Math Problem Statement

Find the final amount of money in an account if $7,900 is deposited at 5% interest compounded semi-annually and the money is left for 5 years.

Solution

This is a compound interest problem where the interest is compounded semi-annually. We can use the compound interest formula to solve it:

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

Where:

  • AA is the final amount.
  • P=7,900P = 7,900 is the principal (initial deposit).
  • r=0.05r = 0.05 (5% annual interest rate).
  • n=2n = 2 because the interest is compounded semi-annually (twice a year).
  • t=5t = 5 years is the time the money is left in the account.

Substituting these values into the formula:

A=7,900(1+0.052)2×5A = 7,900 \left(1 + \frac{0.05}{2}\right)^{2 \times 5}

Let's calculate:

A=7,900(1+0.025)10A = 7,900 \left(1 + 0.025\right)^{10} A=7,900×(1.025)10A = 7,900 \times (1.025)^{10} A=7,900×1.280084A = 7,900 \times 1.280084 A10,112.66A \approx 10,112.66

So, the final amount after 5 years, rounded to two decimal places, is $10,112.66.

Would you like more details or have any questions?

Here are 5 related questions to explore:

  1. What if the interest was compounded quarterly instead of semi-annually?
  2. How does the final amount change if the interest rate increases to 6%?
  3. What if the investment period was only 3 years?
  4. How would the final amount change if the initial deposit was $10,000 instead of $7,900?
  5. What is the difference between simple interest and compound interest in terms of growth over time?

Tip: The more frequently interest is compounded, the higher the final amount will be due to the "compounding effect."

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Math Problem Analysis

Mathematical Concepts

Compound Interest
Exponential Growth
Finance Mathematics

Formulas

A = P(1 + r/n)^(nt)

Theorems

Compound Interest Theorem

Suitable Grade Level

Grades 10-12