Let's solve this problem step-by-step to find the concentration of the unknown ethanol solution:
1. Identify Given Information:
- Solution 1: 20 gallons with an unknown ethanol concentration [tex]\( x \)[/tex].
- Solution 2: 60 gallons with a 12% ethanol concentration.
- Mixture: 80 gallons with a 10% ethanol concentration.
2. Calculate Ethanol in the Mixture:
- Total gallons in the mixture: 80 gallons.
- Ethanol concentration in the mixture: 10%.
- Total ethanol in the mixture:
[tex]\[
\text{Total ethanol in mixture} = 80 \times 0.10 = 8 \text{ gallons}
\][/tex]
3. Calculate Ethanol from Solution 2:
- Total gallons in solution 2: 60 gallons.
- Ethanol concentration in solution 2: 12%.
- Total ethanol in solution 2:
[tex]\[
\text{Total ethanol in solution 2} = 60 \times 0.12 = 7.2 \text{ gallons}
\][/tex]
4. Subtract Ethanol from Solution 2 from the Total Ethanol in the Mixture to Find Ethanol from Solution 1:
[tex]\[
\text{Ethanol from solution 1} = 8 \text{ gallons (total mixture)} - 7.2 \text{ gallons (solution 2)} = 0.8 \text{ gallons}
\][/tex]
5. Determine the Concentration [tex]\( x \)[/tex] of Solution 1:
- Total gallons in solution 1: 20 gallons.
- Ethanol from solution 1: 0.8 gallons.
- Concentration [tex]\( x \)[/tex] of ethanol in solution 1:
[tex]\[
x = \frac{\text{Ethanol from solution 1}}{\text{Total gallons in solution 1}} = \frac{0.8}{20} = 0.04 \text{ or } 4\%
\][/tex]
Thus, the concentration of the unknown ethanol solution is [tex]\( x = 4\% \)[/tex].
