a. To find the accumulated value of Ryan's savings, we use the future value of an ordinary annuity formula for monthly compounding:

$$ FV = P \times \left( \frac{(1 + r/n)^{nt} - 1}{r/n} \right) $$

Where:
- $ P $ is the monthly payment ($80)
- $ r $ is the annual interest rate (5.20% or 0.052)
- $ n $ is the number of times interest is compounded per year (12)
- $ t $ is the number of years (2)

Plugging in the values:

$$ FV = 80 \times \left( \frac{(1 + 0.052/12)^{12 \times 2} - 1}{0.052/12} \right) $$

First, calculate the monthly interest rate:

$$ \frac{0.052}{12} = 0.0043333 $$

Next, calculate the exponent:

$$ 12 \times 2 = 24 $$

Now, calculate the compound factor:

$$ (1 + 0.0043333)^{24} = 1.108243 $$

Then, calculate the numerator:

$$ 1.108243 - 1 = 0.108243 $$

Finally, calculate the future value:

$$ FV = 80 \times \left( \frac{0.108243}{0.0043333} \right) = 80 \times 24.987 = 1,998.96 $$

So, the closest answer is:

$$ \boxed{2,018.79} $$

b. To find the interest earned over the period, subtract the total amount of Ryan's contributions from the accumulated value.

Total contributions:

$$ 80 \times 24 = 1,920 $$

Interest earned:

$$ 2,018.79 - 1,920 = 98.79 $$

So, the interest earned is:

$$ \boxed{98.79} $$

Question

a. To find the accumulated value of Ryan's savings, we use the future value of an ordinary annuity formula for monthly compounding:

$$ FV = P \times \left( \frac{(1 + r/n)^{nt} - 1}{r/n} \right) $$

Where:
- $ P $ is the monthly payment ($80)
- $ r $ is the annual interest rate (5.20% or 0.052)
- $ n $ is the number of times interest is compounded per year (12)
- $ t $ is the number of years (2)

Plugging in the values:

$$ FV = 80 \times \left( \frac{(1 + 0.052/12)^{12 \times 2} - 1}{0.052/12} \right) $$

First, calculate the monthly interest rate:

$$ \frac{0.052}{12} = 0.0043333 $$

Next, calculate the exponent:

$$ 12 \times 2 = 24 $$

Now, calculate the compound factor:

$$ (1 + 0.0043333)^{24} = 1.108243 $$

Then, calculate the numerator:

$$ 1.108243 - 1 = 0.108243 $$

Finally, calculate the future value:

$$ FV = 80 \times \left( \frac{0.108243}{0.0043333} \right) = 80 \times 24.987 = 1,998.96 $$

So, the closest answer is:

$$ \boxed{2,018.79} $$

b. To find the interest earned over the period, subtract the total amount of Ryan's contributions from the accumulated value.

Total contributions:

$$ 80 \times 24 = 1,920 $$

Interest earned:

$$ 2,018.79 - 1,920 = 98.79 $$

So, the interest earned is:

$$ \boxed{98.79} $$

Knowee AI · Accepted Answer