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The problem asks to find the diameter of the smaller wheel. Here's how to solve it:

We are given the following information:

1. The large wheel has a diameter of 20 cm and rotates 135 times to cover a certain distance.
2. The small wheel needs to rotate 180 times to cover the same distance.

To solve this, we use the fact that the distance traveled by the wheel is proportional to its circumference and the number of rotations.

The formula for the circumference of a circle is:

\[
C = \pi \cdot d
\]

Where \( d \) is the diameter. The distance traveled by the wheel is the circumference multiplied by the number of rotations.

Let \( d_s \) be the diameter of the small wheel, and \( d_l = 20 \) cm be the diameter of the large wheel.

Now, the distances traveled by both wheels are equal:

\[
\text{Distance by large wheel} = \text{Distance by small wheel}
\]

\[
135 \times \pi \times d_l = 180 \times \pi \times d_s
\]

Canceling \( \pi \) from both sides:

\[
135 \times d_l = 180 \times d_s
\]

Substitute \( d_l = 20 \) cm:

\[
135 \times 20 = 180 \times d_s
\]

Now solve for \( d_s \):

\[
d_s = \frac{135 \times 20}{180}
\]

\[
d_s = \frac{2700}{180} = 15 \, \text{cm}
\]

So, the diameter of the small wheel is **15 cm**, which corresponds to option C.

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you find the circumference of a circle given its radius?
2. If the large wheel's diameter doubled, how would that affect the number of rotations needed?
3. What is the relationship between diameter and rotations for wheels?
4. How do you calculate the distance traveled if the number of rotations changes?
5. How would the solution change if the number of rotations for the small wheel was different?

**Tip:** When solving geometry problems involving circles and rotations, remember that the distance covered is proportional to the wheel’s circumference, not its diameter directly.

A large wheel has a diameter of 20 cm and can travel a set distance in 135 rotations. A small wheel must rotate 180 times to travel the same distance. What is the diameter of the small wheel?

Learn how to calculate the diameter of a small wheel when given the rotations and diameter of a larger wheel. This problem explores proportional relationships between the distance traveled and the wheel's diameter and rotations. Suitable for students in Grades 6-8.

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