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Let's go through the math problems on the sheet step by step.

### Part 1: Simplify the following ratios
1. **a) $ 9:24 $**
   - To simplify this, divide both numbers by their greatest common divisor (GCD), which is 3:
     \[
     \frac{9}{3} : \frac{24}{3} = 3:8
     \]
   - Answer: $ 3:8 $

2. **b) $ 30:75 $**
   - The GCD of 30 and 75 is 15, so divide both by 15:
     \[
     \frac{30}{15} : \frac{75}{15} = 2:5
     \]
   - Answer: $ 2:5 $

3. **c) $ \frac{1}{3} : \frac{3}{4} $**
   - To simplify ratios involving fractions, multiply both sides by the least common denominator (12):
     \[
     \left( \frac{1}{3} \times 12 \right) : \left( \frac{3}{4} \times 12 \right) = 4:9
     \]
   - Answer: $ 4:9 $

4. **d) $ \frac{5}{6} : \frac{2}{3} $**
   - Multiply both sides by the least common denominator (6):
     \[
     \left( \frac{5}{6} \times 6 \right) : \left( \frac{2}{3} \times 6 \right) = 5:4
     \]
   - Answer: $ 5:4 $

### Part 2: Write each of the following as a simplified rate
1. **a) 300 mL in 5 minutes**
   - To simplify the rate, divide both values by 5:
     \[
     \frac{300}{5} = 60 \text{ mL per minute}
     \]
   - Answer: 60 mL/min

2. **b) $48.60 for 9 kilograms**
   - Divide $48.60 by 9 to get the cost per kilogram:
     \[
     \frac{48.60}{9} = 5.40 \text{ dollars per kilogram}
     \]
   - Answer: $5.40/kg

### Part 3: Convert the following rates into the units given in the brackets
1. **a) 200 g/day (kg/week)**
   - First, convert grams to kilograms: $ 200 \, g = 0.2 \, kg $.
   - Next, convert days to weeks: $ 1 \, week = 7 \, days $, so:
     \[
     0.2 \, \text{kg/day} \times 7 = 1.4 \, \text{kg/week}
     \]
   - Answer: $ 1.4 \, \text{kg/week} $

2. **b) $18/h (c/min)**
   - Convert dollars per hour to cents per hour: $ 18 \, \text{dollars} = 1800 \, \text{cents} $.
   - There are 60 minutes in an hour, so divide 1800 cents by 60:
     \[
     \frac{1800}{60} = 30 \, \text{cents per minute}
     \]
   - Answer: $ 30 \, \text{c/min} $

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Would you like more details on any part of this? Here are five questions you might want to explore:

1. How do you find the greatest common divisor (GCD) of two numbers?
2. How can you simplify ratios involving fractions more efficiently?
3. What are some methods for converting units, such as grams to kilograms?
4. Why is it important to convert units correctly when simplifying rates?
5. What is the difference between rates and ratios in math problems?

**Tip:** When simplifying ratios, always check if both sides share a common divisor. This makes it easier to reduce ratios to their simplest form.

Simplify the following ratios and convert rates.

Learn how to simplify ratios such as 9:24 and 30:75, and how to convert rates like 200 g/day to kg/week. This math problem covers basic concepts of ratio simplification and rate unit conversion, suitable for students in Grades 6-8.

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