Answer :
To determine which equation correctly models the bank's loan offer, we start with the compound interest formula:
[tex]$$
A = P\left(1+\frac{r}{n}\right)^{n t},
$$[/tex]
where
- [tex]$P$[/tex] is the principal (loan amount),
- [tex]$r$[/tex] is the annual interest rate (in decimal),
- [tex]$n$[/tex] is the number of compounding periods per year, and
- [tex]$t$[/tex] is the number of years.
In this problem:
- [tex]$P = 20000$[/tex],
- [tex]$r = 0.0625$[/tex] (which is [tex]$6.25\%$[/tex]), and
- [tex]$n = 4$[/tex] (since the interest is compounded quarterly).
First, we calculate the interest rate per quarter:
[tex]$$
\frac{r}{n} = \frac{0.0625}{4} = 0.015625.
$$[/tex]
This means the amount in one quarter is multiplied by:
[tex]$$
1 + \frac{r}{n} = 1 + 0.015625 = 1.015625.
$$[/tex]
Since there are [tex]$4$[/tex] compounding periods in a year, after [tex]$t$[/tex] years, the exponent becomes [tex]$4t$[/tex]. Thus, the full expression for the amount [tex]$A$[/tex] after [tex]$t$[/tex] years is:
[tex]$$
A = 20000\left(1.015625\right)^{4t}.
$$[/tex]
This expression is equivalent to the equation:
[tex]$$
A = 20000\left(1+\frac{0.0625}{4}\right)^{4t}.
$$[/tex]
Upon reviewing the provided options, we can see that this matches the fourth equation:
[tex]$$
A = 20000\left(1+\frac{0.0625}{4}\right)^{4t}.
$$[/tex]
Therefore, the correct answer is the fourth option.
[tex]$$
A = P\left(1+\frac{r}{n}\right)^{n t},
$$[/tex]
where
- [tex]$P$[/tex] is the principal (loan amount),
- [tex]$r$[/tex] is the annual interest rate (in decimal),
- [tex]$n$[/tex] is the number of compounding periods per year, and
- [tex]$t$[/tex] is the number of years.
In this problem:
- [tex]$P = 20000$[/tex],
- [tex]$r = 0.0625$[/tex] (which is [tex]$6.25\%$[/tex]), and
- [tex]$n = 4$[/tex] (since the interest is compounded quarterly).
First, we calculate the interest rate per quarter:
[tex]$$
\frac{r}{n} = \frac{0.0625}{4} = 0.015625.
$$[/tex]
This means the amount in one quarter is multiplied by:
[tex]$$
1 + \frac{r}{n} = 1 + 0.015625 = 1.015625.
$$[/tex]
Since there are [tex]$4$[/tex] compounding periods in a year, after [tex]$t$[/tex] years, the exponent becomes [tex]$4t$[/tex]. Thus, the full expression for the amount [tex]$A$[/tex] after [tex]$t$[/tex] years is:
[tex]$$
A = 20000\left(1.015625\right)^{4t}.
$$[/tex]
This expression is equivalent to the equation:
[tex]$$
A = 20000\left(1+\frac{0.0625}{4}\right)^{4t}.
$$[/tex]
Upon reviewing the provided options, we can see that this matches the fourth equation:
[tex]$$
A = 20000\left(1+\frac{0.0625}{4}\right)^{4t}.
$$[/tex]
Therefore, the correct answer is the fourth option.
