Priyanka spends `2//15` of her income on clothes, `1//9` of her income on entertianment, `4//15` of her income on rent, `1//3` of income on food and she saves the remaining amount of ₹ `10, 500`. What iis the income of Priyanka ?

Explanation

To find Priyanka's income, we will follow these steps:

### Step 1: Define the total income
Let Priyanka's total income be represented as \( X \).

### Step 2: Calculate the fractions of income spent
1. **Clothes**: She spends \( \frac{2}{15}X \) on clothes.
2. **Entertainment**: She spends \( \frac{1}{9}X \) on entertainment.
3. **Rent**: She spends \( \frac{4}{15}X \) on rent.
4. **Food**: She spends \( \frac{1}{3}X \) on food.

### Step 3: Find a common denominator
To add these fractions, we need a common denominator. The denominators are 15, 9, and 3. The least common multiple (LCM) of these numbers is 45.

### Step 4: Convert each fraction to have the common denominator
1. **Clothes**: \( \frac{2}{15}X = \frac{2 \times 3}{15 \times 3}X = \frac{6}{45}X \)
2. **Entertainment**: \( \frac{1}{9}X = \frac{1 \times 5}{9 \times 5}X = \frac{5}{45}X \)
3. **Rent**: \( \frac{4}{15}X = \frac{4 \times 3}{15 \times 3}X = \frac{12}{45}X \)
4. **Food**: \( \frac{1}{3}X = \frac{1 \times 15}{3 \times 15}X = \frac{15}{45}X \)

### Step 5: Add all the expenditures
Now, we can add all the fractions:
\[ \text{Total Expenditure} = \frac{6}{45}X + \frac{5}{45}X + \frac{12}{45}X + \frac{15}{45}X = \frac{6 + 5 + 12 + 15}{45}X = \frac{38}{45}X \]

### Step 6: Calculate the remaining amount
Priyanka saves the remaining amount, which is ₹10,500. Therefore, we can set up the equation:
\[ X - \frac{38}{45}X = 10,500 \]

### Step 7: Simplify the equation
This can be simplified to:
\[ \frac{7}{45}X = 10,500 \]

### Step 8: Solve for \( X \)
To find \( X \), multiply both sides by \( \frac{45}{7} \):
\[ X = 10,500 \times \frac{45}{7} \]

### Step 9: Calculate the value
Calculating this gives:
\[ X = 10,500 \times \frac{45}{7} = 67,500 \]

### Conclusion
Priyanka's total income is ₹67,500.

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