Answer :
Certainly! Let's go through the detailed, step-by-step solution to determine how much the textile worker spent on house rent and clothing.
### Step 1: Understand the Problem
We are given the following information:
- Monthly income: Rs. 3500
- Cost of living index: 136
- Expenditures on food, fuel and lighting, and miscellaneous are given as Rs. 1400, Rs. 560, and Rs. 630, respectively.
- Group indices for food, clothing, house rent, fuel and lighting, and miscellaneous are given as 180, 150, 100, 110, and 80, respectively.
We need to find out how much the worker spent on house rent and clothing.
### Step 2: Calculate the Proportion of Cost of Living Index
The cost of living index proportion is calculated as:
[tex]\[ \text{Proportion} = \frac{\text{Cost of Living Index}}{100} = \frac{136}{100} = 1.36 \][/tex]
### Step 3: Establish Known Total Expenditures without Clothing and House Rent
The total expenditures on food, fuel and lighting, and miscellaneous can be summed up:
[tex]\[ \text{Total known expenditures} = 1400 + 560 + 630 = 2590 \][/tex]
### Step 4: Calculate the Adjusted Total Expenditures
To find out the actual total expenditure, we divide the total known expenditures by the cost of living proportion:
[tex]\[ \text{Total actual expenditures (excluding clothing and house rent)} = \frac{2590}{1.36} \approx 1904.41 \][/tex]
Since this is less than the total monthly income, we need to calculate the estimated expenditure which includes the uncalculated categories (house rent and clothing).
### Step 5: Determine the Total Actual Expenditure Should be Adjusted by the Cost of Living Proportion
The total amount that should be spent given the cost of living index is:
[tex]\[ \text{Total adjusted expenditures} = 3500 \times 1.36 = 4760 \][/tex]
### Step 6: Calculate the Remaining Expenditures
The difference between the monthly income and the known expenditures gives us the budget left for house rent and clothing:
[tex]\[ \text{Estimated total for house rent and clothing} = 3500 - 2590 = 910 \][/tex]
### Step 7: Distribute the Remaining Expenditures According to Indices
First, we sum up the group indices:
[tex]\[ \text{Sum of Group Indices} = 180 + 150 + 100 + 110 + 80 = 620 \][/tex]
Then calculate the proportions for clothing and house rent based on their respective group indices:
[tex]\[ \text{Proportion for clothing} = \frac{150}{620} \approx 0.2419 \][/tex]
[tex]\[ \text{Proportion for house rent} = \frac{100}{620} \approx 0.1613 \][/tex]
### Step 8: Calculate the Expenditure on Clothing
The expenditure on clothing, based on its proportion, is:
[tex]\[ \text{Expenditure on clothing} = 0.2419 \times 910 \approx 220.16 \][/tex]
### Step 9: Calculate the Expenditure on House Rent
The expenditure on house rent, based on its proportion, is:
[tex]\[ \text{Expenditure on house rent} = 0.1613 \times 910 \approx 146.77 \][/tex]
### Conclusion
The amounts spent by the textile worker on house rent and clothing are as follows:
- House Rent: Rs. 146.77
- Clothing: Rs. 220.16
These subdivisions ensure that the total expenditures align correctly with the given monthly income and the cost of living index.
### Step 1: Understand the Problem
We are given the following information:
- Monthly income: Rs. 3500
- Cost of living index: 136
- Expenditures on food, fuel and lighting, and miscellaneous are given as Rs. 1400, Rs. 560, and Rs. 630, respectively.
- Group indices for food, clothing, house rent, fuel and lighting, and miscellaneous are given as 180, 150, 100, 110, and 80, respectively.
We need to find out how much the worker spent on house rent and clothing.
### Step 2: Calculate the Proportion of Cost of Living Index
The cost of living index proportion is calculated as:
[tex]\[ \text{Proportion} = \frac{\text{Cost of Living Index}}{100} = \frac{136}{100} = 1.36 \][/tex]
### Step 3: Establish Known Total Expenditures without Clothing and House Rent
The total expenditures on food, fuel and lighting, and miscellaneous can be summed up:
[tex]\[ \text{Total known expenditures} = 1400 + 560 + 630 = 2590 \][/tex]
### Step 4: Calculate the Adjusted Total Expenditures
To find out the actual total expenditure, we divide the total known expenditures by the cost of living proportion:
[tex]\[ \text{Total actual expenditures (excluding clothing and house rent)} = \frac{2590}{1.36} \approx 1904.41 \][/tex]
Since this is less than the total monthly income, we need to calculate the estimated expenditure which includes the uncalculated categories (house rent and clothing).
### Step 5: Determine the Total Actual Expenditure Should be Adjusted by the Cost of Living Proportion
The total amount that should be spent given the cost of living index is:
[tex]\[ \text{Total adjusted expenditures} = 3500 \times 1.36 = 4760 \][/tex]
### Step 6: Calculate the Remaining Expenditures
The difference between the monthly income and the known expenditures gives us the budget left for house rent and clothing:
[tex]\[ \text{Estimated total for house rent and clothing} = 3500 - 2590 = 910 \][/tex]
### Step 7: Distribute the Remaining Expenditures According to Indices
First, we sum up the group indices:
[tex]\[ \text{Sum of Group Indices} = 180 + 150 + 100 + 110 + 80 = 620 \][/tex]
Then calculate the proportions for clothing and house rent based on their respective group indices:
[tex]\[ \text{Proportion for clothing} = \frac{150}{620} \approx 0.2419 \][/tex]
[tex]\[ \text{Proportion for house rent} = \frac{100}{620} \approx 0.1613 \][/tex]
### Step 8: Calculate the Expenditure on Clothing
The expenditure on clothing, based on its proportion, is:
[tex]\[ \text{Expenditure on clothing} = 0.2419 \times 910 \approx 220.16 \][/tex]
### Step 9: Calculate the Expenditure on House Rent
The expenditure on house rent, based on its proportion, is:
[tex]\[ \text{Expenditure on house rent} = 0.1613 \times 910 \approx 146.77 \][/tex]
### Conclusion
The amounts spent by the textile worker on house rent and clothing are as follows:
- House Rent: Rs. 146.77
- Clothing: Rs. 220.16
These subdivisions ensure that the total expenditures align correctly with the given monthly income and the cost of living index.
