Find Laspeyre's, Paasche's, and Dorbish-Bowley's price index numbers for the following data.

\begin{tabular}{|c|c|c|c|c|}
\hline
\multirow{2}{*}{Item} & \multicolumn{2}{|c|}{2004} & \multicolumn{2}{c|}{2008} \\
\cline{2-5}
& Price (Rs.) & Quantity & Price (Rs.) & Quantity \\
\hline
A & 10 & 5 & 12 & 4 \\
\hline
B & 15 & 8 & 18 & 7 \\
\hline
C & 6 & 3 & 4 & 5 \\
\hline
D & 3 & 4 & 3 & 5 \\
\hline
\end{tabular}

Certainly! Let's find the Laspeyre's, Paasche's, and Dorbish-Bowley's price index numbers for the given data step by step.

### Data Analysis

We have the following data for items A, B, C, and D:

#### 2004:
- Price (Rs.): [tex]\( P_0 \)[/tex]
- Quantity: [tex]\( Q_0 \)[/tex]

| Item | Price (P_0) | Quantity (Q_0) |
|------|-------------|----------------|
| A | 10 | 5 |
| B | 15 | 8 |
| C | 6 | 3 |
| D | 3 | 4 |

#### 2008:
- Price (Rs.): [tex]\( P_1 \)[/tex]
- Quantity: [tex]\( Q_1 \)[/tex]

| Item | Price (P_1) | Quantity (Q_1) |
|------|-------------|----------------|
| A | 12 | 4 |
| B | 18 | 7 |
| C | 4 | 5 |
| D | 3 | 5 |

### Laspeyre's Price Index

Laspeyre's Price Index is calculated using the formula:
[tex]\[ \text{Laspeyres Index} = \left( \frac{\sum (P_1 \times Q_0)}{\sum (P_0 \times Q_0)} \right) \times 100 \][/tex]

1. Numerator [tex]\( \sum (P_1 \times Q_0) \)[/tex]:
- [tex]\( (12 \times 5) + (18 \times 8) + (4 \times 3) + (3 \times 4) = 60 + 144 + 12 + 12 = 228 \)[/tex]

2. Denominator [tex]\( \sum (P_0 \times Q_0) \)[/tex]:
- [tex]\( (10 \times 5) + (15 \times 8) + (6 \times 3) + (3 \times 4) = 50 + 120 + 18 + 12 = 200 \)[/tex]

3. Laspeyres Index:
- [tex]\[ \text{Laspeyres Index} = \left( \frac{228}{200} \right) \times 100 = 114 \% \][/tex]

### Paasche's Price Index

Paasche's Price Index is calculated using the formula:
[tex]\[ \text{Paasche Index} = \left( \frac{\sum (P_1 \times Q_1)}{\sum (P_0 \times Q_1)} \right) \times 100 \][/tex]

1. Numerator [tex]\( \sum (P_1 \times Q_1) \)[/tex]:
- [tex]\( (12 \times 4) + (18 \times 7) + (4 \times 5) + (3 \times 5) = 48 + 126 + 20 + 15 = 209 \)[/tex]

2. Denominator [tex]\( \sum (P_0 \times Q_1) \)[/tex]:
- [tex]\( (10 \times 4) + (15 \times 7) + (6 \times 5) + (3 \times 5) = 40 + 105 + 30 + 15 = 190 \)[/tex]

3. Paasche Index:
- [tex]\[ \text{Paasche Index} = \left( \frac{209}{190} \right) \times 100 = 110 \% \][/tex]

### Dorbish-Bowley's Price Index

Dorbish-Bowley's Price Index is calculated as the simple average of Laspeyres and Paasche indices:
[tex]\[ \text{Dorbish-Bowley Index} = \frac{\text{Laspeyres Index} + \text{Paasche Index}}{2} \][/tex]

- Dorbish-Bowley Index:
- [tex]\[ \text{Dorbish-Bowley Index} = \frac{114 + 110}{2} = 112 \% \][/tex]

### Summary

The price index numbers based on the given data are:
- Laspeyre's Price Index: [tex]\( 114 \% \)[/tex]
- Paasche's Price Index: [tex]\( 110 \% \)[/tex]
- Dorbish-Bowley's Price Index: [tex]\( 112 \% \)[/tex]