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The problem involves dividing three pieces of cake, weighing:

- \( 4 \frac{1}{2} \) lbs
- \( 6 \frac{3}{4} \) lbs
- \( 7 \frac{1}{5} \) lbs

into parts of equal weights, where each part is as heavy as possible. The question asks for the maximum number of guests that can be entertained, with each guest receiving one part of the cake.

### Step 1: Convert the weights to improper fractions.
We need to express these mixed fractions as improper fractions:

- \( 4 \frac{1}{2} = \frac{9}{2} \)
- \( 6 \frac{3}{4} = \frac{27}{4} \)
- \( 7 \frac{1}{5} = \frac{36}{5} \)

### Step 2: Find the Least Common Divisor (LCD) of these fractions.
To divide the cakes into equal parts, we need to find the greatest common divisor (GCD) of these fractions. First, express the fractions with a common denominator:

- \( \frac{9}{2} = \frac{90}{20} \)
- \( \frac{27}{4} = \frac{135}{20} \)
- \( \frac{36}{5} = \frac{144}{20} \)

### Step 3: Find the GCD of the numerators.
The numerators are 90, 135, and 144. We now find the GCD of these numbers:

- Prime factorization of 90: \( 2 \times 3^2 \times 5 \)
- Prime factorization of 135: \( 3^3 \times 5 \)
- Prime factorization of 144: \( 2^4 \times 3^2 \)

The common prime factor is \( 3^2 = 9 \).

Thus, the GCD of the numerators is 9.

### Step 4: The largest possible part size.
Each part will weigh \( \frac{9}{20} \) lbs.

### Step 5: Calculate the total number of guests.
To find the maximum number of guests, divide the total weight of each cake by the weight of one part \( \frac{9}{20} \):

- For \( \frac{9}{2} \) lbs: \( \frac{\frac{9}{2}}{\frac{9}{20}} = 10 \)
- For \( \frac{27}{4} \) lbs: \( \frac{\frac{27}{4}}{\frac{9}{20}} = 15 \)
- For \( \frac{36}{5} \) lbs: \( \frac{\frac{36}{5}}{\frac{9}{20}} = 16 \)

### Step 6: Total number of guests.
The total number of guests is \( 10 + 15 + 16 = 41 \).

Therefore, the maximum number of guests that can be entertained is **41**, which matches none of the provided options.

Let me know if you'd like further details or have more questions!

Here are 5 related questions:
1. How do we find the least common multiple (LCM) of fractions?
2. How would the solution change if the cake weights were given as whole numbers?
3. What is the difference between GCD and LCM in context of division problems?
4. Can you explain how mixed fractions are converted into improper fractions?
5. How would you solve this if the cakes had irrational weights like \( \sqrt{2} \) lbs?

**Tip:** Always simplify fractions before trying to find common divisors or multiples!

Three pieces of cakes weighing 4 1/2 lbs, 6 3/4 lbs, and 7 1/5 lbs are to be divided into parts of equal weights. Each part must be as heavy as possible. If one such part is served to each guest, what is the maximum number of guests that could be entertained?

Solve the problem of dividing three cakes weighing 4 1/2 lbs, 6 3/4 lbs, and 7 1/5 lbs into equal parts using the greatest common divisor (GCD). Learn how to calculate the maximum number of guests that can be entertained based on fraction division and GCD.

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