In a certain medical test designed to measure carbohydrate tolerance, an adult drinks 9 ounces of a 20% glucose solution. When the test is administered to a child, the glucose concentration must be decreased to 10%. How much 20% glucose solution and how much water should be used to prepare 9 ounces of 10% glucose solution?
To solve this problem, we need to consider the amount of glucose in the 20% solution and the final desired concentration of the solution. Let's first calculate the amount of glucose in the 20% solution:
Amount of glucose = Concentration of glucose × Volume of the solution
= 0.2 × 9 ounces
= 1.8 ounces
Now, let's assume we need x ounces of the 20% glucose solution and y ounces of water to prepare 9 ounces of the 10% glucose solution. The total volume of the solution will be x + y ounces.
Since the glucose amount is preserved when diluting the solution, we can set up an equation for the amount of glucose in the final solution:
1.8 ounces of glucose = 0.1 (x + y) ounces
Now, we have two equations:
1. Equation for the volume of the solution:
x + y = 9 ounces
2. Equation for the amount of glucose:
1.8 ounces = 0.1 (x + y) ounces
We can solve this system of equations to find the values of x and y. Let's rearrange the first equation to solve for y:
y = 9 - x
Substitute this value of y in the second equation:
1.8 ounces = 0.1x + 0.1(9 - x)
1.8 ounces = 0.1x + 0.9 - 0.1x
Simplifying the equation:
1.8 ounces = 0.9 ounces
0 = 0.9 ounces - 1.8 ounces
This equation is not satisfied, and it means that the given scenario is not possible. There is no combination of the 20% glucose solution and water that can be mixed to obtain a 10% glucose solution.