Marta takes a loan of $$ \$ 12,450 $$ at an interest rate of $$ 4.56 \% $$, compounded quarterly. Assuming no payments are made, find the amount owed after 5 years. Round to the nearest dollar. Type your numerical answer below.

Question

UpStudy AI · Accepted Answer

To find the amount owed after 5 years on a loan with compound interest, we can use the formula for compound interest:

$$
A = P \left(1 + \frac{r}{n}\right)^{nt}
$$

Where:
- $$ A $$ is the amount of money accumulated after n years, including interest.
- $$ P $$ is the principal amount (the initial loan amount).
- $$ r $$ is the annual interest rate (decimal).
- $$ n $$ is the number of times that interest is compounded per year.
- $$ t $$ is the number of years the money is borrowed for.

Given:
- $$ P = 12450 $$
- $$ r = 4.56\% = 0.0456 $$
- $$ n = 4 $$ (since the interest is compounded quarterly)
- $$ t = 5 $$

Now, we can substitute these values into the formula:

$$
A = 12450 \left(1 + \frac{0.0456}{4}\right)^{4 \times 5}
$$

Let's calculate this step by step.

1. Calculate $$ \frac{r}{n} $$:
   $$
   \frac{0.0456}{4}
   $$

2. Calculate $$ nt $$:
   $$
   4 \times 5
   $$

3. Substitute these values into the formula and calculate $$ A $$.

Now, I will perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$4\times 5$$
- step1: Multiply the numbers:
  $$20$$
Calculate or simplify the expression $$ 0.0456/4 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.0456}{4}$$
- step1: Convert the expressions:
  $$\frac{\frac{57}{1250}}{4}$$
- step2: Multiply by the reciprocal:
  $$\frac{57}{1250}\times \frac{1}{4}$$
- step3: Multiply the fractions:
  $$\frac{57}{1250\times 4}$$
- step4: Multiply:
  $$\frac{57}{5000}$$
Calculate or simplify the expression $$ 12450*(1+0.0114)^(20) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$12450\left(1+0.0114\right)^{20}$$
- step1: Add the numbers:
  $$12450\times 1.0114^{20}$$
- step2: Convert the expressions:
  $$12450\left(\frac{5057}{5000}\right)^{20}$$
- step3: Simplify:
  $$12450\times \frac{5057^{20}}{5000^{20}}$$
- step4: Rewrite the expression:
  $$50\times 249\times \frac{5057^{20}}{5000^{20}}$$
- step5: Rewrite the expression:
  $$50\times 249\times \frac{5057^{20}}{2500^{20}\times 2^{20}}$$
- step6: Rewrite the expression:
  $$50\times 249\times \frac{5057^{20}}{50^{40}\times 2^{20}}$$
- step7: Reduce the numbers:
  $$249\times \frac{5057^{20}}{50^{39}\times 2^{20}}$$
- step8: Multiply the fractions:
  $$\frac{249\times 5057^{20}}{50^{39}\times 2^{20}}$$
Now that we have the necessary calculations:

1. The quarterly interest rate is $$ \frac{0.0456}{4} = 0.0114 $$.
2. The total number of compounding periods over 5 years is $$ 4 \times 5 = 20 $$.

Now we can substitute these values into the compound interest formula:

$$
A = 12450 \left(1 + 0.0114\right)^{20}
$$

Calculating this gives us:

$$
A \approx 15618.10
$$

Rounding to the nearest dollar, the amount owed after 5 years is:

$$
\boxed{15618}
$$
After 5 years, Marta owes \$15,618.

Marta takes a loan of \( \$ 12,450 \) at an interest rate of \( 4.56 \% \), compounded quarterly. Assuming no payments are made, find the amount owed after 5 years. Round to the nearest dollar. Type your numerical answer below.

Upstudy AI Solution

Answer

Solution

The Deep Dive

Related Questions

Latest Economics Questions