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Part A: How much money will Rachel have in 10 years if the interest is compounded annually?

Part B: How much money will Rachel have in 10 years if the interest is compounded monthly?
Part C: How much money will Rachel have in 10 years if the interest is compounded continuously?

1 Answer

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Final answer:

To calculate the future value of an investment with compound interest, use the formula A = P(1 + r/n)^(n*t). For annual compounding, Rachel needs to put $3855.56 into the bank account. For monthly compounding, she needs to put $3825.92. For continuous compounding, she needs to put $3678.79.

Step-by-step explanation:

To calculate the future value of an investment with compound interest, you can use the formula:

A = P(1 + r/n)^(n*t)

Where:

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

For Part A, since the interest is compounded annually, n would be 1. Plugging in the values, we have:

A = P(1 + 0.10/1)^(1*10) = P(1 + 0.10)^10 = 10000

Simplifying the equation:

1.10^10 = 10000/P

Using algebra to solve for P:

P = 10000 / 1.10^10 = $3855.56

Therefore, Rachel needs to put $3855.56 into the bank account to have $10,000 in ten years if the interest is compounded annually.

For Part B, since the interest is compounded monthly, n would be 12 (12 months in a year). Plugging in the values, we have:

A = P(1 + 0.10/12)^(12*10) = P(1 + 0.00833)^120 = 10000

1.00833^120 = 10000/P

Using algebra to solve for P:

P = 10000 / 1.00833^120 = $3825.92

Therefore, Rachel needs to put $3825.92 into the bank account to have $10,000 in ten years if the interest is compounded monthly.

For Part C, continuous compounding uses the formula:

A = Pe^(r*t)

Where:

  • A is the future value
  • P is the principal amount (the initial investment)
  • e is Euler's number (approximately 2.71828)
  • r is the annual interest rate (written as a decimal)
  • t is the number of years

Plugging in the values, we have:

A = P * e^(0.10*10) = 10000

e^(1) = 10000/P

Using algebra to solve for P:

P = 10000 / e^(1) = $3678.79

Therefore, Rachel needs to put $3678.79 into the bank account to have $10,000 in ten years if the interest is compounded continuously.

User Dlongley
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