A nurse has two solutions that contain different concentrations of a certain medication. One is a 20% concentration and the other is a 5% concentration. How many cubic centimeters of each should he mix to obtain 10 cc of a 14% solution?
If x at 20%, then
.20x + .05(10-x) = .14(10)
To solve this problem, we need to use the concept of the concentration of a solution, which is the amount of solute (medication) present in a given volume of the solution. We'll use the method of allegory, which is commonly used to solve mixture problems.
Let's suppose the nurse needs to mix x cc of the 20% concentration solution and y cc of the 5% concentration solution to obtain 10 cc of a 14% concentration solution.
Step 1: Express the concentrations as decimals
The 20% concentration can be expressed as 0.20, and the 5% concentration as 0.05.
Step 2: Set up the equation
Since we need a total volume of 10 cc of the 14% solution, our equation will be:
0.20x + 0.05y = 0.14 * 10
Step 3: Simplify the equation
0.20x + 0.05y = 1.4
Step 4: Solve the equation
There are multiple ways to solve this equation, but one common method is to use the method of substitution. Since we need to solve for x and y in terms of each other, we can solve for x in terms of y or vice versa.
Let's solve for x in terms of y:
0.20x = 1.4 - 0.05y
x = (1.4 - 0.05y) / 0.20
Step 5: Substitute the value of x in the equation
Substitute x = (1.4 - 0.05y) / 0.20 into the original equation:
0.20((1.4 - 0.05y) / 0.20) + 0.05y = 1.4
Step 6: Simplify and solve for y
Multiply through by 0.2 to eliminate the denominator:
1.4 - 0.05y + 0.05y = 1.4 * 0.2
1.4 = 0.28
Step 7: Check the solution
To check the solution, substitute the values of x and y into the original equation:
0.20x + 0.05y = 0.14 * 10
0.20(0.28) + 0.05(9.72) = 1.4
0.056 + 0.486 = 1.4
0.542 = 1.4
Since 1.4 does not equal 0.542, we made a mistake in our calculations somewhere. Let's analyze the steps and correct the error.
Upon reviewing the steps, we made a mistake in simplifying the equation in Step 3. It should be:
0.20x + 0.05y = 0.14 * 10
0.20x + 0.05y = 1.4
Let's go through the calculations again, using the corrected equation:
0.20x + 0.05y = 1.4
Next, we need to solve for x in terms of y:
0.20x = 1.4 - 0.05y
x = (1.4 - 0.05y) / 0.20
Now, substitute the value of x into the original equation:
0.20((1.4 - 0.05y) / 0.20) + 0.05y = 1.4
Simplify and solve for y:
1.4 - 0.05y + 0.05y = 1.4
1.4 = 1.4
Since the equation is satisfied, our solution is correct. Thus, we have found that y = 10 - x.
We can now solve for the values of x and y:
Substituting y = 10 - x into the equation x = (1.4 - 0.05y) / 0.20:
x = (1.4 - 0.05(10 - x)) / 0.20
x = (1.4 - 0.5 + 0.05x) / 0.20
x = (0.9 + 0.05x) / 0.20
0.20x = 0.9 + 0.05x
0.20x - 0.05x = 0.9
0.15x = 0.9
x = 0.9 / 0.15
x = 6
Substituting x = 6 into the equation y = 10 - x:
y = 10 - 6
y = 4
Therefore, the nurse should mix 6 cc of the 20% concentration solution and 4 cc of the 5% concentration solution to obtain 10 cc of a 14% solution.