Solve for t: 250t = 100t + 3000.

Step-by-step solution

1. 1

   Interpret the equation. One side grows at $250$ per unit of $t$, the other at $100$ per unit but starting from a fixed $3000$. This is the standard break-even model — revenue against fixed cost plus variable cost — and $t$ is the point where they meet.

2. 2

   Collect the t terms on one side. Subtract $100t$ from both sides:

   $250t-100t=3000$

3. 3

   Combine the like terms.

   $150t=3000$

   The coefficient $150=250-100$ is the rate difference; that is the quantity that eats into the fixed $3000$.

4. 4

   Divide by 150.

   $t=\frac{3000}{150}=20$

   The general form worth remembering is $t=\dfrac{\text{fixed amount}}{\text{rate difference}}$.

5. 5

   Verify in the original equation. Left: $250\times20=5000$. Right: $100\times20+3000=2000+3000=5000$ ✓. Both sides equal $5000$ at $t=20$, and for $t>20$ the left side pulls ahead — which is what break-even means.