Understanding Pie Diagrams and Radii Ratios

A pie diagram, or pie chart, is a type of graph that represents data in a circular format. The circle is divided into sectors, where each sector's area is proportional to the quantity it represents compared to the total quantity. When constructing pie diagrams for two different datasets (like the expenditures of two families), the area of each circle should be proportional to the total value (total expenditure) it represents.

The area of a circle is given by the formula $\text{Area} = \pi R^2$, where $R$ is the radius of the circle. If the area of the pie diagram is proportional to the total expenditure, then we can say:

$\text{Area} \propto \text{Total Expenditure}$

So, $\pi R^2 \propto \text{Total Expenditure}$. This implies $R^2 \propto \text{Total Expenditure}$.

Therefore, the ratio of the squares of the radii of two pie diagrams should be equal to the ratio of their respective total expenditures.

Calculating Total Expenditure for Each Family

Let's first calculate the total monthly expenditure for Family A and Family B from the given table.

Total Expenditure for Family A ($T_A$):

$T_A = 3500 + 500 + 1500 + 2000 + 2500 = 10000$ Rs.

Total Expenditure for Family B ($T_B$):

$T_B = 2700 + 800 + 1000 + 1800 + 1800 = 8100$ Rs.

Determining the Ratio of Radii

As established, the ratio of the squares of the radii is equal to the ratio of the total expenditures. Let $R_A$ be the radius for Family A's pie diagram and $R_B$ be the radius for Family B's pie diagram.

$\frac{R_A^2}{R_B^2} = \frac{T_A}{T_B}$

Substitute the calculated total expenditures:

$\frac{R_A^2}{R_B^2} = \frac{10000}{8100} = \frac{100}{81}$

To find the ratio of the radii, we take the square root of both sides:

$\sqrt{\frac{R_A^2}{R_B^2}} = \sqrt{\frac{100}{81}}$

$\frac{R_A}{R_B} = \frac{\sqrt{100}}{\sqrt{81}}$

$\frac{R_A}{R_B} = \frac{10}{9}$

Thus, the ratio of the radii of the circles for Family A and Family B should be 10 : 9.

Summary of Radius Ratio Calculation

• Calculate total expenditure for Family A ($T_A$).
• Calculate total expenditure for Family B ($T_B$).
• The ratio of the areas of the pie charts is equal to the ratio of total expenditures: $\text{Area}_A / \text{Area}_B = T_A / T_B$.
• The area of a circle is $\pi R^2$. So, $(\pi R_A^2) / (\pi R_B^2) = T_A / T_B$, which simplifies to $R_A^2 / R_B^2 = T_A / T_B$.
• The ratio of radii is the square root of the ratio of total expenditures: $R_A / R_B = \sqrt{T_A / T_B}$.
• Substitute $T_A = 10000$ and $T_B = 8100$ to get $R_A / R_B = \sqrt{10000 / 8100} = \sqrt{100/81} = 10/9$.

Revision Table: Key Concepts

Additional Information: Pie Chart Construction

When constructing a single pie chart for one family's expenditure, each item's expenditure is represented as a sector. The angle of each sector is calculated as:

$\text{Sector Angle} = (\text{Item Expenditure} / \text{Total Expenditure}) \times 360^\circ$

For example, for Family A's Food expenditure:

Angle for Food (Family A) $= (3500 / 10000) \times 360^\circ = 0.35 \times 360^\circ = 126^\circ$.

Similarly, angles for all items in both families' expenditures could be calculated. However, this problem specifically asks about the ratio of the radii of the two entire pie diagrams, not the angles of individual sectors within one pie chart. The choice of radius affects the overall size of the pie chart, which should reflect the total magnitude of the data being represented.