To find the initial amount (principal) that the company needs to deposit today to accumulate $30,000 in 5 years with 10% annual interest compounded quarterly, we can use the compound interest formula:

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

Where:

• $A$ = future amount (total after interest), which is $30,000,
• $P$ = initial principal (the amount we need to find),
• $r$ = annual interest rate, which is 10% or 0.10,
• $n$ = number of times interest is compounded per year, which is quarterly (4 times per year),
• $t$ = time in years, which is 5 years.

Step 1: Set up the equation

Plugging in the values we know:

$30000 = P \left(1 + \frac{0.10}{4}\right)^{4 \times 5}$

Step 2: Simplify the expression

1. Calculate the quarterly interest rate: $\frac{0.10}{4} = 0.025$.
2. Calculate the total number of compounding periods: $4 \times 5 = 20$.

So the equation becomes:

$30000 = P \left(1 + 0.025\right)^{20}$

Step 3: Calculate $(1 + 0.025)^{20}$

$(1 + 0.025)^{20} \approx 1.63862$

Step 4: Solve for $P$

Now, divide both sides by 1.63862 to isolate $P$:

$P = \frac{30000}{1.63862} \approx 18312.20$

Final Answer

The company should invest approximately $18,312 today to reach $30,000 in 5 years with 10% interest compounded quarterly.

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Would you like more details on any step or have questions?

Here are some related questions to explore:

1. What if the interest were compounded monthly instead of quarterly?
2. How would the required investment change if the interest rate were different?
3. What if the investment period were extended to 10 years?
4. How much would the investment grow if compounded continuously?
5. How does quarterly compounding compare to simple interest over the same period?

Tip: When calculating compound interest, breaking down each component of the formula step-by-step can help avoid errors and clarify the process.