If the sum of the diameters of two circles is 30 cm and the difference of their radii is 5 cm, then the ratio of area of the smaller circle to that of the bigger circle is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 1 : 4
A diameter is twice the radius, so the sum of the diameters gives the sum of the radii. The sum and the difference of the radii then give each radius, and the areas of circles are in the ratio of the squares of their radii.
Let the radii be \(R\) (bigger) and \(r\) (smaller):
\(2R+2r=30\), so \(R+r=15\) (1)
\(R-r=5\) (2)
Adding (1) and (2), \(r\) cancels: \(2R=20\), so \(R=10\) cm and \(r=5\) cm
Ratio of areas \(\displaystyle =\frac{\pi r^2}{\pi R^2}=\frac{25}{100}=\frac{1}{4}\), that is, 1 : 4