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Let's break down and solve the problem step-by-step:

### Given:
- Harley has **6 pints** of lemonade.
- Rosie has **\(3 \frac{2}{3}\)** times as many pints of lemonade as Harley.

### Part (a): How many pints of lemonade does Rosie have?
1. First, convert \(3 \frac{2}{3}\) to an improper fraction:
   \[
   3 \frac{2}{3} = \frac{3 \times 3 + 2}{3} = \frac{9 + 2}{3} = \frac{11}{3}
   \]

2. Multiply this fraction by the amount of lemonade Harley has:
   \[
   \text{Rosie’s pints} = 6 \times \frac{11}{3} = \frac{6 \times 11}{3} = \frac{66}{3} = 22
   \]

   Rosie has **22 pints** of lemonade.

### Part (b): How many more pints of lemonade does Rosie have than Harley?
1. Subtract the amount of lemonade Harley has from the amount Rosie has:
   \[
   22 - 6 = 16
   \]

   Rosie has **16 more pints** of lemonade than Harley.

### Final Answers:
- **(a)** Rosie has **22 pints** of lemonade.
- **(b)** Rosie has **16 more pints** of lemonade than Harley.

Would you like a detailed explanation of how these calculations were made or have any questions? 

Here are 5 related questions to further expand on this topic:
1. If Harley had 8 pints of lemonade instead, how many pints would Rosie have?
2. What if Rosie had 4 times as many pints as Harley instead of \(3 \frac{2}{3}\) times?
3. If Rosie gave 5 pints of lemonade to Harley, how many pints would each have then?
4. How many pints of lemonade would Rosie have if the ratio was instead \(2 \frac{1}{2}\) times Harley's amount?
5. Can you represent the situation using an equation involving variables for both Harley's and Rosie's amounts?

**Tip:** When multiplying mixed numbers by whole numbers, it's often easiest to first convert the mixed number into an improper fraction for straightforward multiplication.

Harley has 6 pints of lemonade. Rosie has 3 2/3 times as many pints of lemonade as Harley. 
a) How many pints of lemonade does Rosie have?
b) How many more pints of lemonade does Rosie have than Harley?

Learn how to solve a word problem involving multiplying a mixed number by a whole number. In this problem, Harley has 6 pints of lemonade, and Rosie has 3 2/3 times as many pints. This exercise involves converting mixed numbers to improper fractions and performing basic arithmetic operations.

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