1. Problem: A machine fills 680 bottles in 5 hours. How many bottles will it fill in 3 hours? Formula: Bottles filled are directly proportional to time. Calculation: $$\text{Bottles per hour} = \frac{680}{5} = 136$$ So, in 3 hours: $$136 \times 3 = 408$$ bottles. 2. Problem: Kim types 108 words in 6 minutes. How many words in 30 minutes? Formula: Words typed are directly proportional to time. Calculation: $$\text{Words per minute} = \frac{108}{6} = 18$$ In 30 minutes: $$18 \times 30 = 540$$ words. 3. Problem: A bus travels 40 km in 30 minutes. How far in 3 hours? Formula: Distance is proportional to time at constant speed. Convert 3 hours to minutes: $$3 \times 60 = 180$$ minutes. Speed: $$\frac{40}{30} = \frac{4}{3} \text{ km/min}$$ Distance in 180 minutes: $$\frac{4}{3} \times 180 = 240$$ km. 4. Problem: A water tank casts a 21 m shadow; a 9.5 m tree casts 8 m shadow. Find tank height. Formula: Heights and shadows are proportional. Set ratio: $$\frac{\text{Tank height}}{21} = \frac{9.5}{8}$$ Calculate tank height: $$\text{Tank height} = \frac{9.5}{8} \times 21 = 24.9375$$ m. 5. Problem: Bacteria enlarged 60000 times is 3 cm. Find length if enlarged 10000 times. Formula: Length is proportional to enlargement factor. Original length: $$\frac{3}{60000} = 0.00005$$ cm. Length at 10000 times: $$0.00005 \times 10000 = 0.5$$ cm. 6. Problem: Map scale 1 cm = 20 km. Raj drives 70 km. Find distance on map. Formula: Distance on map = Actual distance / scale factor. Calculation: $$\frac{70}{20} = 3.5$$ cm. Final answers: 1. 408 bottles 2. 540 words 3. 240 km 4. 24.9375 m 5. 0.5 cm 6. 3.5 cm