Question

The ratio of the circumferences of two circles is 5:7; then the ratio of their radii is

• A. 7:5
• B. 5:7
• C. 25:49
• D. 49:25

Hint:

For circles, the ratio of their radii, the ratio of their diameters, and the ratio of their circumferences are all the same. If the ratio of radii is a:b, the ratio of circumferences is also a:b.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
The circumference of a circle is directly proportional to its radius. This question tests the relationship between the ratio of circumferences and the ratio of radii.

Step 2: Key Formula or Approach:
The circumference of a circle, \(C = 2\pi r\).
Let the radii of the two circles be \(r_1\) and \(r_2\), and their circumferences be \(C_1\) and \(C_2\).
The ratio of their circumferences is \(\frac{C_1}{C_2} = \frac{2\pi r_1}{2\pi r_2}\).

Step 3: Detailed Explanation:
We are given the ratio of the circumferences:
\[ \frac{C_1}{C_2} = \frac{5}{7} \] Using the formula, we have:
\[ \frac{2\pi r_1}{2\pi r_2} = \frac{5}{7} \] The term \(2\pi\) is a common factor in the numerator and the denominator, so it cancels out.
\[ \frac{r_1}{r_2} = \frac{5}{7} \] So, the ratio of the radii is 5:7.

Step 4: Final Answer:
The ratio of the radii of the two circles is 5:7. This matches option (B).