A pharmacist has 80 milliliters of a $25 \%$ saline solution. Which expression can be used to find the percentage concentration of the final solution if she adds $x$ milliliters of a $10 \%$ saline solution?
$y=\frac{0.25(80)+0.1 x}{x} \cdot 100$
$y=\frac{0.25(80)+0.1 x}{80} \cdot 100$
$y=\frac{0.25(80)+0.1 x}{80+0.1 x} \cdot 100$
$y=\frac{0.25(80)+0.1 x}{80+x} \cdot 100$
Answer (2)
Calculate the amount of saline in the initial solution: 0.25 × 80 = 20 ml.
Calculate the amount of saline added: 0.10 × x = 0.1 x ml.
Calculate the total amount of saline in the final solution: 20 + 0.1 x ml.
Calculate the total volume of the final solution: 80 + x ml.
Calculate the percentage concentration of the final solution: 80 + x 0.25 ( 80 ) + 0.1 x × 100 .
Explanation
Problem Analysis Let's analyze the problem. We have an initial saline solution and we are adding another saline solution to it. We need to find the expression that represents the percentage concentration of the final solution.
Saline in Initial Solution The initial solution has a volume of 80 ml and a concentration of 25%. The amount of saline in the initial solution is:
Calculating Initial Saline 0.25 × 80 = 20 ml
Saline Added We are adding x ml of a 10% saline solution. The amount of saline added is:
Calculating Added Saline 0.10 × x = 0.1 x ml
Total Saline The total amount of saline in the final solution is the sum of the saline in the initial solution and the saline added:
Calculating Total Saline 20 + 0.1 x ml
Total Volume The total volume of the final solution is the sum of the initial volume and the volume added:
Calculating Total Volume 80 + x ml
Final Concentration The concentration of the final solution is the total amount of saline divided by the total volume:
Calculating Concentration 80 + x 20 + 0.1 x
Percentage Concentration To express the concentration as a percentage, we multiply by 100:
Calculating Percentage 80 + x 20 + 0.1 x × 100 = 80 + x 0.25 ( 80 ) + 0.1 x × 100
Final Expression Therefore, the expression for the percentage concentration of the final solution is:
Final Answer y = 80 + x 0.25 ( 80 ) + 0.1 x × 100
Examples
Imagine you're mixing different fruit juices to create a new blend. You start with 80 ml of a juice that's 25% sugar, and you want to add some juice that's only 10% sugar. The expression we found helps you calculate the final sugar concentration of your blend, depending on how much of the 10% juice you add. This is useful in cooking, chemistry, or any situation where you're mixing solutions with different concentrations and need to know the final concentration.
The correct expression to find the percentage concentration of the final saline solution is y = 80 + x 0.25 ( 80 ) + 0.1 x × 100 . This accounts for both the saline from the initial solution and the added saline solution. Therefore, the answer choice is clearly defined as the third option in the list provided.
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