Dr. Bhatia mixes 2 liters of a [tex]$5 \%$[/tex] saline solution with [tex]$x$[/tex] liters of a 9% saline solution. The table shows the amount of each solution used.
Saline Solution
Amount (liters)
Concentration
Total
[tex]$5\%$[/tex] Saline
2
0.05
0.1
[tex]$9\%$[/tex] Saline
[tex]$x$[/tex]
0.09
0.09x
Mixture
[tex]$2+x$[/tex]
0.08
0.64
What is the value of [tex]$x$[/tex]?
A. 3
B. 4
C. 5
D. 6
Answer (1)
Calculate the amount of saline in the 5% solution: 2 × 0.05 = 0.1 .
Calculate the amount of saline in the 9% solution: 0.09 x .
Set up the equation: 0.1 + 0.09 x = 0.08 ( 2 + x ) .
Solve for x : x = 6 . The value of x is 6 .
Explanation
Understanding the Problem We are given a mixture problem where 2 liters of a 5% saline solution are mixed with x liters of a 9% saline solution. The resulting mixture is ( 2 + x ) liters of an 8% saline solution. We need to find the value of x .
Calculating Saline Amounts The amount of saline in the 5% solution is 2 × 0.05 = 0.1 liters. The amount of saline in the 9% solution is x × 0.09 = 0.09 x liters. The amount of saline in the mixture is ( 2 + x ) × 0.08 liters.
Setting up the Equation The total amount of saline in the mixture is the sum of the amounts of saline in the two original solutions. Therefore, we can set up the following equation: 0.1 + 0.09 x = 0.08 ( 2 + x ) Now, we solve for x .
Solving for x Expanding the right side of the equation, we get: 0.1 + 0.09 x = 0.16 + 0.08 x Subtracting 0.08 x from both sides, we have: 0.1 + 0.01 x = 0.16 Subtracting 0.1 from both sides, we get: 0.01 x = 0.06 Dividing both sides by 0.01 , we find: x = 0.01 0.06 = 6
Final Answer Therefore, the value of x is 6.
Examples
Mixture problems are commonly used in chemistry and pharmacy to determine the correct proportions of different solutions to achieve a desired concentration. For example, a pharmacist might need to mix different concentrations of a drug to create a specific dosage for a patient. Similarly, in cooking, you might mix different ingredients with varying concentrations of sugar or salt to achieve the desired taste.
