Noor and Ethan ride bikes on a trail after school. Noor first completed her homework, so she starts the bike ride 0.25 hours later than Ethan. Noor rides her bike at a constant rate of 12 miles per hour Ethan rides his bike at a constant rate of 9 miles per hour. How long will Noor ride her bike before she catches up with Ethan?

To find out how long Noor rides her bike before catching up with Ethan, we can set up an equation. Let \( t \) be the time Noor rides her bike in hours. During this time, Ethan has already been riding for \( t + 0.25 \) hours. The distance both have traveled will be the same when Noor catches up to Ethan. Therefore, we can set up the equation based on their distances: For Noor: Distance = Speed × Time = \( 12t \) For Ethan: Distance = Speed × Time = \( 9(t + 0.25) \) Setting the distances equal to each other: \[ 12t = 9(t + 0.25) \] Expanding and solving for \( t \): \[ 12t = 9t + 2.25 \] Subtracting \( 9t \) from both sides gives: \[ 3t = 2.25 \] Now, dividing both sides by 3: \[ t = 0.75 \] So, Noor will ride her bike for 0.75 hours (or 45 minutes) before she catches up with Ethan. Now let's look at some interesting elements related to biking and speed! Noor's speed of 12 miles per hour is quite brisk, especially when you consider that most casual bikers ride at around 10-15 miles per hour. This makes biking a fabulous way to travel, as it's not just good exercise, but it's also an efficient means of transport in urban areas. Imagine zipping past traffic jams while soaking up nature's beauty! Not only is biking fun, but it's also eco-friendly! By choosing to bike instead of drive, you significantly reduce your carbon footprint. For those interested in further exploring biking, local bike advocacy groups often host events that promote sustainable transport and safety on the roads. Plus, they provide valuable resources like bike repair workshops and ride events, making cycling an even more delightful adventure!