Question

You would like to have $74,000 available in 15 years. There are two options. Account A has a rate of 7.5% compounded once a year. Account B has a
rate of 7% compounded daily. How much would you have to deposit in each account to reach your goal?

Answer 1

Answer:

Account A = $25,009.49

Account B = $25,898.00

Step-by-step explanation:

To find out how much would need to be deposited in each account to reach the goal of $74,000 in 15 years, we can use the compound interest formula:

$\boxed{\begin{array}{l}\underline{\textsf{Compound Interest Formula}}\\\\A=P\left(1+\dfrac{r}{n}\right)^{nt}\\\\\textsf{where:}\\\phantom{ww}\bullet\;\;\textsf{ A is the final amount.}\\\phantom{ww}\bullet\;\;\textsf{ P is the principal amount.}\\\phantom{ww}\bullet\;\;\textsf{ r is the interest rate (in decimal form).}\\\phantom{ww}\bullet\;\;\textsf{ n is the number of times interest is applied per year.}\\\phantom{ww}\bullet\;\;\textsf{ t is the time (in years).}\end{array}}$

Account A

Account A has a rate of 7.5% compounded once a year. Therefore:

• A = $74,000
• r = 7.5% = 0.075
• n = 1
• t = 15 years

Substitute these values into the equation and solve for P:

$74000=P\left(1+\dfrac{0.075}{1}\right)^{1\cdot 15}$

$74000=P\left(1.075\right)^{15}$

$P=\dfrac{74000}{(1.075)^{15}}$

$P=25009.485415...$

$P=\ 25009.49$

Therefore, we would have to deposit $25,009.49 into account A to reach an account balance of $74,000 in 15 years.

Account B

Account B has a rate of 7% compounded daily. Therefore:

• A = $74,000
• r = 7% = 0.07
• n = 365
• t = 15 years

Substitute these values into the equation and solve for P:

$74000=P\left(1+\dfrac{0.07}{365}\right)^{365\cdot 15}$

$74000=P\left(1+\dfrac{0.07}{365}\right)^{5475}$

$P=\dfrac{74000}{\left(1+\dfrac{0.07}{365}\right)^{5475}}$

$P=25898.000508...$

$P=\ 25898.00$

Therefore, we would have to deposit $25898.00 into account B to reach an account balance of $74,000 in 15 years.