9. You can type 250 letters in 60 seconds. Your friend can type 375 letters in 90 seconds. Do these rates form a proportion? Explain.

Yes, these rates **do form a proportion**. Let's break down why: ### Understanding the Rates 1. **Your Typing Rate:** - **Letters Typed:** 250 - **Time Taken:** 60 seconds - **Rate:** \( \frac{250 \text{ letters}}{60 \text{ seconds}} \) 2. **Your Friend's Typing Rate:** - **Letters Typed:** 375 - **Time Taken:** 90 seconds - **Rate:** \( \frac{375 \text{ letters}}{90 \text{ seconds}} \) ### Simplifying the Rates Let's simplify both rates to see if they are equivalent. 1. **Your Rate Simplified:** \[ \frac{250}{60} = \frac{25}{6} \approx 4.1667 \text{ letters per second} \] 2. **Friend's Rate Simplified:** \[ \frac{375}{90} = \frac{25}{6} \approx 4.1667 \text{ letters per second} \] ### Comparing the Rates Both simplified rates are equal: \[ \frac{25}{6} = \frac{25}{6} \] This equality confirms that **the rates are proportional**, meaning both you and your friend type letters at the same rate of approximately 4.1667 letters per second. ### Another Way to Confirm: Cross-Multiplication To verify proportions, you can use cross-multiplication: \[ \frac{250}{60} = \frac{375}{90} \] Cross-multiply: \[ 250 \times 90 = 375 \times 60 \] \[ 22,500 = 22,500 \] Since both sides are equal, the original statement is a true proportion. ### Conclusion Both you and your friend have typing rates that are directly proportional. This means that for every second spent typing, both of you produce the same number of letters, maintaining a consistent and equal rate.