Explanation:

Laspeyres' index and Paasche's index are two methods of calculating price indices. Laspeyres' index uses the base year quantities, while Paasche's index uses the current year quantities.

Calculation of Laspeyre's Index:

To calculate Laspeyres' index, we use the base year quantities. The formula for Laspeyres' index is:

Laspeyres' index = (Sum of base year prices x base year quantities) / (Sum of base year prices x base year quantities)

Here, the sum of base year prices = 10 x 2 + 5 x 1 = 20 + 5 = 25

Laspeyres' index = (25 x 10) / 25 = 10

Calculation of Paasche's Index:

To calculate Paasche's index, we use the current year quantities. The formula for Paasche's index is:

Paasche's index = (Sum of current year prices x current year quantities) / (Sum of base year prices x current year quantities)

Here, the sum of current year prices = 2/P x 5 + 28/27 x 2 = 10/P + 56/27

Paasche's index = (10/P + 56/27 x 2) / (25 x 5/P + 25 x 2)

Paasche's index = (10/P + 56/27 x 2) / (125/P + 50)

Calculation of the missing figure:

We are given that the quantity for commodity X in the current year is 5. We can use this information to find P.

Using the formula for Paasche's index, we get:

Paasche's index = (10/P + 56/27 x 2) / (125/P + 50)

5P/125 + 2P = 10

7P = 10 x 125

P = 178.57

Therefore, the missing figure in the table is P = 178.57.

Explanation for the calculation:

The calculation involves finding the Laspeyres' index and Paasche's index for the given data. We use the formulae for these indices to calculate their values. We are given the quantity for commodity X in the current year, and we use this information to find the value of P that makes the Paasche's index equal to 2. We substitute this value of P in the Laspeyres' index formula to find the missing figure in the table.