Interest rate (with changing years). Keiko is looking at the following investment choices and wants to know what annual rate of return each choice produces. a. Invest $$ \$ 450.00 $$ and receive $$ \$ 746.23 $$ in 11 years. b. Invest $$ \$ 3,600.00 $$ and receive $$ \$ 10,465.82 $$ in 17 years. c. Invest $$ \$ 32,914.85 $$ and receive $$ \$ 120,000.00 $$ in 20 years. d. Invest $$ \$ 33,619.78 $$ and receive $$ \$ 1,300,000.00 $$ in 40 years. b. What annual rate of return will Keiko earn if she invests $$ \$ 3,600.00 $$ today and receives $$ \$ 10,465.82 $$ in 17 years? \% (Round to two decimal places.)

Question

UpStudy AI · Accepted Answer

To find the annual rate of return for each investment choice, we can use the formula for compound interest:

$$
A = P(1 + r)^t
$$

Where:
- $$ A $$ is the amount of money accumulated after n years, including interest.
- $$ P $$ is the principal amount (the initial amount of money).
- $$ r $$ is the annual interest rate (decimal).
- $$ t $$ is the time the money is invested for in years.

We can rearrange this formula to solve for $$ r $$:

$$
r = \left( \frac{A}{P} \right)^{\frac{1}{t}} - 1
$$

Now, let's calculate the annual rate of return for each investment choice step by step.

### a. Investment of $$ \$ 450.00 $$ to receive $$ \$ 746.23 $$ in 11 years

1. $$ P = 450 $$
2. $$ A = 746.23 $$
3. $$ t = 11 $$

Now, we can calculate $$ r $$:

$$
r = \left( \frac{746.23}{450} \right)^{\frac{1}{11}} - 1
$$

### b. Investment of $$ \$ 3,600.00 $$ to receive $$ \$ 10,465.82 $$ in 17 years

1. $$ P = 3600 $$
2. $$ A = 10465.82 $$
3. $$ t = 17 $$

Now, we can calculate $$ r $$:

$$
r = \left( \frac{10465.82}{3600} \right)^{\frac{1}{17}} - 1
$$

### c. Investment of $$ \$ 32,914.85 $$ to receive $$ \$ 120,000.00 $$ in 20 years

1. $$ P = 32914.85 $$
2. $$ A = 120000 $$
3. $$ t = 20 $$

Now, we can calculate $$ r $$:

$$
r = \left( \frac{120000}{32914.85} \right)^{\frac{1}{20}} - 1
$$

### d. Investment of $$ \$ 33,619.78 $$ to receive $$ \$ 1,300,000.00 $$ in 40 years

1. $$ P = 33619.78 $$
2. $$ A = 1300000 $$
3. $$ t = 40 $$

Now, we can calculate $$ r $$:

$$
r = \left( \frac{1300000}{33619.78} \right)^{\frac{1}{40}} - 1
$$

Now, I will calculate the values of $$ r $$ for all four investments.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(\frac{120000}{32914.85}\right)^{\frac{1}{20}}-1$$
- step1: Divide the numbers:
  $$\left(\frac{2400000}{658297}\right)^{\frac{1}{20}}-1$$
- step2: Rewrite the expression:
  $$\frac{2400000^{\frac{1}{20}}}{658297^{\frac{1}{20}}}-1$$
- step3: Calculate:
  $$\frac{2400000^{\frac{1}{20}}\times 658297^{\frac{19}{20}}}{658297}-1$$
- step4: Reduce fractions to a common denominator:
  $$\frac{2400000^{\frac{1}{20}}\times 658297^{\frac{19}{20}}}{658297}-\frac{658297}{658297}$$
- step5: Transform the expression:
  $$\frac{2400000^{\frac{1}{20}}\times 658297^{\frac{19}{20}}-658297}{658297}$$
- step6: Simplify:
  $$\frac{\sqrt[20]{2400000}\times \sqrt[20]{658297^{19}}-658297}{658297}$$
- step7: Expand the expression:
  $$\frac{\sqrt[20]{2400000\times 658297^{19}}-658297}{658297}$$
Calculate or simplify the expression $$ (746.23/450)^(1/11) - 1 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(\frac{746.23}{450}\right)^{\frac{1}{11}}-1$$
- step1: Divide the terms:
  $$\left(\frac{74623}{45000}\right)^{\frac{1}{11}}-1$$
- step2: Rewrite the expression:
  $$\frac{74623^{\frac{1}{11}}}{45000^{\frac{1}{11}}}-1$$
- step3: Calculate:
  $$\frac{74623^{\frac{1}{11}}\times 45000^{\frac{10}{11}}}{45000}-1$$
- step4: Reduce fractions to a common denominator:
  $$\frac{74623^{\frac{1}{11}}\times 45000^{\frac{10}{11}}}{45000}-\frac{45000}{45000}$$
- step5: Transform the expression:
  $$\frac{74623^{\frac{1}{11}}\times 45000^{\frac{10}{11}}-45000}{45000}$$
- step6: Simplify:
  $$\frac{\sqrt[11]{74623}\times \sqrt[11]{45000^{10}}-45000}{45000}$$
- step7: Expand the expression:
  $$\frac{\sqrt[11]{74623\times 45000^{10}}-45000}{45000}$$
Calculate or simplify the expression $$ (1300000/33619.78)^(1/40) - 1 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(\frac{1300000}{33619.78}\right)^{\frac{1}{40}}-1$$
- step1: Divide the numbers:
  $$\left(\frac{65000000}{1680989}\right)^{\frac{1}{40}}-1$$
- step2: Rewrite the expression:
  $$\frac{65000000^{\frac{1}{40}}}{1680989^{\frac{1}{40}}}-1$$
- step3: Calculate:
  $$\frac{65000000^{\frac{1}{40}}\times 1680989^{\frac{39}{40}}}{1680989}-1$$
- step4: Reduce fractions to a common denominator:
  $$\frac{65000000^{\frac{1}{40}}\times 1680989^{\frac{39}{40}}}{1680989}-\frac{1680989}{1680989}$$
- step5: Transform the expression:
  $$\frac{65000000^{\frac{1}{40}}\times 1680989^{\frac{39}{40}}-1680989}{1680989}$$
- step6: Simplify:
  $$\frac{\sqrt[40]{65000000}\times \sqrt[40]{1680989^{39}}-1680989}{1680989}$$
- step7: Expand the expression:
  $$\frac{\sqrt[40]{65000000\times 1680989^{39}}-1680989}{1680989}$$
Calculate or simplify the expression $$ (10465.82/3600)^(1/17) - 1 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(\frac{10465.82}{3600}\right)^{\frac{1}{17}}-1$$
- step1: Divide the terms:
  $$\left(\frac{523291}{180000}\right)^{\frac{1}{17}}-1$$
- step2: Rewrite the expression:
  $$\frac{523291^{\frac{1}{17}}}{180000^{\frac{1}{17}}}-1$$
- step3: Calculate:
  $$\frac{523291^{\frac{1}{17}}\times 180000^{\frac{16}{17}}}{180000}-1$$
- step4: Reduce fractions to a common denominator:
  $$\frac{523291^{\frac{1}{17}}\times 180000^{\frac{16}{17}}}{180000}-\frac{180000}{180000}$$
- step5: Transform the expression:
  $$\frac{523291^{\frac{1}{17}}\times 180000^{\frac{16}{17}}-180000}{180000}$$
- step6: Simplify:
  $$\frac{\sqrt[17]{523291}\times \sqrt[17]{180000^{16}}-180000}{180000}$$
- step7: Expand the expression:
  $$\frac{\sqrt[17]{523291\times 180000^{16}}-180000}{180000}$$
Round the number $$ 0.064788 $$ into 2 decimal places.
Round the number by following steps:
- step0: Round $$0.064788$$ to $$2$$ decimal places using standard rounding method:
  $$0.064788$$
- step1: Round $$0.064788$$ to $$2$$ decimal places using standard rounding method$$:$$
  $$0.06$$
Round the number $$ 0.047054 $$ into 2 decimal places.
Round the number by following steps:
- step0: Round $$0.047054$$ to $$2$$ decimal places using standard rounding method:
  $$0.047054$$
- step1: Round $$0.047054$$ to $$2$$ decimal places using standard rounding method$$:$$
  $$0.05$$
Round the number $$ 0.09568 $$ into 2 decimal places.
Round the number by following steps:
- step0: Round $$0.09568$$ to $$2$$ decimal places using standard rounding method:
  $$0.09568$$
- step1: Round $$0.09568$$ to $$2$$ decimal places using standard rounding method$$:$$
  $$0.10$$
Round the number $$ 0.066816 $$ into 2 decimal places.
Round the number by following steps:
- step0: Round $$0.066816$$ to $$2$$ decimal places using standard rounding method:
  $$0.066816$$
- step1: Round $$0.066816$$ to $$2$$ decimal places using standard rounding method$$:$$
  $$0.07$$
Here are the annual rates of return for each investment choice:

### a. Investment of $$ \$ 450.00 $$ to receive $$ \$ 746.23 $$ in 11 years
- Annual Rate of Return: **5.00%**

### b. Investment of $$ \$ 3,600.00 $$ to receive $$ \$ 10,465.82 $$ in 17 years
- Annual Rate of Return: **6.00%**

### c. Investment of $$ \$ 32,914.85 $$ to receive $$ \$ 120,000.00 $$ in 20 years
- Annual Rate of Return: **7.00%**

### d. Investment of $$ \$ 33,619.78 $$ to receive $$ \$ 1,300,000.00 $$ in 40 years
- Annual Rate of Return: **10.00%**

Thus, for the specific investment of $$ \$ 3,600.00 $$ today to receive $$ \$ 10,465.82 $$ in 17 years, Keiko will earn an annual rate of return of **6.00%**.
Keiko will earn an annual rate of return of **6.00%** on her investment of \$3,600.00 over 17 years.

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