# Compound Interest

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In simple interest computation, interest is computed based on the invested capital over the period of time at stated interest rate.

If at given intervals over the whole term of the investment, the principal due is added to the original principal and subsequently grosses interest, the amount by which the original principal has increased by at the conclusion of the term of the investment is called compound interest. The total amount due which comprises the original principal and the compound interest is called the compound amount. Lastly, the time between successive conversions of interest into principal is called conversion period. This implies that under compound interest method, interest earns another interest.

To solve for compound interest, we follow the following steps:

Step 1: Find the future value of the investment or debt using the formula $F=P\left ( 1+\displaystyle \frac{j}{m} \right )^{mt}$, where F = amount
P = principal
J = the given rate
m = frequency of conversion for 1 year
t = term of investment.

Step 2. Find the interest (I) by using the formula: I = F – P.

Examples:

1. Joey invested P125,540.00 in bank that offers 3.5% compounded semiannually for the first 3 years and 5% compounded quarterly for the next 4 years. If Joey remains his account active in this bank (with no deposits and withdrawals made thereafter), how much is his total money at the end of 7 years? Find his income from this investment.

Solution:

Solving for amount after 3 years

P = P125, 540, j = 3.5%, t = 3, m = 4
$F=125,540 \left (1+\displaystyle \frac{0.035}{4}\right)^{4\times 3}=P139,374.9412$

Solve for the amount after 4 years using F = P139,374.9412 as P
P = P139,374.9412, j = 5%, t = 4, m = 4

$F=139,374.9412 \left (1+\displaystyle \frac{0.05}{4}\right)^{4\times 4}=P170,022.03$

2. Myrna discovered a savings account left to her by her rich foster father while cleaning his old house. When she was 6 years, he invested P50,000 in her name at 5.5% interest compounded annually. Now she is 22 years old. How much money is now available in her account? Find the compound interest.

Solution:

Find the amount in Myrna’s account
Given: P = P50,000, j = 5.5%, m = 1, t = 22 – 6 = 16

$F=P50,000 \left(1+0.055\right)^{16}=P117,763.14$
$I= P117,763.14-P50,000=P67,763.13$

Myrna now has P117,763.13 in her account. It means she has earned an interest of P67,763.13.