Out of her income, Dhara spends `1/8` th on rent and `1/4` th on other expenses. From remaining she keeps Rs x in her bank account and the rest she keeps as cash. Out of her income Amu spends `1/6` th on rent and `1/3` rd on other expenses. From remaining she keeps Rs x in her bank account and the rest she keeps as cash. Amount kept as cash with Dhara was equal to that kept with Amu. What is the respective ratio between the incomes of Dhara and Amu?

Options

A. `2:5`
B. `4:5`
C. `5:6`
D. None of these

Answer

To solve the problem, we will denote Dhara's income as \( D \) and Amu's income as \( A \).

### Step 1: Calculate Dhara's expenses
Dhara spends \( \frac{1}{8} \) of her income on rent and \( \frac{1}{4} \) on other expenses.

- Rent:
\[ \text{Rent} = \frac{1}{8} D \]
- Other expenses:
\[ \text{Other expenses} = \frac{1}{4} D = \frac{2}{8} D \]

### Step 2: Calculate Dhara's total expenses
Now, we can calculate Dhara's total expenses:
\[ \text{Total expenses} = \text{Rent} + \text{Other expenses} = \frac{1}{8} D + \frac{2}{8} D = \frac{3}{8} D \]

### Step 3: Calculate Dhara's remaining income
The remaining income after expenses is:
\[ \text{Remaining income} = D - \text{Total expenses} = D - \frac{3}{8} D = \frac{5}{8} D \]

### Step 4: Calculate Dhara's cash and bank savings
From the remaining income, Dhara keeps Rs \( x \) in her bank account. Therefore, the cash she keeps is:
\[ \text{Cash with Dhara} = \frac{5}{8} D - x \]

### Step 5: Calculate Amu's expenses
Amu spends \( \frac{1}{6} \) of her income on rent and \( \frac{1}{3} \) on other expenses.

- Rent:
\[ \text{Rent} = \frac{1}{6} A \]
- Other expenses:
\[ \text{Other expenses} = \frac{1}{3} A = \frac{2}{6} A \]

### Step 6: Calculate Amu's total expenses
Now, we can calculate Amu's total expenses:
\[ \text{Total expenses} = \text{Rent} + \text{Other expenses} = \frac{1}{6} A + \frac{2}{6} A = \frac{3}{6} A = \frac{1}{2} A \]

### Step 7: Calculate Amu's remaining income
The remaining income after expenses is:
\[ \text{Remaining income} = A - \text{Total expenses} = A - \frac{1}{2} A = \frac{1}{2} A \]

### Step 8: Calculate Amu's cash and bank savings
From the remaining income, Amu keeps Rs \( x \) in her bank account. Therefore, the cash she keeps is:
\[ \text{Cash with Amu} = \frac{1}{2} A - x \]

### Step 9: Set the cash amounts equal
According to the problem, the cash kept by Dhara is equal to the cash kept by Amu:
\[ \frac{5}{8} D - x = \frac{1}{2} A - x \]

### Step 10: Solve for the ratio of incomes
By simplifying the equation, we can eliminate \( x \):
\[ \frac{5}{8} D = \frac{1}{2} A \]
To eliminate the fractions, multiply both sides by 8:
\[ 5D = 4A \]
Thus, the ratio of Dhara's income to Amu's income is:
\[ \frac{D}{A} = \frac{4}{5} \]

### Final Answer
The respective ratio between the incomes of Dhara and Amu is \( 4:5 \).