Understanding Sanjay's Income Problem

This question asks us to figure out Sanjay's total income. We know how much he spends on two main categories: house rent and household expenses. A key detail is that the household expenses are calculated based on the amount left *after* paying the house rent. We are also given the final amount he saves, which is ₹1,800.

• Spending on house rent: 20% of total income.
• Spending on household expenses: 70% of the *remaining* income.
• Amount saved: ₹1,800.

Step-by-Step Income Calculation

Let's use a variable, say \(I\), to represent Sanjay's total income in rupees.

Step 1: Calculate House Rent Spending

Sanjay spends 20% of his total income on house rent. We can write this using LaTeX:

House Rent = \(20\% \times I = \frac{20}{100} \times I = 0.20 \times I\)

Step 2: Calculate Remaining Income After Rent

After paying the house rent, the income left is the total income minus the house rent spending.

Remaining Income = \(I - (0.20 \times I) = (1 - 0.20) \times I = 0.80 \times I\)

So, 80% of his total income is left after paying the house rent.

Step 3: Calculate Household Expenses

Sanjay spends 70% of this remaining income on household expenses.

Household Expenses = \(70\% \times (\text{Remaining Income})\)

Household Expenses = \(\frac{70}{100} \times (0.80 \times I) = 0.70 \times 0.80 \times I = 0.56 \times I\)

This means household expenses amount to 56% of his total income.

Step 4: Calculate Total Spending

His total spending is the sum of money spent on house rent and household expenses.

Total Spending = House Rent + Household Expenses

Total Spending = \((0.20 \times I) + (0.56 \times I) = (0.20 + 0.56) \times I = 0.76 \times I\)

So, Sanjay spends 76% of his total income in total.

Step 5: Relate Savings to Total Income

The amount Sanjay saves is what's left after all his spending. Savings = Total Income - Total Spending.

Savings = \(I - (0.76 \times I) = (1 - 0.76) \times I = 0.24 \times I\)

This shows that Sanjay saves 24% of his total income.

Step 6: Solve for Total Income

We are given that Sanjay saves ₹1,800. We can set up an equation using the savings calculated in the previous step.

\(0.24 \times I = 1800\)

To find the total income (\(I\)), we need to isolate \(I\):

\(I = \frac{1800}{0.24}\)

To make the division easier, we can multiply the numerator and denominator by 100:

\(I = \frac{1800 \times 100}{0.24 \times 100} = \frac{180000}{24}\)

Now, we simplify the fraction:

\(I = \frac{180000}{24} = 7500\)

Therefore, Sanjay's total income is ₹7,500.

Final Answer Verification

Let's check if this total income works with the given information:

• Total Income: ₹7,500
• House Rent (20% of ₹7,500): \(0.20 \times 7500 = ₹1,500\)
• Remaining Income after rent: \(7500 - 1500 = ₹6,000\)
• Household Expenses (70% of ₹6,000): \(0.70 \times 6000 = ₹4,200\)
• Total Spending: \(1500 + 4200 = ₹5,700\)
• Savings: Total Income - Total Spending = \(7500 - 5700 = ₹1,800\)

The calculated savings of ₹1,800 match the amount given in the question. This confirms our calculation for the total income is correct.