Four friends, Ann, Bob, Charlie and David together have $200 . The amount with Charlie is twice of that with Bob . The amount with Ann and Bob together is $40 less than the amount with David. Charlie has $30 less than what David has. What is the amount with Ann?

Options

A. 20
B. 30
C. 40
D. 60

Answer

To solve the problem step by step, we will define variables for the amounts each friend has and set up equations based on the information given.

### Step 1: Define Variables
Let:
- Bob's amount = \( X \)
- Charlie's amount = \( 2X \) (since Charlie has twice the amount of Bob)
- Ann's amount = \( Y \)
- David's amount = \( Z \)

### Step 2: Set Up Equations
From the problem statement, we have the following information:
1. The total amount of money is $200:
\[ X + 2X + Y + Z = 200 \]
This simplifies to:
\[ 3X + Y + Z = 200 \quad \text{(Equation 1)} \]

2. The amount with Ann and Bob together is $40 less than the amount with David:
\[ Y + X = Z - 40 \quad \text{(Equation 2)} \]

3. Charlie has $30 less than David:
\[ 2X = Z - 30 \quad \text{(Equation 3)} \]

### Step 3: Solve for Z
From Equation 3, we can express \( Z \) in terms of \( X \):
\[ Z = 2X + 30 \]

### Step 4: Substitute Z in Equation 2
Now we substitute \( Z \) from Equation 3 into Equation 2:
\[ Y + X = (2X + 30) - 40 \]
This simplifies to:
\[ Y + X = 2X - 10 \]
Rearranging gives:
\[ Y = 2X - X - 10 \]
Thus,
\[ Y = X - 10 \quad \text{(Equation 4)} \]

### Step 5: Substitute Y and Z in Equation 1
Now substitute \( Y \) and \( Z \) in Equation 1:
\[ 3X + (X - 10) + (2X + 30) = 200 \]
This simplifies to:
\[ 3X + X - 10 + 2X + 30 = 200 \]
Combining like terms:
\[ 6X + 20 = 200 \]

### Step 6: Solve for X
Now, isolate \( X \):
\[ 6X = 200 - 20 \]
\[ 6X = 180 \]
\[ X = \frac{180}{6} = 30 \]

### Step 7: Find Y
Now that we have \( X \), we can find \( Y \):
\[ Y = X - 10 = 30 - 10 = 20 \]

### Conclusion
The amount with Ann is \( Y = 20 \).