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 bigger and smaller circles be $r_1$ and $r_2$ respectively. Given: - Sum of diameters: $d_1 + d_2 = 40$ cm, where $d = 2r$ implies $2r_1 + 2r_2 = 40 \implies r_1 + r_2 = 20$. - Difference of radii: $r_1 - r_2 = 6$. Solving the system: \[ r_1 + r_2 = 20, \quad r_1 - r_2 = 6 \] Adding both: \[ 2r_1 = 26 \implies r_1 = 13 \] Subtracting: \[ 2r_2 = 14 \implies r_2 = 7 \] The area of a circle is proportional to the square of its radius. Hence, \[ \text{Ratio of areas} = \frac{\pi r_2^2}{\pi r_1^2} = \frac{7^2}{13^2} = \frac{49}{169} \]