Question

NEED HELP ASAP ALSO I’M GIVING A TON OF POINTS AND THANKS! :)

asked Jun 5, 2023 23.3k views

2 Answers

JediKnight · Answer 1 · 2023-06-08T01:58:44+0000

Answer:

$\begin{array}c\cline{1-3} \vphantom{\frac12} \sf koooAccount& b^t& \sf Annual\;Percentage\;rate \\\cline{1-3} \vphantom{\frac12} \sf compounded\;annually& (1.03425)^t& 3.425\%\\\cline{1-3} \vphantom{\frac12} \sf compounded\;semiannually & \left(1.03454\right)^t &3.454\% \\\cline{1-3} \vphantom{\frac12} \sf compounded\;quarterly & \left(1.03469\right)^t& 3.469\%\\\cline{1-3} \vphantom{\frac12} \sf compounded\; daily& \left(1.03484\right)^t& 3.484\%\\\cline{1-3} \end{array}$

Explanation:

$\boxed{\begin{minipage}{8.5 cm}\underline{Compound Interest Formula}\\\\$ A=P\left(1+(r)/(n)\right)^(nt)$\\\\where:\\\\ \phantom{ww}$\bullet$ $A =$ final amount \\ \phantom{ww}$\bullet$ $P =$ principal amount \\ \phantom{ww}$\bullet$ $r =$ interest rate (in decimal form) \\ \phantom{ww}$\bullet$ $n =$ number of times interest is applied per year \\ \phantom{ww}$\bullet$ $t =$ time (in years) \\ \end{minipage}}$

From inspection of the compound interest formula:

$\implies b^t=\left[\left(1+(r)/(n)\right)^n\right]^t$

To find the annual percentage rate:

$\implies APR=(b-1)* 100$

Compounded annually

Given:

r = 3.425% = 0.03425
n = 1

Substitute the values into the formula for
b^t :

$\implies b^t=\left[\left(1+(0.03425)/(1)\right)^1\right]^t$

$\implies b^t=\left(1.03425\right)^t$

Therefore, the annual percentage rate is:

$\implies APR=(1-1.03425) * 100$

$\implies APR=0.03425 * 100$

$\implies APR=3.425\%$

Compounded semi-annually

Given:

r = 3.425% = 0.03425
n = 2

Substitute the values into the formula for
b^t :

$\implies b^t=\left[\left(1+(0.03425)/(2)\right)^2\right]^t$

$\implies b^t=\left(1.03454326...\right)^t$

$\implies b^t=\left(1.03454\right)^t$

Therefore, the annual percentage rate is:

$\implies APR=(1-1.03454) * 100$

$\implies APR=0.03454 * 100$

$\implies APR=3.454\%$

Compounded quarterly

Given:

r = 3.425% = 0.03425
n = 4

Substitute the values into the formula for
b^t :

$\implies b^t=\left[\left(1+(0.03425)/(4)\right)^4\right]^t$

$\implies b^t=\left(1.03469241...\right)^t$

$\implies b^t=\left(1.03469\right)^t$

Therefore, the annual percentage rate is:

$\implies APR=(1-1.03469) * 100$

$\implies APR=0.03469 * 100$

$\implies APR=3.469\%$

Compounded daily

Given:

r = 3.425% = 0.03425
n = 365

Substitute the values into the formula for
b^t :

$\implies b^t=\left[\left(1+(0.03425)/(365)\right)^(365)\right]^t$

$\implies b^t=\left(1.03484162...\right)^t$

$\implies b^t=\left(1.03484\right)^t$

Therefore, the annual percentage rate is:

$\implies APR=(1-1.03484) * 100$

$\implies APR=0.03484 * 100$

$\implies APR=3.484\%$

Note: There is an error in the given table: the given "compounded daily" rate is incorrect.

Amir Beygi · Answer 2 · 2023-06-10T14:43:10+0000

Answer:

Compounded Semiannually:
(1.03454)^t and APR = 3.454%

Compounded Quarterly:
(1.03469)^t and APR = 3.469%

Explanation:

Compound Interest Formula:

The Compound Interest Formula is as follows:
A = P(1 + (r)/(n))^(nt) , which can also be written as:
A = P((1 + (r)/(n))^n)^t , using properties of exponents. This is crucial since we want one value inside the parenthesis to calculate the APR (Annual Percentage Rate). Since we want the APR, the only thing we really care about is the
, but this is going to be our APR + 1.

In the question, it's formatted as:
b^t , and this inside part is going to be our b, so:
b = (1 + (r)/(n))^n , and to calculate the APR we simply subtract 1.

Calculating APR:

Compounded semiannually means the interest is compounded twice a year, so n = 2. We are also given the interest to be 3.425% which in decimal form is 0.03425

Plugging this into the formula stated above we get:
$b = (1 + (0.03425)/(2))^2 \approx 1.03454$