The pie diagrams on the monthly expenditure of two families A and B are drawn with radii of two circles taken in the ratio 16 : 9 to compare their expenditures.
Rs. 25,600 and Rs. 8,100
Analyzing Expenditure Comparison with Pie Diagrams and Radius Ratios
When comparing different quantities using pie diagrams, the total quantity being represented by each diagram is proportional to the area of the circle. The area of a circle is calculated using the formula \(\text{Area} = \pi r^2\), where \(r\) is the radius of the circle.
In this question, we are comparing the monthly expenditures of two families, A and B, using pie diagrams. The radii of the two circles drawn for these diagrams are in the ratio 16 : 9. This means if the radius of the pie diagram for Family A is \(r_A\) and for Family B is \(r_B\), then:
\(\frac{r_A}{r_B} = \frac{16}{9}\)
Since the total expenditure is represented by the area of the circle, and the area is proportional to the square of the radius (\(r^2\)), the ratio of the expenditures will be the square of the ratio of the radii.
Let \(E_A\) be the total expenditure of Family A and \(E_B\) be the total expenditure of Family B. Then:
\(\frac{E_A}{E_B} = \frac{\text{Area}_A}{\text{Area}_B} = \frac{\pi r_A^2}{\pi r_B^2} = \frac{r_A^2}{r_B^2} = \left(\frac{r_A}{r_B}\right)^2\)
Substituting the given ratio of radii:
\(\frac{E_A}{E_B} = \left(\frac{16}{9}\right)^2 = \frac{16^2}{9^2} = \frac{256}{81}\)
So, the appropriate data for the total expenditures of families A and B should be in the ratio 256 : 81.
Let's examine the given options to find which pair of expenditures has a ratio of 256 : 81.
Option 1: Rs. 16,000 and Rs. 9,000
Ratio = \(\frac{16000}{9000} = \frac{16}{9}\). This is the ratio of radii, not the required ratio of expenditures.
Option 2: Rs. 8,000 and Rs. 4,500
Ratio = \(\frac{8000}{4500} = \frac{80}{45}\). Dividing both by 5, we get \(\frac{16}{9}\). This is also the ratio of radii, not the required ratio of expenditures.
Option 3: Rs. 25,600 and Rs. 8,100
Ratio = \(\frac{25600}{8100} = \frac{256}{81}\). This ratio matches the calculated ratio of expenditures required for the radii ratio of 16:9.
Option 4: Rs. 4,000 and Rs. 3,000
Ratio = \(\frac{4000}{3000} = \frac{4}{3}\). This ratio does not match the required ratio of expenditures.
Based on the analysis, only the expenditures Rs. 25,600 and Rs. 8,100 have the correct ratio (256:81) corresponding to the radii ratio (16:9) for comparing total expenditures using pie diagrams.
The data used for the two pie diagrams should represent total expenditures in the ratio that is the square of the ratio of their radii.
Revision Table: Pie Diagram Ratios
| Concept | Relationship | Formula/Ratio |
|---|---|---|
| Area of Circle | Proportional to square of radius | \(\text{Area} = \pi r^2 \implies \text{Area} \propto r^2\) |
| Total Quantity in Pie Chart | Proportional to Area of circle | Total Quantity \(\propto\) Area |
| Ratio of Total Quantities | Square of the ratio of Radii | \(\frac{\text{Quantity}_1}{\text{Quantity}_2} = \left(\frac{r_1}{r_2}\right)^2\) |
| Ratio of Radii | Square root of the ratio of Total Quantities | \(\frac{r_1}{r_2} = \sqrt{\frac{\text{Quantity}_1}{\text{Quantity}_2}}\) |
Additional Information: Comparing Data with Pie Charts
Pie charts are commonly used to show how a whole is divided into parts. Each sector of the pie represents a proportion of the total. When comparing two different totals using separate pie charts, it is important that the areas of the circles accurately reflect the ratio of the total amounts being compared.
If the radii of the circles are simply made proportional to the totals, it leads to a misleading comparison because the area (which represents the total amount) grows with the square of the radius, not linearly with the radius. Therefore, to maintain the correct visual proportion based on area, the radius should be proportional to the square root of the total amount.
Conversely, if the radii are in a specific ratio, the total amounts they represent for accurate comparison must be in the ratio of the squares of those radii. This ensures that the area of each circle is proportional to the total value it depicts.
In this problem, the radii ratio is 16:9. To make the areas represent the total expenditures correctly, the expenditure ratio must be \((16)^2 : (9)^2\), which is 256 : 81.
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