Let's break down the problem step by step.

Part A: Maximum number of books on each shelf

We are given that the total length of the bookshelf is $60 \frac{2}{5} \, \text{cm}$, which can be converted to a decimal: $60 \frac{2}{5} = 60 + \frac{2}{5} = 60 + 0.4 = 60.4 \, \text{cm}$ Now, we'll find the maximum number of books for each shelf.

1. Top Shelf:

   • Thickness of the first book: $2 \, \text{cm}$
   • Maximum number of books on the top shelf: $\text{Number of books} = \frac{60.4 \, \text{cm}}{2 \, \text{cm}} = 30.2$ Since we can't have a fraction of a book, the maximum number is 30 books.

2. Middle Shelf:

   • Thickness of the second book: $3 \frac{1}{5} \, \text{cm} = 3.2 \, \text{cm}$
   • Maximum number of books on the middle shelf: $\text{Number of books} = \frac{60.4 \, \text{cm}}{3.2 \, \text{cm}} = 18.875$ So, the maximum number of books is 18.

3. Bottom Shelf:

   • Thickness of the third book: $2 \frac{4}{5} \, \text{cm} = 2.8 \, \text{cm}$
   • Maximum number of books on the bottom shelf: $\text{Number of books} = \frac{60.4 \, \text{cm}}{2.8 \, \text{cm}} = 21.571$ So, the maximum number of books is 21.

Part B: Ratio

The ratio of the maximum number of books on the top shelf to the middle shelf to the bottom shelf is: $\text{Ratio} = 30:18:21$ To simplify this ratio, divide each term by their greatest common divisor (GCD), which is 3: $\text{Simplified ratio} = \frac{30}{3}:\frac{18}{3}:\frac{21}{3} = 10:6:7$

Final Answers:

• Top Shelf: 30 books
• Middle Shelf: 18 books
• Bottom Shelf: 21 books
• Ratio (simplified): 10:6:7

Would you like any more details or further clarifications?

Here are 5 related questions to expand this topic:

1. How do we find the GCD of numbers to simplify a ratio?
2. What is the total number of books that can fit on all three shelves?
3. If the length of the shelf increased by 10 cm, how would that affect the number of books on each shelf?
4. How would the problem change if the thickness of the books varied within the same shelf?
5. What would be the effect on the ratio if one of the book thicknesses was slightly different?

Tip: When working with ratios, always simplify them by finding the greatest common divisor of the terms!