Answer
Compound interest problem on future value is computed based on original principal and also on the accumulated interest. Amount of interest depends on the stated principal, the interest rate, which is given as a percent and varies from bank to bank, and the length of time for which the money is deposited.
Work Step by Step
Suppose a person deposited \[\$2000\]at a bank in the savings account offered, which is offering a rate of \[6%\]. It is required to compute the amount, which is represented by A, of money in the account post 3 years when interest is compounded once a year. The following formula can be used:
\[\begin{align}
& A=\ P{{\left( 1+r \right)}^{t}} \\
\end{align}\]
I represents the interest; P is the principal, which is \[\$2000\];r is the interest rate, which is \[6%\]; and t is the time for which the money is deposited, which is \[3\].The amount A is called the account’s future value that has been calculated as \[\$2382\].
Compound interest problem on present value is computed on the original principal as well as on the accumulated interest.The amount of interest depends on the principal, the interest rate, which is given as a percent and varies from bank to bank, and the length of time for which the money is deposited.
Example:
Suppose a person wants to accumulate\[\$20,000\] after 5 years by investing at a bank that is offering a savings account, at a rate of \[6%\]. It is required to compute the amount, which is represented by A, of money in the account post 3 years when interest is compounded once a year. The following formula can be used:
\[\begin{align}
& P\text{ }=\text{ }\frac{A}{{{\left( 1+\frac{r}{n} \right)}^{nt}}} \\
& =\ \frac{\$20,000}{{{\left(1+\frac{0.06}{12}\right)}^{12\times5}}}\\&=\frac{\$20,000}{{{\left(1+0.005\right)}^{60}}}\end{align}\]
Further simplifying:
\[\begin{align}
& P=\frac{\$20,000}{1.348}\\&=\\$14,827.45\end{align}\]
I represents the interest; A is the amount of future value that is required, which is \[\$20,000\];r is the interest rate, which is \[6%\]; and t is the time for which the money is deposited, which is \[60\].The amount P is called the account’s present value that has been calculated as \[\$14,827.45\].This is the amount that should be invested today so that the person earns \[\$20,000\] in 5 years.
The key difference between a compound interest problem involving future value and compound interest problem involving present value is that to compute the future value the amount is compounded and to compute the present value the amount is discounted.
