Question

Fawzia is going to invest $440 and leave it in an account for 9 years. Assuming the interest is compounded monthly, what interest rate, to the nearest hundredth of a percent, would be required in order for Fawzia to end up with $560?

Answer 1

Fawzia would need an interest rate of approximately 22.2% (to the nearest hundredth of a percent) in order to end up with $560 after 9 years with monthly compounding.

The compound interest formula is given by:

`\[ A = P \left(1 + (r)/(n)\right)^(nt) \]`

where:

A is the future value of the investment/loan, including interest.

P is the principal amount (initial investment).

r is the annual interest rate (as a decimal).

n is the number of times that interest is compounded per year.

t is the time the money is invested or borrowed for, in years.

In this case, Fawzia is investing $440 for 9 years, and she wants to end up with $560. We need to find the interest rate
`(\( r \))`.

`\[ 560 = 440 \left(1 + (r)/(12)\right)^(12 * 9) \]`

Now, let's solve for
`\( r \)`:

`\[ (560)/(440) = \left(1 + (r)/(12)\right)^(108) \]`

`\[ (56)/(44) = \left(1 + (r)/(12)\right)^(108) \]`

`\[ (14)/(11) = \left(1 + (r)/(12)\right)^(108) \]`

Now, take the 108th root of both sides:

`\[ \left(1 + (r)/(12)\right) = \left((14)/(11)\right)^{(1)/(108)} \]`

`\[ 1 + (r)/(12) = 1.0185 \]`

Now, solve for
`\( r \)`:

`\[ (r)/(12) = 0.0185 \]`

`\[ r = 0.0185 * 12 \]`

`\[ r \approx 0.222 \]`

To find the interest rate as a percentage, multiply by 100:

`\[ r \approx 22.2\% \]`

Answer 2

To find the interest rate, we can use the formula for compound interest:

A = P(1 + r/n)^(nt)

Where:

A = Final amount

P = Principal amount (initial investment)

r = Annual interest rate (as a decimal)

n = Number of times interest is compounded per year

t = Number of years

In this case, we have:

P = $440

A = $560

t = 9 years

n = 12 (compounded monthly)

We can rearrange the formula to solve for r:

r = (A/P)^(1/(nt)) - 1

Plugging in the values, we get:

r = ($560/$440)^(1/(9*12)) - 1

Calculating this expression gives us the interest rate required to end up with $560 after 9 years of monthly compounding, which is approximately 1.65%

Your welcome :)