Suppose $$ \$ 5,000 $$ is deposited into an account paying $$ 7.5 \% $$ interest, compounded annually. How much money is in the account after four years if no withdrawals or additional deposits are made?

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UpStudy AI · Accepted Answer

To find the amount of money in the account after four years with no withdrawals or additional deposits, we can use the formula for compound interest:

$$ A = P \left( 1 + \frac{r}{100} \right)^n $$

where:
- $$ A $$ is the amount of money in the account after $$ n $$ years,
- $$ P $$ is the principal amount (initial deposit),
- $$ r $$ is the annual interest rate,
- $$ n $$ is the number of years.

Given:
- Principal amount ($$ P $$) = $5,000
- Annual interest rate ($$ r $$) = 7.5%
- Number of years ($$ n $$) = 4

Substitute the given values into the formula:

$$ A = 5000 \left( 1 + \frac{7.5}{100} \right)^4 $$

Now, we can calculate the amount of money in the account after four years.
Calculate the value by following steps:
- step0: Calculate:
  $$5000\left(1+\frac{7.5}{100}\right)^{4}$$
- step1: Divide the terms:
  $$5000\left(1+\frac{3}{40}\right)^{4}$$
- step2: Add the numbers:
  $$5000\left(\frac{43}{40}\right)^{4}$$
- step3: Simplify:
  $$5000\times \frac{43^{4}}{40^{4}}$$
- step4: Rewrite the expression:
  $$40\times 125\times \frac{43^{4}}{40^{4}}$$
- step5: Reduce the numbers:
  $$125\times \frac{43^{4}}{40^{3}}$$
- step6: Rewrite the expression:
  $$125\times \frac{43^{4}}{5^{3}\times 8^{3}}$$
- step7: Rewrite the expression:
  $$5^{3}\times \frac{43^{4}}{5^{3}\times 8^{3}}$$
- step8: Reduce the numbers:
  $$1\times \frac{43^{4}}{8^{3}}$$
- step9: Multiply:
  $$\frac{43^{4}}{8^{3}}$$
- step10: Simplify:
  $$\frac{43^{4}}{512}$$
The amount of money in the account after four years is $6677.345703125.
After four years, the account will have $6,677.35.

Suppose is deposited into an account paying interest,
compounded annually.
How much money is in the account after four years if no withdrawals or
additional deposits are made?

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Suppose is deposited into an account paying interest, compounded annually. How much money is in the account after four years if no withdrawals or additional deposits are made?