A family's ratio of savings to expenditure for last month was 2 : 13. This month, due to unforeseen expenditure, savings fell to ` 50% ` of the amount saved last month. Salary, last month was Rs 10,000. This month there was an increase of `15%` in the salary. How much did the family spend this month?

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Ratio of Savings to Expenditure
The ratio of savings to expenditure is given as 2:13. This means that for every 2 parts of savings, there are 13 parts of expenditure.

### Step 2: Calculate the Total Units
The total parts in the ratio is \(2 + 13 = 15\) units.

### Step 3: Calculate the Value of One Unit
The total salary last month was Rs 10,000. To find the value of one unit:
\[ \text{Value of one unit} = \frac{\text{Total Salary}}{\text{Total Units}} = \frac{10,000}{15} = \frac{2000}{3} \text{ Rs} \]

### Step 4: Calculate Last Month's Savings
Since savings are 2 units:
\[ \text{Savings} = 2 \times \frac{2000}{3} = \frac{4000}{3} \text{ Rs} \]

### Step 5: Calculate Last Month's Expenditure
Since expenditure is 13 units:
\[ \text{Expenditure} = 13 \times \frac{2000}{3} = \frac{26000}{3} \text{ Rs} \]

### Step 6: Calculate This Month's Salary
This month, the salary increased by 15%. Therefore:
\[ \text{New Salary} = 10,000 \times 1.15 = 11,500 \text{ Rs} \]

### Step 7: Calculate This Month's Savings
This month, savings fell to 50% of last month's savings:
\[ \text{New Savings} = 0.5 \times \frac{4000}{3} = \frac{2000}{3} \text{ Rs} \]

### Step 8: Calculate This Month's Expenditure
To find the expenditure for this month, subtract the new savings from the new salary:
\[ \text{Expenditure} = \text{New Salary} - \text{New Savings} = 11,500 - \frac{2000}{3} \]
To perform the subtraction, convert 11,500 into a fraction:
\[ 11,500 = \frac{34,500}{3} \]
Now, perform the subtraction:
\[ \text{Expenditure} = \frac{34,500}{3} - \frac{2000}{3} = \frac{34,500 - 2000}{3} = \frac{32,500}{3} \text{ Rs} \]

### Step 9: Convert to Decimal
To express the expenditure in decimal form:
\[ \frac{32,500}{3} \approx 10,833.33 \text{ Rs} \]

### Final Answer
The family spent approximately Rs 10,833.33 this month.

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