If `sum Iomega = 3510 , sum omega =23+ x ` and index number is 135, find the value of x.

Answer

To solve the problem, we will use the given information and the formula for the index number. Here are the steps to find the value of \( x \):

### Step 1: Write down the formula for the index number.
The formula for the index number (cost of living index) is given by:

\[ \text{Index} = \frac{\sum I\omega}{\sum \omega} \]

### Step 2: Substitute the given values into the formula.
From the question, we have:

- \(\sum I\omega = 3510\)
- \(\sum \omega = 23 + x\)
- Index = 135

Substituting these values into the formula gives:

\[ 135 = \frac{3510}{23 + x} \]

### Step 3: Cross multiply to eliminate the fraction.
Cross multiplying gives us:

\[ 135 \times (23 + x) = 3510 \]

### Step 4: Distribute the 135.
Distributing \( 135 \) on the left side results in:

\[ 135 \times 23 + 135x = 3510 \]

Calculating \( 135 \times 23 \):

\[ 3105 + 135x = 3510 \]

### Step 5: Isolate \( x \).
Now, we will isolate \( x \) by subtracting \( 3105 \) from both sides:

\[ 135x = 3510 - 3105 \]

Calculating the right side:

\[ 135x = 405 \]

### Step 6: Solve for \( x \).
Now, divide both sides by \( 135 \):

\[ x = \frac{405}{135} \]

Calculating this gives:

\[ x = 3 \]

### Final Answer:
The value of \( x \) is \( 3 \).

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