The DECI-Pizza Company has a special pizza that has 10 slices. Two of the
slices are each 1
6 of the whole pizza, two are each 1
8 , four are each 1
12 , and two are
each 1
24 . A group of n friends share the pizza by distributing all of these slices.
They do not cut any of the slices. Each of the n friends receives, in total, an
equal fraction of the whole pizza. For what values of n > 1 is this possible

Let's start by calculating the total fraction of the pizza based on the sizes of the slices.

### Step 1: Calculate the Total Pizza Fraction

The pizza has 10 slices, broken down as follows:

- **Two slices, each 1/6** of the whole pizza.
- **Two slices, each 1/8** of the whole pizza.
- **Four slices, each 1/12** of the whole pizza.
- **Two slices, each 1/24** of the whole pizza.

Now, calculate the sum of these slices:

\[
2 \times \frac{1}{6} + 2 \times \frac{1}{8} + 4 \times \frac{1}{12} + 2 \times \frac{1}{24}
\]

Breaking this down:

\[
= \frac{2}{6} + \frac{2}{8} + \frac{4}{12} + \frac{2}{24}
\]

Simplifying each term:

\[
= \frac{1}{3} + \frac{1}{4} + \frac{1}{3} + \frac{1}{12}
\]

### Step 2: Find a Common Denominator

The least common denominator of 3, 4, and 12 is 12. Let's rewrite each fraction with a denominator of 12:

\[
\frac{1}{3} = \frac{4}{12}, \quad \frac{1}{4} = \frac{3}{12}, \quad \frac{1}{3} = \frac{4}{12}, \quad \frac{1}{12} = \frac{1}{12}
\]

Now, sum these fractions:

\[
\frac{4}{12} + \frac{3}{12} + \frac{4}{12} + \frac{1}{12} = \frac{12}{12} = 1
\]

So, the total of the slices is exactly 1 whole pizza.

### Step 3: Distribute the Pizza Equally

Each of the \(n\) friends must receive an equal fraction of the whole pizza, so each friend must receive \(\frac{1}{n}\) of the pizza.

Now we need to check for which values of \(n\), where \(n > 1\), it is possible to divide the slices exactly. This means that the sum of the fractions that each friend receives must be a multiple of \(\frac{1}{n}\).

### Step 4: Look for Values of \(n\)

The slice sizes are:

- \( \frac{1}{6}, \frac{1}{8}, \frac{1}{12}, \frac{1}{24} \)

To determine which values of \(n\) work, \(n\) must divide all these slice sizes in such a way that each friend can get exactly \(\frac{1}{n}\) of the pizza. The common multiple of all slice sizes should divide evenly into 1.

Thus, the smallest possible values of \(n\) that work are those that divide the total amount of pizza and match the denominators of the slices.

### Step 5: Conclusion

For \(n\) to divide all slices evenly, it should be a divisor of the least common multiple of the denominators involved (6, 8, 12, 24). By trial, possible values of \(n\) include \(n = 2, 3, 4, 6, 8\).

These values of \(n\) allow the pizza to be shared equally among the friends without cutting any of the slices. Therefore, \(n = 2, 3, 4, 6, 8\) are possible solutions.

---

Would you like further details or have any questions?

Here are 5 related questions:
1. How do you find the least common multiple of a set of numbers?
2. How do you approach distributing fractions equally among groups?
3. What strategies can be used for dividing fractions of items?
4. Can this type of problem be extended to larger groups and more slices?
5. How can the concept of common multiples be applied to other sharing scenarios?

**Tip:** When dividing fractions, always seek the least common multiple to ensure even distribution among groups.

The DECI-Pizza Company has a special pizza that has 10 slices. Two of the slices are each 1/6 of the whole pizza, two are each 1/8, four are each 1/12, and two are each 1/24. A group of n friends share the pizza by distributing all of these slices. They do not cut any of the slices. Each of the n friends receives, in total, an equal fraction of the whole pizza. For what values of n > 1 is this possible?

Learn how to solve a fraction distribution problem where a special pizza with 10 slices of different sizes is shared equally among friends. This problem involves calculating the sum of fractions, finding the least common multiple, and determining the suitable values of n > 1. Ideal for students in Grades 6-8.

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