n}100% 100%

## Examples of problems with compound interest

Example 1.

Find the profit from $ 30,000 deposit for 3 years at 10% per annum, if at the end of each year interest is added to deposit.
**Solution.** Use the formula for calculating compound interest:

P = 30,000(1 + | 10% | )^{3} = 30,000 · 1.1^{3} = 39,930 |

100% |

**Answer:** profit is $ 9,930.

Example 2.

Knowing that the annual interest rate is 12%, find an equivalent monthly interest rate.
**Solution.**

If you put in the bank A dollars, then after a year we get:

P = A(1 + | 12% | ) |

100% |

If the interest accrued each month with an interest rate of x, then after a year (12 months)

P = A(1 + | x | )^{12} |

100% |

Equating these values we obtain the equation, the solution of which will determine the monthly interest rate

A(1 + | 12% | ) = A(1 + | x | )^{12} |

100% | 100% |

1.12 = (1 + | x | )^{12} |

100% |

^{12}√

1.12

- 1)·100% ≈ 0.9488792934583046%
**Answer:** monthly interest rate is equal 0.9488792934583046%.

**N.B. From this task you can see that the monthly interest rate is not equal to the annual interest rate divided by 12.**

Percent
Percent Definition. Main information
Decimal to Percent. Percent to Decimal
The method of solving problems with percentage

The most common types of problems with percent:
Find P percent of A
B is P percent of what?
B is what percent of A?
Find the number which is greater (less) than a given number by P percent
Find the number A if known number B which greater (less) than A by P percent

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