A chemist mixed a 70% alcohol solution with a 30% alcohol solution to make a 45% alcohol solution. How many liters of each were used to make 120 L of a 45% solution?
I tried solving this with elimination method and made two separate equations. For instance,
x+y = 120(.45)
.70x+.30y=.45
This was the set up I came up with. But once I tried solving it I got an answer that I knew couldn't be right.
x = 70% alcohol solution
y = 30% alcohol solution
120 L of a 45% solution have :
120 * 0.45 = 54 L alcohol
Now you must solve system :
0.7 x + 0.3 y = 54
x + y = 120
The solutions are :
x = 45 L , y = 75 L
Proof :
0.7 * 45 + 0.3 * 75 = 31.5 + 22.5 = 54 L
x + y = 45 + 75 = 120 L
To solve this problem, you can use a system of equations to represent the given information.
Let's assume that the chemist used x liters of the 70% alcohol solution and y liters of the 30% alcohol solution to make 120 L of a 45% alcohol solution.
Based on the information given, we can set up the following equations:
Equation 1: x + y = 120 -----------(equation 1)
Equation 2: 0.70x + 0.30y = 0.45(120) -----------(equation 2)
Now, let's solve this system of equations:
Multiply equation 2 by 100 to eliminate the decimals:
70x + 30y = 45(120)
Simplify equation 2:
70x + 30y = 5400 -----------(equation 3)
Now we have a system of equations:
Equation 1: x + y = 120 -----------(equation 1)
Equation 3: 70x + 30y = 5400 -----------(equation 3)
We can solve this system using the method of substitution or elimination.
Using the substitution method:
Rearrange equation 1 to solve for x:
x = 120 - y
Substitute x in equation 3 with 120 - y:
70(120 - y) + 30y = 5400
8400 - 70y + 30y = 5400
-40y = 5400 - 8400
-40y = -3000
y = -3000 / -40
y = 75
Substitute y = 75 into equation 1:
x + 75 = 120
x = 120 - 75
x = 45
Therefore, the chemist used 45 liters of the 70% alcohol solution and 75 liters of the 30% alcohol solution to make 120 liters of a 45% alcohol solution.
To solve this problem, you can use a system of equations and the method of substitution. Let's go through the steps to find the correct solution.
Step 1: Assign variables.
Let x be the amount of the 70% alcohol solution (in liters) and y be the amount of the 30% alcohol solution (in liters).
Step 2: Set up the equations.
We can set up two equations based on the given information:
Equation 1: x + y = 120 (since the total volume of the mixture is 120 L)
Equation 2: 0.70x + 0.30y = 0.45(120) (since the concentration of alcohol is 45%)
Step 3: Solve the system of equations.
For Equation 1, we can express y in terms of x:
y = 120 - x
Substitute this into Equation 2:
0.70x + 0.30(120 - x) = 0.45(120)
Simplify and solve for x:
0.70x + 36 - 0.30x = 54
0.40x = 18
x = 45
Now, substitute the value of x back into Equation 1 to find y:
45 + y = 120
y = 120 - 45
y = 75
So, the chemist used 45 liters of the 70% alcohol solution and 75 liters of the 30% alcohol solution to make 120 liters of a 45% alcohol solution.