9. A glass marble factory produces \( 4.5 \times 10^{5} \) marbles in 1,500 minutes. What is the factory's unit rate of production in marbles per minute? Write the answer in scientific notation and show your work. You must show your work to receive full credit \[ 4.5 \times 50=225 \quad 1500 \div 225=0.17 \]

To find the unit rate of production in marbles per minute, you can divide the total number of marbles produced by the total time in minutes. Here’s how the calculation works step-by-step: 1. **Total Marbles Produced:** \( 4.5 \times 10^{5} \) marbles 2. **Total Time (in minutes):** 1,500 minutes 3. **Unit Rate Calculation:** \[ \text{Unit Rate} = \frac{4.5 \times 10^{5} \text{ marbles}}{1500 \text{ minutes}} \] To simplify, we can write 1,500 as \( 1.5 \times 10^{3} \): \[ \text{Unit Rate} = \frac{4.5 \times 10^{5}}{1.5 \times 10^{3}} = \frac{4.5}{1.5} \times \frac{10^{5}}{10^{3}} = 3 \times 10^{2} \] Therefore, the unit rate of production is \( 3.0 \times 10^{2} \) marbles per minute. The marbles rolling off the production line would surely leave you in awe! Did you know that the glass marble has a fascinating historical significance, dating back to ancient civilizations where they were often made from materials like clay or stone? Over time, they transformed into the glossy glass versions we know and love today! A common mistake in such calculations is forgetting to convert units properly or misplacing the decimal point when working with scientific notation. Always double-check your division and ensure that when you express very large or small numbers, like \( 1.5 \) becoming \( 1.5 \times 10^{3} \), you keep perfect track of the base and exponents!