In a city, the total income of all people with salary below Rs. 10000 per annum is less than the total income of all people with salary above Rs. 10000 per annum. If the salaries of people in the first group increases by 5% and the salaries of people in the second group decreases by 5% then the average income of all people

Options

A. increases
B. decreases
C. remains the same
D. cannot be determined from the data

Answer

To solve the problem step by step, we will analyze the situation mathematically.

### Step 1: Define the Groups and Their Salaries
Let:
- Group 1 (people with salary below Rs. 10,000):
- Number of people = \( x \)
- Average salary = \( a \) (where \( a < 10,000 \))
- Total income of Group 1 = \( XA \)

- Group 2 (people with salary above Rs. 10,000):
- Number of people = \( y \)
- Average salary = \( b \) (where \( b > 10,000 \))
- Total income of Group 2 = \( yb \)

### Step 2: Understand the Given Condition
According to the problem, the total income of Group 1 is less than the total income of Group 2:
\[ xa < yb \]

### Step 3: Calculate the New Incomes After Changes
- For Group 1, the salaries increase by 5%:
- New total income of Group 1 = \( xa + 0.05(xa) = xa(1 + 0.05) = 1.05xa \)

- For Group 2, the salaries decrease by 5%:
- New total income of Group 2 = \( yb - 0.05(yb) = yb(1 - 0.05) = 0.95yb \)

### Step 4: Calculate the New Average Income
The new total income of all people combined is:
\[ \text{New Total Income} = 1.05xa + 0.95yb \]

The total number of people is:
\[ \text{Total Number of People} = x + y \]

Thus, the new average income is:
\[ \text{New Average Income} = \frac{1.05xa + 0.95yb}{x + y} \]

### Step 5: Analyze the Impact on Average Income
We need to determine whether the new average income is greater than, less than, or equal to the old average income:
- Old total income = \( xa + yb \)
- Old average income = \( \frac{xa + yb}{x + y} \)

Now, we compare:
- New total income = \( 1.05xa + 0.95yb \)
- Old total income = \( xa + yb \)

### Step 6: Compare the New and Old Totals
To see if the average increases or decreases, we can compare:
\[ 1.05xa + 0.95yb \quad \text{and} \quad xa + yb \]

Rearranging gives:
\[ (1.05xa + 0.95yb) - (xa + yb) = 0.05xa - 0.05yb \]
\[ = 0.05(xa - yb) \]

Since we know from the earlier condition that \( xa < yb \), it follows that:
\[ xa - yb < 0 \]

Thus:
\[ 0.05(xa - yb) < 0 \]

This means:
\[ 1.05xa + 0.95yb < xa + yb \]

### Conclusion
The new average income is less than the old average income. Therefore, the average income of all people decreases.