To find how many years $t$ it will take for an investment of $1,000 to grow to $1,500 with a 5% annual interest rate, we can use the formula for compound interest, assuming it's compounded annually:

$A = P(1 + r)^t$

where:

• $A$ is the final amount ($1,500),
• $P$ is the initial principal ($1,000),
• $r$ is the annual interest rate (5%, or 0.05),
• $t$ is the time in years, which we want to solve for.

Substitute the known values:

$1500 = 1000(1 + 0.05)^t$

Simplify this to form the equation:

$1.5 = (1.05)^t$

Now, to solve for $t$, you can use logarithms:

$t = \frac{\ln(1.5)}{\ln(1.05)}$

This equation will give you the number of years needed for the investment to reach $1,500.

Would you like a step-by-step calculation or have any further questions?

---

1. How would the equation change if the interest was compounded monthly?
2. What would be the time if the interest rate was 7% instead?
3. How much would the investment grow to in 10 years at 5% interest?
4. How does simple interest calculation differ from compound interest in this scenario?
5. How long would it take for the investment to double at 5% interest?

Tip: Logarithms are helpful for solving equations where the unknown is an exponent.