Andy has twice as many marbles as Pandy, and Sandy has half as many has Andy and Pandy put together. If Andy has `75` marbles more than Sandy. How many does each of them have?

Answer

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

### Step 1: Define the variables
Let the number of marbles Pandy has be represented by \( x \).

### Step 2: Express Andy's marbles in terms of \( x \)
According to the question, Andy has twice as many marbles as Pandy. Therefore, we can express Andy's marbles as:
\[ \text{Andy’s marbles} = 2x \]

### Step 3: Express Sandy's marbles in terms of \( x \)
Sandy has half as many marbles as Andy and Pandy put together. The total number of marbles Andy and Pandy have together is:
\[ \text{Andy’s marbles} + \text{Pandy’s marbles} = 2x + x = 3x \]
Thus, Sandy's marbles can be expressed as:
\[ \text{Sandy’s marbles} = \frac{1}{2} \times (3x) = \frac{3x}{2} \]

### Step 4: Set up the equation based on the information about Andy and Sandy
The problem states that Andy has 75 marbles more than Sandy. This can be written as:
\[ 2x = \frac{3x}{2} + 75 \]

### Step 5: Solve the equation
To solve for \( x \), first, we will eliminate the fraction by multiplying the entire equation by 2:
\[ 2(2x) = 2\left(\frac{3x}{2} + 75\right) \]
This simplifies to:
\[ 4x = 3x + 150 \]

Now, subtract \( 3x \) from both sides:
\[ 4x - 3x = 150 \]
This simplifies to:
\[ x = 150 \]

### Step 6: Find the number of marbles each person has
Now that we have \( x \), we can find the number of marbles for each person.

- **Pandy's marbles**:
\[ \text{Pandy's marbles} = x = 150 \]

- **Andy’s marbles**:
\[ \text{Andy’s marbles} = 2x = 2 \times 150 = 300 \]

- **Sandy’s marbles**:
\[ \text{Sandy’s marbles} = \frac{3x}{2} = \frac{3 \times 150}{2} = \frac{450}{2} = 225 \]

### Final Answer:
- Pandy has 150 marbles.
- Andy has 300 marbles.
- Sandy has 225 marbles.