Using the data below, calculate the index numbers for question 13 using Laspeyre's and Paasche's methods.

Here's the solution for question 13.

To construct the index numbers, we first need to calculate the necessary sums.
Let $P_0$ and $Q_0$ be the price and quantity in the base year (2010), and $P_1$ and $Q_1$ be the price and quantity in the current year (2012).

Step 1: Create a table to calculate the required products.

$$\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline \textbf{Items} & \textbf{P}_0 & \textbf{Q}_0 & \textbf{P}_1 & \textbf{Q}_1 & \textbf{P}_0 \textbf{Q}_0 & \textbf{P}_1 \textbf{Q}_0 & \textbf{P}_0 \textbf{Q}_1 & \textbf{P}_1 \textbf{Q}_1 \\ \hline \text{A} & 6 & 50 & 10 & 56 & 300 & 500 & 336 & 560 \\ \text{B} & 2 & 100 & 2 & 120 & 200 & 200 & 240 & 240 \\ \text{C} & 4 & 60 & 6 & 60 & 240 & 360 & 240 & 360 \\ \text{D} & 10 & 30 & 12 & 24 & 300 & 360 & 240 & 288 \\ \text{E} & 8 & 40 & 12 & 36 & 320 & 480 & 288 & 432 \\ \hline \textbf{Total} & & & & & \textbf{1360} & \textbf{1900} & \textbf{1344} & \textbf{1880} \\ \hline \end{array}$$

From the table, we have:
$\sum P_0 Q_0 = 1360$
$\sum P_1 Q_0 = 1900$
$\sum P_0 Q_1 = 1344$
$\sum P_1 Q_1 = 1880$

a) Laspeyre's Method:
The formula for Laspeyre's Price Index ($P_{01}^L$) is:

$$P_{01}^L = \frac{\sum P_1 Q_0}{\sum P_0 Q_0} \times 100$$

Substitute the values:

$$P_{01}^L = \frac{1900}{1360} \times 100$$ $$P_{01}^L = 1.3970588 \times 100$$ $$P_{01}^L = \boxed{139.71}$$

b) Paasche's Method:
The formula for Paasche's Price Index ($P_{01}^P$) is:

$$P_{01}^P = \frac{\sum P_1 Q_1}{\sum P_0 Q_1} \times 100$$

Substitute the values:

$$P_{01}^P = \frac{1880}{1344} \times 100$$ $$P_{01}^P = 1.3988095 \times 100$$ $$P_{01}^P = \boxed{139.88}$$

c) Fisher's Method:
The formula for Fisher's Ideal Price Index ($P_{01}^F$) is the geometric mean of Laspeyre's and Paasche's indices:

$$P_{01}^F = \sqrt{P_{01}^L \times P_{01}^P}$$

Alternatively:

$$P_{01}^F = \sqrt{\frac{\sum P_1 Q_0}{\sum P_0 Q_0} \times \frac{\sum P_1 Q_1}{\sum P_0 Q_1}} \times 100$$

Using the calculated Laspeyre's and Paasche's indices:

$$P_{01}^F = \sqrt{139.70588 \times 139.88095}$$ $$P_{01}^F = \sqrt{19542.06}$$ $$P_{01}^F = \boxed{139.79}$$