1. **Problem:** Given 10 yellow marbles correspond to 45 green marbles, find the number of yellow marbles when there are 9 green marbles. **Formula:** Use the ratio and proportion formula $$\frac{a}{b} = \frac{c}{d}$$ where $a$ and $b$ are known quantities, and $c$ or $d$ is unknown. **Step:** Set up the proportion $$\frac{10}{45} = \frac{x}{9}$$. **Operation:** Cross multiply: $$10 \times 9 = 45 \times x$$. **Solve:** $$90 = 45x \Rightarrow x = \frac{90}{45} = 2$$. **Scale factor:** The green marbles decreased from 45 to 9, scale factor $$\frac{9}{45} = \frac{1}{5}$$, so yellow marbles scale down by $$\frac{1}{5}$$ from 10 to 2. 2. **Problem:** Convert 24 inches to feet and find the equivalent of 25 inches in feet. **Formula:** 1 foot = 12 inches. **Step:** Convert 24 inches to feet: $$\frac{24}{12} = 2$$ feet. **Find:** For 25 inches, $$\frac{25}{12} = 2.0833$$ feet. **Scale factor:** Inches to feet scale factor is $$\frac{1}{12}$$. 3. **Problem:** Given 250 miles in 4 hours, find the time for 100 miles. **Formula:** Use rate formula $$\text{Rate} = \frac{\text{Distance}}{\text{Time}}$$. **Step:** Calculate rate: $$\frac{250}{4} = 62.5$$ miles/hour. **Find:** Time for 100 miles: $$\frac{100}{62.5} = 1.6$$ hours. **Scale factor:** Distance scaled by $$\frac{100}{250} = 0.4$$, time scales similarly. 4. **Problem:** Convert 127 centimeters to inches and verify 74.46 inches in centimeters. **Formula:** 1 inch = 2.54 centimeters. **Step:** Convert 127 cm to inches: $$\frac{127}{2.54} = 50$$ inches. **Verify:** Convert 74.46 inches to cm: $$74.46 \times 2.54 = 189.2$$ cm. **Scale factor:** Centimeters to inches scale factor is $$\frac{1}{2.54}$$. 5. **Problem:** Given 160 feet in 20 seconds, find time for 20 feet. **Formula:** Rate = distance/time. **Step:** Rate = $$\frac{160}{20} = 8$$ feet/second. **Find:** Time for 20 feet: $$\frac{20}{8} = 2.5$$ seconds. **Scale factor:** Distance scaled by $$\frac{20}{160} = 0.125$$, time scales accordingly. 6. **Problem:** Given 72 red marbles correspond to 150 yellow marbles, find yellow marbles for 216 red marbles. **Formula:** $$\frac{72}{150} = \frac{216}{x}$$. **Operation:** Cross multiply: $$72x = 150 \times 216$$. **Solve:** $$x = \frac{150 \times 216}{72} = 450$$ yellow marbles. **Scale factor:** Red marbles scaled by $$\frac{216}{72} = 3$$, yellow marbles scale by 3 from 150 to 450. **Final answers:** 1. 2 yellow marbles 2. 2.0833 feet 3. 1.6 hours 4. 50 inches (127 cm), 189.2 cm (74.46 inches) 5. 2.5 seconds 6. 450 yellow marbles