Beaker A contains sugаr solution with 18 percent sugar. beaker B contains sugar solution with 12 percent sugar. How much of each must he mixed together to get solution of 16 percent sugar weighing 240 gm of it?

Answer

To solve the problem of mixing two sugar solutions to achieve a desired concentration, we can follow these steps:

### Step 1: Define Variables
Let:
- \( x \) = weight of Beaker A (18% sugar solution)
- \( y \) = weight of Beaker B (12% sugar solution)

### Step 2: Set Up the First Equation
According to the problem, the total weight of the mixture is 240 grams. Therefore, we can write the first equation as:
\[ x + y = 240 \]

### Step 3: Set Up the Second Equation
Next, we need to account for the sugar content in the mixtures. The total sugar in the final solution should be equal to 16% of 240 grams. Thus, we can write the second equation as:
\[ 0.18x + 0.12y = 0.16 \times 240 \]

### Step 4: Simplify the Second Equation
Calculating \( 0.16 \times 240 \):
\[ 0.16 \times 240 = 38.4 \]
So the second equation becomes:
\[ 0.18x + 0.12y = 38.4 \]

### Step 5: Eliminate Decimals
To eliminate decimals, we can multiply the entire second equation by 100:
\[ 18x + 12y = 3840 \]

### Step 6: Rearrange the First Equation
From the first equation \( x + y = 240 \), we can express \( x \) in terms of \( y \):
\[ x = 240 - y \]

### Step 7: Substitute in the Second Equation
Now, substitute \( x = 240 - y \) into the second equation:
\[ 18(240 - y) + 12y = 3840 \]

### Step 8: Distribute and Combine Like Terms
Distributing gives:
\[ 4320 - 18y + 12y = 3840 \]
Combine like terms:
\[ 4320 - 6y = 3840 \]

### Step 9: Solve for \( y \)
Rearranging gives:
\[ -6y = 3840 - 4320 \]
\[ -6y = -480 \]
Dividing both sides by -6:
\[ y = 80 \]

### Step 10: Solve for \( x \)
Now substitute \( y = 80 \) back into the first equation:
\[ x + 80 = 240 \]
\[ x = 240 - 80 \]
\[ x = 160 \]

### Conclusion
The weights of the solutions to be mixed are:
- Weight of Beaker A (18% solution) = 160 grams
- Weight of Beaker B (12% solution) = 80 grams