A forester is making a gasoline-oil mixture for a chainsaw engine. The forester has 3 gallons of a mixture that is 27 parts gasoline and 1 part oil. How many gallons of gasoline should the forester add to bring the mixture to 42 parts gasoline and 1 part oil? (Round your answer to two decimal places.)

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!

Learn how to solve a real-world problem involving the adjustment of a gasoline-oil mixture ratio from 27:1 to 42:1 for a chainsaw engine. This step-by-step solution covers key algebraic concepts and ratio calculations, suitable for students in Grades 9-12.

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