Four friends Arun, Amit, Ankur and Ajay went a shop. Arun's total money was found to be `1"/"9` times the sum of the rest. Amit's total money was found to be `1"/"4` times the sum of the rest while Ajay's total money was found to be `2"/"3` times the sum of the rest. What percentage of the total money belonged to Ankur?

Options

A. `36%`
B. `40%`
C. `25%`
D. `30%`

Answer

To solve the problem step by step, we need to set up equations based on the information provided about the amounts of money each friend has.

### Step 1: Define Variables
Let:
- Arun's money = \( X \)
- Amit's money = \( Y \)
- Ankur's money = \( W \)
- Ajay's money = \( Z \)

### Step 2: Set Up Equations Based on Given Conditions
1. **Arun's Condition**: Arun's total money is \( \frac{1}{9} \) times the sum of the rest.
\[ X = \frac{1}{9}(Y + Z + W) \]
Multiplying both sides by 9 gives:
\[ 9X = Y + Z + W \quad \text{(Equation 1)} \]

2. **Amit's Condition**: Amit's total money is \( \frac{1}{4} \) times the sum of the rest.
\[ Y = \frac{1}{4}(X + Z + W) \]
Multiplying both sides by 4 gives:
\[ 4Y = X + Z + W \quad \text{(Equation 2)} \]

3. **Ajay's Condition**: Ajay's total money is \( \frac{2}{3} \) times the sum of the rest.
\[ Z = \frac{2}{3}(X + Y + W) \]
Multiplying both sides by 3 gives:
\[ 3Z = 2(X + Y + W) \quad \text{(Equation 3)} \]

### Step 3: Rearranging the Equations
From Equation 1:
\[ Y + Z + W = 9X \]

From Equation 2:
\[ X + Z + W = 4Y \]

From Equation 3:
\[ 3Z = 2(X + Y + W) \]

### Step 4: Substitute and Solve
Now we can express \( Y \), \( Z \), and \( W \) in terms of \( X \).

1. From Equation 1, we can express \( W \):
\[ W = 9X - Y - Z \]

2. Substitute \( W \) into Equation 2:
\[ X + Z + (9X - Y - Z) = 4Y \]
Simplifying gives:
\[ 10X - Y = 4Y \]
Rearranging gives:
\[ 10X = 5Y \quad \Rightarrow \quad Y = 2X \quad \text{(Equation 4)} \]

3. Substitute \( Y \) into Equation 1:
\[ W = 9X - 2X - Z = 7X - Z \]

4. Substitute \( W \) into Equation 3:
\[ 3Z = 2(X + 2X + (7X - Z)) \]
Simplifying gives:
\[ 3Z = 2(10X - Z) \quad \Rightarrow \quad 3Z + 2Z = 20X \quad \Rightarrow \quad 5Z = 20X \quad \Rightarrow \quad Z = 4X \quad \text{(Equation 5)} \]

### Step 5: Find \( W \)
Now substitute \( Z \) back into the equation for \( W \):
\[ W = 7X - 4X = 3X \]

### Step 6: Calculate Total Money
Now we have:
- \( X \) (Arun) = \( X \)
- \( Y \) (Amit) = \( 2X \)
- \( Z \) (Ajay) = \( 4X \)
- \( W \) (Ankur) = \( 3X \)

Total money:
\[ \text{Total} = X + 2X + 4X + 3X = 10X \]

### Step 7: Calculate Percentage of Ankur's Money
Ankur's money is \( W = 3X \). The percentage of the total money that Ankur has is:
\[ \text{Percentage} = \left( \frac{W}{\text{Total}} \right) \times 100 = \left( \frac{3X}{10X} \right) \times 100 = 30\% \]

### Final Answer
Ankur has **30%** of the total money.

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