Math Problem Statement

a. The cost of a new automobile is $10,300. If the interest rate is 8%, how much would you have to set aside now to provide this sum in eight years? 

Solution

To solve this problem, we will use the formula for the present value of a sum of money. The present value (PV) is the amount you need to invest now to achieve a future value (FV) given a certain interest rate and time period. The formula is:

PV=FV(1+r)tPV = \frac{FV}{(1 + r)^t}

Where:

  • FVFV is the future value, which is $10,300.
  • rr is the annual interest rate, which is 8% or 0.08.
  • tt is the time period, which is 8 years.

Now, let's substitute the values into the formula:

PV=10,300(1+0.08)8PV = \frac{10,300}{(1 + 0.08)^8}

PV=10,300(1.08)8PV = \frac{10,300}{(1.08)^8}

PV=10,3001.850935,565.17PV = \frac{10,300}{1.85093} \approx 5,565.17

So, you would need to set aside approximately $5,565.17 now to provide $10,300 in 8 years at an interest rate of 8%.

Do you want further details on how this calculation works or have any other questions?

Here are 5 related questions you may want to explore:

  1. How does the present value change with different interest rates?
  2. What is the future value if you invest $5,565.17 at 8% for 8 years?
  3. How do compounding frequencies affect the present value calculation?
  4. What if the interest is compounded monthly instead of annually?
  5. How much would you need to invest for a 10-year period instead of 8 years?

Tip: When dealing with interest rates and time periods, be mindful of how frequently the interest is compounded, as this can significantly affect the result.

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Math Problem Analysis

Mathematical Concepts

Finance
Compound Interest
Present Value

Formulas

Present Value Formula: PV = FV / (1 + r)^t

Theorems

Present Value Theorem

Suitable Grade Level

Grades 10-12