A, B and C have 40, x and y balls with them respectively. If B gives 20 balls to A,. He is left with half as many balls as C. If together they had 60 more balls, each of them would have had 100 balls on an average. What is the value of `x:y`?

Options

A. `3:2`
B. `2:3`
C. `2:1`
D. `3:4`

Answer

To solve the problem step by step, we will set up equations based on the information provided in the question.

### Step 1: Set up the initial conditions
Let:
- A has 40 balls
- B has x balls
- C has y balls

### Step 2: Analyze the first condition
When B gives 20 balls to A:
- A will have: \( 40 + 20 = 60 \) balls
- B will have: \( x - 20 \) balls

According to the problem, after giving away the balls, B is left with half as many balls as C:
\[ x - 20 = \frac{y}{2} \]
Multiplying both sides by 2 to eliminate the fraction:
\[ 2(x - 20) = y \]
Expanding this gives:
\[ 2x - 40 = y \quad \text{(Equation 1)} \]

### Step 3: Analyze the second condition
The problem states that if together they had 60 more balls, each would have had 100 balls on average. Therefore, the total number of balls they would have is:
\[ 40 + x + y + 60 = 300 \]
This simplifies to:
\[ x + y + 100 = 300 \]
Thus, we can rearrange this to find:
\[ x + y = 200 \quad \text{(Equation 2)} \]

### Step 4: Solve the equations
Now we have two equations:
1. \( 2x - y = 40 \)
2. \( x + y = 200 \)

We can solve these equations simultaneously. From Equation 2, we can express y in terms of x:
\[ y = 200 - x \]
Now, substitute this expression for y into Equation 1:
\[ 2x - (200 - x) = 40 \]
Simplifying this gives:
\[ 2x - 200 + x = 40 \]
Combining like terms:
\[ 3x - 200 = 40 \]
Adding 200 to both sides:
\[ 3x = 240 \]
Dividing by 3:
\[ x = 80 \]

### Step 5: Find the value of y
Now substitute \( x = 80 \) back into Equation 2 to find y:
\[ 80 + y = 200 \]
Subtracting 80 from both sides:
\[ y = 120 \]

### Step 6: Find the ratio x:y
Now we can find the ratio \( x:y \):
\[ x:y = 80:120 \]
Simplifying this ratio by dividing both sides by 40:
\[ x:y = 2:3 \]

### Final Answer
The value of \( x:y \) is \( 2:3 \).