The index number of the following data for the year 2019, taking 2018 as base year was found to be 116. The simple aggregate method was used for calculation. Find numerical value of a, b, if the sum of prices is, the year 2019 is ₹ 203.
`{:("Item","Price (in ₹) for year 2019","Price in ₹ for year 2018"),(A,30,10),(B,25,20),(C,45,25),(D,15,30),(E,35,a),(F,b,50):}`

Answer

To solve the problem, we need to find the values of \( a \) and \( b \) given the index number for the year 2019 and the prices for the year 2018. We will use the simple aggregate method for index number calculation.

### Step-by-Step Solution:

1. **Understanding the Given Information:**
- The index number for 2019 (taking 2018 as the base year) is given as 116.
- The sum of prices for the year 2019 is ₹ 203.
- The prices for the year 2018 are provided for items A, B, C, D, E, and F.

2. **Using the Formula for Simple Aggregate Index:**
The formula for the simple aggregate index number is:
\[ P01 = \frac{\sum P1}{\sum P0} \times 100 \]
Where:
- \( P01 \) is the index number (116 in this case).
- \( \sum P1 \) is the sum of prices in the current year (2019), which is ₹ 203.
- \( \sum P0 \) is the sum of prices in the base year (2018).

3. **Setting Up the Equation:**
From the formula, we can rearrange it to find \( \sum P0 \):
\[ 116 = \frac{203}{\sum P0} \times 100 \]
Rearranging gives:
\[ \sum P0 = \frac{203 \times 100}{116} \]

4. **Calculating \( \sum P0 \):**
\[ \sum P0 = \frac{20300}{116} \approx 175 \]

5. **Finding the Value of \( a \):**
The sum of prices in the base year (2018) is:
\[ 10 + 20 + 25 + 30 + a + 50 = 135 + a \]
Setting this equal to \( \sum P0 \):
\[ 135 + a = 175 \]
Solving for \( a \):
\[ a = 175 - 135 = 40 \]

6. **Finding the Value of \( b \):**
The sum of prices in the current year (2019) is:
\[ 30 + 25 + 45 + 15 + 35 + b = 150 + b \]
Setting this equal to \( \sum P1 \):
\[ 150 + b = 203 \]
Solving for \( b \):
\[ b = 203 - 150 = 53 \]

### Final Values:
- \( a = 40 \)
- \( b = 53 \)