UPSC NDA 2 2017 Mathematics Solved Paper

Concept:In a pie chart, the area of each circle is proportional to the total expenditure it represents.Thus, the radii are chosen so that the areas of the circles are in the same ratio as the total expenditures.Explanation:First, compute the total monthly expenditure for each family.Family A: $3500 + 500 + 1500 + 2000 + 2500 = 10000$ Rs.Family B: $2700 + 800 + 1000 + 1800 + 1800 = 8100$ Rs.The area of a circle is $\pi r^2$.So, the ratio of areas equals the ratio of total expenditures:$\frac{\text{Area of circle A}}{\text{Area of circle B}} = \frac{10000}{8100} = \frac{100}{81}$.This gives $\frac{\pi r_a^2}{\pi r_b^2} = \frac{100}{81}$.Cancel $\pi$ and take square roots: $\frac{r_a}{r_b} = \sqrt{\frac{100}{81}} = \frac{10}{9}$.Hence, the ratio of the radii is $10 : 9$.Answer:The radii of the circles should be chosen in the ratio $10 : 9$, which corresponds to option B.