Peo needs R30 835,42 to buy a second hand trailer. Approximately how long will it take him to save towards this amount if he deposits R25 000 now into a savings account earning 10,5% interest per year compounded weekly?

a.

2 weeks

b.

104 weeks

c.

52 weeks

d.

24 weeks

To solve this problem, we need to use the formula for compound interest:

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

Where:
- \( A \) is the amount of money accumulated after \( t \) years, including interest.
- \( P \) is the principal amount (initial investment).
- \( r \) is the annual interest rate (decimal).
- \( n \) is the number of times that interest is compounded per year.
- \( t \) is the time the money is invested or borrowed for, in years.

Given:
- \( A = 30\,835.42 \) (the amount Peo needs)
- \( P = 25\,000 \) (the initial deposit)
- \( r = 10.5\% = 0.105 \) (annual interest rate)
- \( n = 52 \times 7 = 364 \) (compounded weekly, i.e., 52 weeks/year × 7 days/week)
  
We need to find \( t \).

1. **Rearrange the formula to solve for \( t \):**

\[
t = \frac{\log\left(\frac{A}{P}\right)}{n \cdot \log\left(1 + \frac{r}{n}\right)}
\]

2. **Substitute the given values into the equation:**

\[
t = \frac{\log\left(\frac{30\,835.42}{25\,000}\right)}{364 \cdot \log\left(1 + \frac{0.105}{364}\right)}
\]

Now, let's calculate \( t \).The calculation gives approximately \( t = 103.91 \) weeks.

Therefore, the correct answer is:

**b. 104 weeks**

Would you like more details or have any other questions?

Here are 8 related questions to further explore this topic:
1. How does the frequency of compounding affect the total amount of interest earned?
2. What would happen if the interest were compounded monthly instead of weekly?
3. How would the time change if Peo deposited R30,000 instead of R25,000?
4. How does compound interest differ from simple interest in this context?
5. What would the time be if the interest rate were 12% instead of 10.5%?
6. Can you explain how to derive the formula for compound interest?
7. How much interest will Peo have earned by the end of the saving period?
8. What if Peo withdraws some money before the end of the saving period? How would that affect the time?

**Tip:** Compound interest grows faster with more frequent compounding periods, making it a powerful tool for growing investments over time.

Learn how to calculate savings growth over time using compound interest. Solve the problem of determining the time required to save a specific amount with weekly compounding at 10.5% interest rate. Suitable for high school students.

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