How many ounces of a solution that is 30 percent salt must be added to a 50-ounce solution that is 10 percent salt so that the resulting solution is 20 percent salt?

Options

A. 20
B. 30
C. 40
D. 50

Answer

To solve the problem step by step, we need to determine how many ounces of a 30% salt solution must be added to a 50-ounce solution that is 10% salt so that the resulting solution is 20% salt.

### Step 1: Define the variables
Let \( X \) be the number of ounces of the 30% salt solution that we need to add.

### Step 2: Calculate the total volume of the new solution
After adding \( X \) ounces of the 30% solution to the existing 50 ounces, the total volume of the solution becomes:
\[ X + 50 \text{ ounces} \]

### Step 3: Calculate the amount of salt in each solution
1. **Salt in the 30% solution**: The amount of salt in the \( X \) ounces of the 30% solution is:
\[ 0.3X \text{ ounces of salt} \]
2. **Salt in the 10% solution**: The amount of salt in the 50 ounces of the 10% solution is:
\[ 0.1 \times 50 = 5 \text{ ounces of salt} \]

### Step 4: Set up the equation for the resulting solution
The total amount of salt in the new solution (after adding both solutions) should equal the amount of salt in the resulting solution, which is 20% of the total volume:
\[ \text{Total salt} = 0.3X + 5 \]
The amount of salt in the resulting solution (which is 20% of the total volume \( X + 50 \)) is:
\[ 0.2(X + 50) \]

### Step 5: Set the equation
Now we can set up the equation:
\[ 0.3X + 5 = 0.2(X + 50) \]

### Step 6: Simplify and solve the equation
Expanding the right side:
\[ 0.3X + 5 = 0.2X + 10 \]
Now, subtract \( 0.2X \) from both sides:
\[ 0.3X - 0.2X + 5 = 10 \]
This simplifies to:
\[ 0.1X + 5 = 10 \]
Now, subtract 5 from both sides:
\[ 0.1X = 5 \]
Finally, divide by 0.1:
\[ X = \frac{5}{0.1} = 50 \]

### Conclusion
Thus, the number of ounces of the 30% salt solution that must be added is:
\[ \boxed{50} \]