To solve this problem, we need to adjust the gasoline-oil ratio from the current 27:1 to the desired 42:1 by adding more gasoline to the existing mixture. Here's a step-by-step solution:

Step 1: Determine the current amount of gasoline and oil

The forester has 3 gallons of a mixture with a ratio of 27 parts gasoline to 1 part oil. This means the total number of parts in the mixture is:

$27 + 1 = 28 \text{ parts}$

The fraction of gasoline in the mixture is:

$\frac{27}{28}$

So, the amount of gasoline in the current 3-gallon mixture is:

$3 \times \frac{27}{28} = \frac{81}{28} \text{ gallons}$

Similarly, the amount of oil is:

$3 \times \frac{1}{28} = \frac{3}{28} \text{ gallons}$

Step 2: Set up the equation for the desired ratio

Let $x$ be the number of gallons of gasoline to add. After adding $x$ gallons of gasoline, the new amount of gasoline will be:

$\frac{81}{28} + x \text{ gallons}$

The amount of oil remains the same at $\frac{3}{28}$ gallons.

We want the new gasoline-oil ratio to be 42:1, so we set up the ratio equation:

$\frac{\frac{81}{28} + x}{\frac{3}{28}} = 42$

Step 3: Solve for $x$

Simplify the equation by multiplying both sides by $\frac{3}{28}$:

$\frac{81}{28} + x = 42 \times \frac{3}{28}$

$\frac{81}{28} + x = \frac{126}{28}$

Subtract $\frac{81}{28}$ from both sides:

$x = \frac{126}{28} - \frac{81}{28}$

$x = \frac{126 - 81}{28} = \frac{45}{28} \approx 1.61 \text{ gallons}$

Thus, the forester should add approximately 1.61 gallons of gasoline to achieve the desired 42:1 gasoline-oil ratio.

Would you like more details or have any questions?

Here are some related questions:

1. How would the solution change if the initial ratio were different?
2. What if the forester only had 2 gallons of the initial mixture?
3. How does the ratio affect engine performance in a chainsaw?
4. Can you derive a general formula for adjusting ratios by adding a component?
5. What happens if oil is added instead of gasoline?

Tip: Always cross-check your ratios after solving to ensure the desired proportions are achieved!