Ten percent of the black balls were added to twenty percent of the white balls and the balls were 24. Three times the number of black balls exceeds the number of white balls by 20. Find the number of black balls and also, the number of white balls.

Answer

To solve the problem step by step, we will set up equations based on the information provided and then solve them.

### Step 1: Define the Variables
Let:
- \( x \) = number of black balls
- \( y \) = number of white balls

### Step 2: Set Up the First Equation
According to the problem, 10% of the black balls added to 20% of the white balls equals 24. We can write this as:
\[ 0.1x + 0.2y = 24 \]
To eliminate the decimals, we can multiply the entire equation by 10:
\[ x + 2y = 240 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
The problem states that three times the number of black balls exceeds the number of white balls by 20. This can be expressed as:
\[ 3x - y = 20 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
We now have a system of two equations:
1. \( x + 2y = 240 \)
2. \( 3x - y = 20 \)

To solve these equations, we can use the elimination method. First, we can multiply Equation 1 by 3 to align the coefficients of \( x \):
\[ 3(x + 2y) = 3(240) \]
This gives us:
\[ 3x + 6y = 720 \quad \text{(Equation 3)} \]

### Step 5: Subtract Equation 2 from Equation 3
Now we will subtract Equation 2 from Equation 3:
\[ (3x + 6y) - (3x - y) = 720 - 20 \]
This simplifies to:
\[ 3x + 6y - 3x + y = 700 \]
Thus, we have:
\[ 7y = 700 \]
Dividing both sides by 7 gives:
\[ y = 100 \]

### Step 6: Substitute \( y \) back to find \( x \)
Now that we have \( y \), we can substitute it back into Equation 1 to find \( x \):
\[ x + 2(100) = 240 \]
This simplifies to:
\[ x + 200 = 240 \]
Subtracting 200 from both sides gives:
\[ x = 40 \]

### Final Answer
The number of black balls is \( x = 40 \) and the number of white balls is \( y = 100 \).