imple interest is given by the formula
A

To solve this problem, we will use the formula for simple interest:

$A = P + P \cdot r \cdot t$

Where:

• $A$ is the future balance of the account.
• $P$ is the principal (the amount Colby needs to deposit now).
• $r$ is the annual interest rate as a decimal (in this case, 4%, or 0.04).
• $t$ is the time in years (15 years).

We are given:

• $A = 30,000$ (the future balance Colby wants to have),
• $r = 0.04$ (4% interest rate),
• $t = 15$ years.

The goal is to find $P$, the initial deposit.

Step 1: Rearranging the formula to solve for $P$

We start with the equation: $A = P + P \cdot r \cdot t$

Factor $P$ out of the terms on the right-hand side: $A = P(1 + r \cdot t)$

Now, solve for $P$: $P = \frac{A}{1 + r \cdot t}$

Step 2: Plug in the known values

Substitute $A = 30,000$, $r = 0.04$, and $t = 15$ into the equation: $P = \frac{30,000}{1 + 0.04 \cdot 15}$

Step 3: Simplify

First, calculate $0.04 \cdot 15 = 0.6$, so: $P = \frac{30,000}{1 + 0.6} = \frac{30,000}{1.6}$

Now, divide: $P = \frac{30,000}{1.6} = 18,750$

Final Answer

Colby will need to invest $18,750 now in order to have a balance of $30,000 in his savings account after 15 years.

Would you like more details or have any questions about the calculation?

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Here are 5 related questions to extend your understanding:

1. How would the initial investment change if the interest rate was 5% instead of 4%?
2. What happens to the final balance if Colby decides to invest for 20 years instead of 15?
3. How would the result change if the interest was compounded annually instead of being simple interest?
4. If Colby wanted to have $50,000 after 15 years, how much would he need to invest at 4%?
5. What is the total interest earned over the 15 years with this investment?

Tip: To easily convert a percentage to a decimal, divide the percentage by 100 (e.g., 4% becomes 0.04).