Investment Requirements for Two Accounts

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Question:

Question content area top Part 1 You would like to have ​$73 comma 000 available in 15 years. There are two options. Account A has a rate of 6.5​% compounded once a year. Account B has a rate of 6​% compounded daily. How much would you have to deposit in each account to reach your​ goal? Assume 365 days in a year. LOADING... Click the icon to view some finance formulas. Question content area bottom Part 1 You must invest ​$    in Account A. ​(Round up to the nearest​ dollar.) Part 2 You must invest ​$    in Account B. ​(Round up to the nearest​ dollar.)

Answer:

To determine how much you need to deposit in each account to reach your goal of $73,000 in 15 years, we can use the formula for compound interest:

For Account A (compounded annually):

The formula for compound interest is:

[ A = P(1 + r)^n ]

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).
  • ( n ) is the number of years the money is invested or borrowed.

For Account A:

  • ( A = 73,000 )
  • ( r = 6.5% = 0.065 )
  • ( n = 15 )

Rearranging the formula to solve for ( P ):

[ P = \frac{A}{(1 + r)^n} ]

Substituting the values:

[ P = \frac{73,000}{(1 + 0.065)^{15}} ]

Calculating:

[ P = \frac{73,000}{(1.065)^{15}} \approx \frac{73,000}{2.4546} \approx 29,707.57 ]

Rounding up to the nearest dollar, you must invest $29,708 in Account A.

For Account B (compounded daily):

The formula for compound interest when compounded daily is:

[ A = P\left(1 + \frac{r}{m}\right)^{mt} ]

Where:

  • ( m ) is the number of times that interest is compounded per year (365 for daily).
  • ( t ) is the number of years.

For Account B:

  • ( A = 73,000 )
  • ( r = 6% = 0.06 )
  • ( m = 365 )
  • ( t = 15 )

Rearranging the formula to solve for ( P ):

[ P = \frac{A}{\left(1 + \frac{r}{m}\right)^{mt}} ]

Substituting the values:

[ P = \frac{73,000}{\left(1 + \frac{0.06}{365}\right)^{365 \times 15}} ]

Calculating:

[ P = \frac{73,000}{\left(1 + 0.000164384\right)^{5475}} \approx \frac{73,000}{(1.06)^{15}} \approx \frac{73,000}{2.3966} \approx 30,487.73 ]

Rounding up to the nearest dollar, you must invest $30,488 in Account B.

Summary:

  • You must invest $29,708 in Account A.
  • You must invest $30,488 in Account B.