Question

A bicycle shop designs a new gear system for their bikes. They use a 4-inch gear for bikes with a 22-inch wheel diameter. If the ratio of the gear to wheel diameter remains constant, what is the size of the gear for a bike with a 26-inch wheel?

Answer 1

To solve this problem, we first need to determine the ratio of the gear size to the wheel diameter for the initial bike setup.

Given:
- Gear size = 4 inches
- Wheel diameter = 22 inches

The ratio of the gear size to the wheel diameter can be calculated as follows:

\[
\text{Ratio} = \frac{\text{Gear size}}{\text{Wheel diameter}} = \frac{4 \text{ inches}}{22 \text{ inches}} = \frac{4}{22} = \frac{2}{11}
\]

Now, we want to find the size of the gear for a bike with a 26-inch wheel diameter while maintaining the same ratio. Let \( x \) represent the size of the gear for the 26-inch wheel. The equation can be set up as follows:

\[
\frac{x}{26} = \frac{2}{11}
\]

To find \( x \), we will cross multiply:

\[
11x = 2 \times 26
\]

Calculating the right side:

\[
11x = 52
\]

Next, we solve for \( x \) by dividing both sides by 11:

\[
x = \frac{52}{11} \approx 4.727 \text{ inches}
\]

Thus, the size of the gear for a bike with a 26-inch wheel diameter, maintaining the same ratio, is approximately **4.73 inches** (or exactly \( \frac{52}{11} \) inches).