Math Problem Statement

Complete the table and find the balance A if

​$45004500

is invested at an annual percentage rate of

33​%

for 10 years and compounded n times a year.

Solution

To complete the table and find the balance AA, we use the compound interest formula:

A=P(1+rn)ntA = P \left(1 + \frac{r}{n}\right)^{nt}

Where:

  • PP is the principal amount ($4500)
  • rr is the annual interest rate (33% or 0.33)
  • nn is the number of times interest is compounded per year
  • tt is the time in years (10 years)
  • AA is the amount after time tt

Let's calculate AA for different values of nn:

1. n=1n = 1 (Compounded Annually):

A=4500(1+0.331)1×10=4500(1.33)10A = 4500 \left(1 + \frac{0.33}{1}\right)^{1 \times 10} = 4500 \left(1.33\right)^{10}

2. n=4n = 4 (Compounded Quarterly):

A=4500(1+0.334)4×10=4500(1+0.0825)40=4500(1.0825)40A = 4500 \left(1 + \frac{0.33}{4}\right)^{4 \times 10} = 4500 \left(1 + 0.0825\right)^{40} = 4500 \left(1.0825\right)^{40}

3. n=12n = 12 (Compounded Monthly):

A=4500(1+0.3312)12×10=4500(1+0.0275)120=4500(1.0275)120A = 4500 \left(1 + \frac{0.33}{12}\right)^{12 \times 10} = 4500 \left(1 + 0.0275\right)^{120} = 4500 \left(1.0275\right)^{120}

4. n=365n = 365 (Compounded Daily):

A=4500(1+0.33365)365×10=4500(1+0.0009041)3650=4500(1.0009041)3650A = 4500 \left(1 + \frac{0.33}{365}\right)^{365 \times 10} = 4500 \left(1 + 0.0009041\right)^{3650} = 4500 \left(1.0009041\right)^{3650}

Now, let's compute these values.

I'll proceed to calculate these balances.Here are the balances AA for the different compounding frequencies:

  1. Compounded Annually (n=1n = 1): A=$77,934.35A = \$77,934.35
  2. Compounded Quarterly (n=4n = 4): A=$107,233.06A = \$107,233.06
  3. Compounded Monthly (n=12n = 12): A=$116,689.61A = \$116,689.61
  4. Compounded Daily (n=365n = 365): A=$121,825.11A = \$121,825.11

These values demonstrate how the frequency of compounding affects the final balance over time.

Do you want more details or have any questions?

Here are five related questions to consider:

  1. How would the balance change if the interest rate was lower?
  2. What happens to the balance if the investment period is reduced to 5 years?
  3. How does continuous compounding compare to daily compounding?
  4. What is the effect of doubling the initial investment amount?
  5. How would a semi-annual compounding frequency compare to the others?

Tip: The more frequently interest is compounded, the higher the final balance will be.

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Math Problem Analysis

Mathematical Concepts

Compound Interest
Exponential Growth

Formulas

Compound Interest Formula: A = P (1 + r/n)^(nt)

Theorems

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Suitable Grade Level

High School