Final answer:
To obtain 5 gallons of a 12% alcohol mixture, you would need 2 gallons of 15% alcohol and 3 gallons of 10% alcohol.
Step-by-step explanation:
To solve this problem, we can set up a system of equations using the information provided. Let's say we need x gallons of 15% alcohol and y gallons of 10% alcohol. From the given information, we know that the total volume of the mixture is 5 gallons and the concentration of alcohol in the mixture is 12%. We can use the formula for finding the concentration of a mixture to set up the first equation: 0.15x + 0.10y = 0.12(5). The second equation is the equation for the total volume: x + y = 5. We can solve this system of equations to find the values of x and y, which will give us the number of gallons of each type of alcohol needed.
To solve the system of equations, we can use substitution or elimination. Let's solve it using substitution. From the second equation, we can express x in terms of y: x = 5 - y. Substituting this expression for x into the first equation, we have: 0.15(5 - y) + 0.10y = 0.12(5). Simplifying this equation, we get: 0.75 - 0.15y + 0.10y = 0.60. Combining like terms, we get: -0.05y = -0.15. Dividing both sides by -0.05, we find: y = 3. Thus, we need 3 gallons of 10% alcohol.
Substituting this value of y back into the second equation, we can find the value of x: x = 5 - 3 = 2. Therefore, we need 2 gallons of 15% alcohol. In summary, to obtain 5 gallons of a 12% alcohol mixture, we need 2 gallons of 15% alcohol and 3 gallons of 10% alcohol.