We are tasked with solving the equation:

$\frac{0.5(100+t) - 5 \times 10^8 (100+t)^{-5}}{100+t} = 0.3$

Let's break down the steps to solve for $t$.

Step 1: Simplify the equation

We begin by simplifying the left-hand side of the equation. The expression can be rewritten as:

$\frac{0.5(100+t)}{100+t} - \frac{5 \times 10^8 (100+t)^{-5}}{100+t} = 0.3$

Notice that $\frac{0.5(100+t)}{100+t} = 0.5$, so the equation simplifies to:

$0.5 - \frac{5 \times 10^8 (100+t)^{-5}}{100+t} = 0.3$

Step 2: Isolate the remaining term

Now subtract $0.5$ from both sides:

$-\frac{5 \times 10^8 (100+t)^{-5}}{100+t} = 0.3 - 0.5$

This simplifies to:

$-\frac{5 \times 10^8 (100+t)^{-5}}{100+t} = -0.2$

Multiply both sides by $-1$ to eliminate the negative sign:

$\frac{5 \times 10^8 (100+t)^{-5}}{100+t} = 0.2$

Step 3: Eliminate the denominator

Multiply both sides of the equation by $100+t$ to get rid of the denominator:

$5 \times 10^8 (100+t)^{-5} = 0.2(100+t)$

Step 4: Solve for $t$

Now we have the equation:

$5 \times 10^8 (100+t)^{-5} = 0.2(100+t)$

To solve this, we need to multiply both sides by $(100+t)^5$, and we will expand and solve it. I'll solve this next.The solutions for $t$ are:

$t \approx -136.84 \quad \text{or} \quad t \approx -63.16$

There are also complex solutions:

$t \approx -118.42 \pm 31.90i \quad \text{and} \quad t \approx -81.58 \pm 31.90i$

Since $t$ is likely a real number in this context, the relevant real solutions are approximately:

$t \approx -136.84 \quad \text{or} \quad t \approx -63.16$

Would you like more details on this process, or have any specific questions?

Here are 5 related questions:

1. How does complex number arise in equations like this?
2. How would the behavior of the equation change with different constants?
3. What is the role of exponents and roots in solving equations like this?
4. How can we apply this method to other real-world problems involving large constants?
5. Why is isolating terms important in solving algebraic equations?

Tip: When solving equations with large constants, always consider whether approximating the values simplifies the process.