There are some chocolates with Tom and Jerry. If Tom gives certain number of chocolates to Jerry, then the number of chocolates with them will be interchanged. Instead, if Jerry gives same number of chocolates to Tom, then the number of chocolates with jerry will be one-fourth of the number of chocolates that Tom has. If the total number of chocolates with them is 100, then find the number of chocolates with Tom

Answer

To solve the problem step by step, we will define variables for the number of chocolates Tom and Jerry have, set up equations based on the conditions given, and then solve those equations.

### Step 1: Define Variables
Let:
- \( x \) = number of chocolates Tom has
- \( y \) = number of chocolates Jerry has

From the problem, we know that the total number of chocolates is 100. Therefore, we can express Jerry's chocolates in terms of Tom's:
\[ y = 100 - x \]

### Step 2: Set Up the First Condition
According to the first condition, if Tom gives \( k \) chocolates to Jerry, their chocolates will be interchanged. This gives us the equations:
- Tom's chocolates after giving \( k \): \( x - k \)
- Jerry's chocolates after receiving \( k \): \( y + k \)

Since their chocolates are interchanged:
\[ x - k = y + k \]

### Step 3: Substitute \( y \) in the First Condition
Substituting \( y = 100 - x \) into the equation:
\[ x - k = (100 - x) + k \]
Simplifying this:
\[ x - k = 100 - x + k \]
\[ x + x - k - k = 100 \]
\[ 2x - 2k = 100 \]
Dividing by 2:
\[ x - k = 50 \quad \text{(Equation 1)} \]

### Step 4: Set Up the Second Condition
Now, according to the second condition, if Jerry gives the same \( k \) chocolates to Tom, then Jerry's chocolates will be one-fourth of what Tom has:
- Tom's chocolates after receiving \( k \): \( x + k \)
- Jerry's chocolates after giving \( k \): \( y - k \)

The condition states:
\[ y - k = \frac{1}{4}(x + k) \]

### Step 5: Substitute \( y \) in the Second Condition
Substituting \( y = 100 - x \) into the equation:
\[ (100 - x) - k = \frac{1}{4}(x + k) \]
Multiplying through by 4 to eliminate the fraction:
\[ 4(100 - x) - 4k = x + k \]
Expanding this:
\[ 400 - 4x - 4k = x + k \]
Rearranging gives:
\[ 400 - 4x - x - k = 4k \]
\[ 400 - 5x = 5k \]
Dividing by 5:
\[ 80 - x = k \quad \text{(Equation 2)} \]

### Step 6: Solve the Equations
Now we have two equations:
1. \( x - k = 50 \)
2. \( 80 - x = k \)

Substituting Equation 2 into Equation 1:
\[ x - (80 - x) = 50 \]
Simplifying:
\[ x - 80 + x = 50 \]
\[ 2x - 80 = 50 \]
\[ 2x = 130 \]
\[ x = 65 \]

### Step 7: Find \( y \)
Now substitute \( x \) back to find \( y \):
\[ y = 100 - x = 100 - 65 = 35 \]

### Final Answer
Thus, the number of chocolates with Tom is:
\[ \boxed{65} \]