Final answer:
The ratio of the circumferences of two circles is the same as the ratio of their radii because the circumference formula (C = 2πr) is directly proportional to the radius. Hence, the answer to the student's question is (a) 4:5.
Step-by-step explanation:
The ratio of the radii of two circles is given as 4:5. To determine the ratio of their circumferences, we can use the formula for the circumference of a circle, which is C = 2πr, where π is a constant (approximately 3.14159) and r is the radius of the circle. Because the circumference is directly proportional to the radius, the ratio of the circumferences will be identical to the ratio of the radii.
Therefore, if the first circle has a radius 4 times some unit, its circumference would be C1 = 2π(4u) = 8πu. Similarly, if the second circle has a radius 5 times the same unit, its circumference would be C2 = 2π(5u) = 10πu. Dividing C1 by C2, we get the same ratio 4:5, which means the correct answer is (a) 4:5.