Five years ago Peter invested $20000 into a 5 year GIC at an annual rate of 4.75% compounded monthly. His brother Troy put some money into a CD five years ago that earned him an annual rate of 3.95% compounded daily. If both the investments are the same value after five years, how much did Troy invest in the CD? 

To find out how much Troy invested in the CD, we need to compare the value of Peter's GIC investment after 5 years with the value of Troy's CD investment after 5 years, given that they are equal at the end of the period.

### Step 1: Peter's Investment in the GIC

The formula for compound interest is:

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

where:
- $ A $ is the amount after the investment period,
- $ P $ is the principal (initial investment),
- $ r $ is the annual interest rate (in decimal form),
- $ n $ is the number of times the interest is compounded per year,
- $ t $ is the time in years.

For Peter:
- $ P = 20000 $,
- $ r = 0.0475 $,
- $ n = 12 $ (monthly compounding),
- $ t = 5 $ years.

Substituting these values into the formula:

\[
A_P = 20000 \left(1 + \frac{0.0475}{12}\right)^{12 \times 5}
\]

### Step 2: Troy's Investment in the CD

Let $ P_T $ be the amount Troy invested. For Troy's CD, the same compound interest formula applies:

\[
A = P_T \left(1 + \frac{r_T}{n_T}\right)^{n_T \times t}
\]

For Troy:
- $ A = A_P $ (since both investments are equal after 5 years),
- $ r_T = 0.0395 $,
- $ n_T = 365 $ (daily compounding),
- $ t = 5 $ years.

The equation becomes:

\[
A_P = P_T \left(1 + \frac{0.0395}{365}\right)^{365 \times 5}
\]

### Step 3: Solve for Troy's Initial Investment $ P_T $

We will first calculate $ A_P $ (Peter's amount after 5 years) and then use it to find $ P_T $, Troy's initial investment.

Let me calculate these values.After 5 years, Peter's investment grows to approximately **$25,349.61**. For Troy's CD to reach the same value, he must have initially invested around **$20,806.68**.

Would you like more details on any part of the solution?

Here are 5 related questions you might explore:
1. How would Troy's investment change if his interest rate were compounded monthly instead of daily?
2. What would happen to both investments if the interest rates were swapped?
3. How does increasing the investment duration to 10 years affect both amounts?
4. What if Troy's CD had an annual interest rate of 4.75% instead of 3.95%?
5. How much would Troy need to invest if Peter's interest rate was only 3%?

**Tip:** Compounding more frequently (e.g., daily vs. monthly) can make a small but noticeable difference in the final amount, especially over longer periods.

Five years ago Peter invested $20000 into a 5 year GIC at an annual rate of 4.75% compounded monthly. His brother Troy put some money into a CD five years ago that earned him an annual rate of 3.95% compounded daily. If both the investments are the same value after five years, how much did Troy invest in the CD?

Learn how to calculate the initial investment Troy made in a CD that earned 3.95% interest compounded daily over 5 years, compared to Peter's $20,000 investment in a GIC at 4.75% compounded monthly. This step-by-step solution explains compound interest and exponential growth concepts, ideal for students in Grades 10-12.

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