A dehydrated patient needs a 7.9% saline IV. Unfortunately, the hospital only has bags of 3% and 10% saline solutions. How many liters of each of these solutions should be mixed together to yield 3 liters of the desired concentration?
x = liters of 3% solution
3 - x = liters of 10% solution
.03x = value 3% solution
.10(3 - x) = value 10% solution
3 = liters of 7.9% mixture
.079(3) = 0.237 = value 7.9% mixture
.03x + .10(3 - x) = 0.237
Solve for x, which is number of liters of 3% solution.
(3 - x) = number of liters of 10% solution
To solve this problem, we can use the method of alligation or mixture. It involves finding a ratio that represents the proportion of the two different concentrations needed to achieve the desired concentration.
Let's assign variables:
Let x represent the amount of the 3% saline solution in liters.
Then, (3 - x) represents the amount of the 10% saline solution in liters, since we want a total of 3 liters in the end.
Now, we can set up an equation using the principle that the concentration is directly proportional to the amount of the solution:
(0.03 * x) + (0.1 * (3 - x)) = 0.079 * 3
Let's solve the equation step by step:
0.03x + 0.3 - 0.1x = 0.237
0.03x - 0.1x = 0.237 - 0.3
-0.07x = -0.063
x = -0.063 / -0.07
x = 0.9
So, we need 0.9 liters of the 3% saline solution and (3 - 0.9) = 2.1 liters of the 10% saline solution to yield 3 liters of the desired concentration.