Question

Sophia is going to invest in an account paying an interest rate of 3.4% compounded monthly. How much would Sophia need to invest, to the nearest dollar, for the value of the account to reach $1,450 in 14 years?

Answer 1

To find the amount Sophia needs to invest in an account paying an interest rate of 3.4% compounded monthly to reach $1,450 in 14 years, we can use the formula for compound interest. By substituting the given values into the formula, we find that Sophia needs to invest approximately $714.07.

Step-by-step explanation:

To find out how much Sophia would need to invest, we can use the formula for compound interest:

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

Where:

• A = The future value of the account
• P = The principal amount (initial investment)
• r = The annual interest rate (as a decimal)
• n = The number of times that interest is compounded per year
• t = The number of years

In this case, we have:

• A = $1,450
• r = 3.4% = 0.034
• n = 12 (monthly compounding)
• t = 14 years

Now we can substitute these values into the formula and solve for P:

$1,450 = P(1 + 0.034/12)^(12*14)

To find the value of P, we can rearrange the equation:

P = $1,450 / (1.002833^168)

Calculating this expression, we find that Sophia would need to invest approximately $714.07 to the nearest dollar.

Answer 2

To calculate the amount Sophia needs to invest to reach a certain value in a specific time, given an interest rate compounded monthly, we can use the formula:

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

Where:

A = the future value of the investment

P = the initial principal or investment amount

r = the annual interest rate (expressed as a decimal)

n = the number of times the interest is compounded per year

t = the number of years the investment is held

Since the interest is compounded monthly, we need to divide the annual rate by 12 to find the monthly rate. So:

r/12 = 0.034/12 = 0.002833

Now we can plug in the values into the formula:

A = P(1 + 0.002833)^(12*14)

A = P(1.002833)^(168)

We know that A = $1,450 and we want to find P (the initial principal or investment amount).

So we can rearrange the equation to find P:

P = A/(1.002833)^(168)

Solving for P, we get:

P ≈ $973.99

So Sophia would need to invest around $973.99 to reach $1,450 in 14 years with an annual interest rate of 3.4% compounded monthly.

Step-by-step explanation: