Ethanol Fuel Mixture: Find the Right Blend

What This Problem Teaches

Picture This

Solution: Method 1 — The Component Balance

We track the amount of pure ethanol from each source. This is the standard approach for mixture problems because it focuses on what we're actually trying to control: the concentration of the active ingredient.

Step 1 — Define the variable

Let x = gallons of E85 fuel (85% ethanol) needed.

Since the tank holds 20 gallons total: 20 - x = gallons of E25 fuel (25% ethanol) needed.

Step 2 — Write the ethanol equation

The pure ethanol from both sources must equal the pure ethanol in our target mixture:

Step 3 — Solve the equation

First, simplify the right side:

Distribute on the left:

Combine like terms:

Step 4 — Find both amounts

E85 needed: x = 8⅓ gallons

E25 needed: 20 - x = 20 - 8⅓ = 11⅔ gallons

Solution: Method 2 — The Alligation Cross

This visual method finds the mixing ratio by measuring how far each concentration is from the target. It's particularly elegant when you need to mix exactly two components.

Step 1 — Draw the alligation diagram

Set up the concentrations and target in a cross pattern:

Step 2 — Calculate the differences

Find how far each concentration is from the target:

Step 3 — The ratio principle

The amount of each fuel needed is inversely proportional to its distance from the target:

This means for every 5 parts E85, we need 7 parts E25.

Step 4 — Convert to actual gallons

Total ratio parts: 5 + 7 = 12 parts = 20 gallons

Verification

Let's check that our mixture produces exactly 50% ethanol:

Pure ethanol from E85:0.85 × 8⅓ = 0.85 × 25/3 = 85/12 gallons

Pure ethanol from E25:0.25 × 11⅔ = 0.25 × 35/3 = 35/12 gallons

Total pure ethanol:85/12 + 35/12 = 120/12 = 10 gallons

Final concentration:10 gallons ÷ 20 gallons = 0.50 = 50% ✓

Also verify the amounts sum correctly: 8⅓ + 11⅔ = 20 gallons ✓

Common Pitfalls

Reality Check

Our answer shows Sofia needs more of the low-ethanol fuel (11⅔ gallons E25) than the high-ethanol fuel (8⅓ gallons E85). Does this make sense?

Absolutely. The target concentration (50%) is much closer to the E25 concentration (25%) than to the E85 concentration (85%). To "pull down" the high concentration of E85 to 50%, you need a lot of the lower concentration fuel.

If Sofia had wanted a 65% ethanol mixture (closer to E85), she would need more E85 and less E25. The closer your target is to one component, the more you need of that component.

How to Spot This Problem Type

Look for these telltale signs that signal a mixture problem:

• "Mix," "blend," or "combine" two or more substances
• Different concentrations of the same ingredient (percentages, ratios, or "parts")
• Target concentration that lies between the starting concentrations
• Total volume constraint (tank capacity, recipe amount, etc.)
• Real-world contexts: fuel blending, medication preparation, alloy composition, solution chemistry

The problem structure is always the same: you're controlling the amount of some active ingredient (ethanol, medicine, salt, etc.) by mixing different concentrations.

The Pattern Behind This

All two-component mixture problems follow this general structure:

With the volume constraint: amount₁ + amount₂ = total amount

For this specific problem:

The general alligation formula: When mixing concentrations A and B to get target T, the ratio is:

Notice how the ratios are inverted — the component farther from the target gets used in smaller proportions. This pattern appears everywhere from cooking recipes to pharmaceutical compounding.