Austin invested $$ \$ 13,000 $$ in an account paying an interest rate of $$ 6 \frac{3}{4} \% $$ compounded monthly. Easton invested $$ \$ 13,000 $$ in an account paying an interest rate of $$ 7 \frac{1}{4} \% $$ compounded daily. After 19 years, how much more money would Easton have in his account than Austin, to the nearest dollar?

Question

UpStudy AI · Accepted Answer

To solve this problem, we will calculate the future value of both Austin's and Easton's investments after 19 years, and then find the difference between the two amounts.

### Step 1: Calculate Austin's Future Value

Austin's investment details:
- Principal ($$ P_A $$): $$ 13000 $$
- Annual interest rate ($$ r_A $$): $$ 6 \frac{3}{4} \% = 6.75\% = 0.0675 $$
- Compounding frequency ($$ n_A $$): monthly, so $$ n_A = 12 $$
- Time ($$ t $$): 19 years

The formula for future value with compound interest is:
$$
FV = P \left(1 + \frac{r}{n}\right)^{nt}
$$

Substituting Austin's values:
$$
FV_A = 13000 \left(1 + \frac{0.0675}{12}\right)^{12 \times 19}
$$

### Step 2: Calculate Easton's Future Value

Easton's investment details:
- Principal ($$ P_E $$): $$ 13000 $$
- Annual interest rate ($$ r_E $$): $$ 7 \frac{1}{4} \% = 7.25\% = 0.0725 $$
- Compounding frequency ($$ n_E $$): daily, so $$ n_E = 365 $$
- Time ($$ t $$): 19 years

Using the same future value formula:
$$
FV_E = 13000 \left(1 + \frac{0.0725}{365}\right)^{365 \times 19}
$$

### Step 3: Calculate the Future Values

Now, we will calculate both future values using the formulas derived above.

Let's perform the calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$13000\left(1+\frac{0.0725}{365}\right)^{365\times 19}$$
- step1: Divide the terms:
  $$13000\left(1+\frac{29}{146000}\right)^{365\times 19}$$
- step2: Add the numbers:
  $$13000\left(\frac{146029}{146000}\right)^{365\times 19}$$
- step3: Multiply the numbers:
  $$13000\left(\frac{146029}{146000}\right)^{6935}$$
- step4: Simplify:
  $$13000\times \frac{146029^{6935}}{146000^{6935}}$$
- step5: Rewrite the expression:
  $$1000\times 13\times \frac{146029^{6935}}{146000^{6935}}$$
- step6: Rewrite the expression:
  $$1000\times 13\times \frac{146029^{6935}}{1000^{6935}\times 146^{6935}}$$
- step7: Reduce the numbers:
  $$13\times \frac{146029^{6935}}{1000^{6934}\times 146^{6935}}$$
- step8: Multiply the fractions:
  $$\frac{13\times 146029^{6935}}{1000^{6934}\times 146^{6935}}$$
Calculate or simplify the expression $$ 13000 * (1 + 0.0675/12)^(12*19) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$13000\left(1+\frac{0.0675}{12}\right)^{12\times 19}$$
- step1: Divide the terms:
  $$13000\left(1+\frac{9}{1600}\right)^{12\times 19}$$
- step2: Add the numbers:
  $$13000\left(\frac{1609}{1600}\right)^{12\times 19}$$
- step3: Multiply the numbers:
  $$13000\left(\frac{1609}{1600}\right)^{228}$$
- step4: Simplify:
  $$13000\times \frac{1609^{228}}{1600^{228}}$$
- step5: Rewrite the expression:
  $$200\times 65\times \frac{1609^{228}}{1600^{228}}$$
- step6: Rewrite the expression:
  $$200\times 65\times \frac{1609^{228}}{200^{228}\times 8^{228}}$$
- step7: Reduce the numbers:
  $$65\times \frac{1609^{228}}{200^{227}\times 8^{228}}$$
- step8: Rewrite the expression:
  $$5\times 13\times \frac{1609^{228}}{200^{227}\times 8^{228}}$$
- step9: Rewrite the expression:
  $$5\times 13\times \frac{1609^{228}}{25^{227}\times 8^{227}\times 8^{228}}$$
- step10: Rewrite the expression:
  $$5\times 13\times \frac{1609^{228}}{5^{454}\times 8^{227}\times 8^{228}}$$
- step11: Reduce the numbers:
  $$13\times \frac{1609^{228}}{5^{453}\times 8^{227}\times 8^{228}}$$
- step12: Multiply the fractions:
  $$\frac{13\times 1609^{228}}{5^{453}\times 8^{455}}$$
### Step 4: Future Values Calculation Results

From the calculations, we have:

- Austin's future value ($$ FV_A $$):
  $$
  FV_A \approx 46705.21
  $$

- Easton's future value ($$ FV_E $$):
  $$
  FV_E \approx 51537.65
  $$

### Step 5: Calculate the Difference

Now, we will find the difference between Easton's and Austin's future values:
$$
\text{Difference} = FV_E - FV_A
$$

Substituting the values:
$$
\text{Difference} \approx 51537.65 - 46705.21
$$

Let's calculate this difference.
Calculate the value by following steps:
- step0: Calculate:
  $$51537.647966-46705.213463$$
- step1: Subtract the numbers:
  $$4832.434503$$
### Step 6: Final Calculation of the Difference

The difference between Easton's and Austin's future values is approximately:
$$
\text{Difference} \approx 4832.43
$$

### Conclusion

To the nearest dollar, Easton would have approximately $$ \$ 4,832 $$ more in his account than Austin after 19 years.
Easton would have \$4,832 more than Austin after 19 years.

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