Question

If the sum of the diameters of two circles is 40 cm and the difference of their radii is 6 cm, then the ratio of the area of the smaller circle to that of the bigger circle is:

• A. 49:16
• B. 49:169
• C. 169:49
• D. 1:4

Hint:

Convert diameters to radii, form simultaneous equations, solve for radii, then compare the squares of radii to find area ratios.

Answer 1

The Correct Option is B

Let the radii of the larger and smaller circles be $r_1$ and $r_2$, respectively. The given conditions are: - The sum of their diameters is 40 cm. Since diameter $d = 2r$, this translates to $2r_1 + 2r_2 = 40$, which simplifies to $r_1 + r_2 = 20$. - The difference between their radii is 6 cm, so $r_1 - r_2 = 6$. Solving these two equations simultaneously: \[r_1 + r_2 = 20 \quad \text{and} \quad r_1 - r_2 = 6\]Adding the two equations yields: \[2r_1 = 26 \implies r_1 = 13\]Subtracting the second equation from the first yields: \[2r_2 = 14 \implies r_2 = 7\]The area of a circle is proportional to the square of its radius. Therefore, the ratio of the areas is: \[\text{Ratio of areas} = \frac{\text{Area of smaller circle}}{\text{Area of larger circle}} = \frac{\pi r_2^2}{\pi r_1^2} = \frac{7^2}{13^2} = \frac{49}{169}\]