the students decided to cut one cake into sixths, one cake into sevenths, one cake into  eighths, and one cake into ninths. what is the mean value of these three numbers: the mean, median, and mode amount of cake that each student received

To solve the problem, we first need to determine the mean, median, and mode for the amounts of cake each student received from the different cuts, and then find the mean of these three values.

### Step 1: Determine the amounts of cake each student received.

- **Cake cut into sixths:** Each student receives \( \frac{1}{6} \) of a cake.
- **Cake cut into sevenths:** Each student receives \( \frac{1}{7} \) of a cake.
- **Cake cut into eighths:** Each student receives \( \frac{1}{8} \) of a cake.
- **Cake cut into ninths:** Each student receives \( \frac{1}{9} \) of a cake.

### Step 2: Calculate the mean amount of cake.
The mean amount of cake is the average of the four fractions:
\[
\text{Mean} = \frac{1/6 + 1/7 + 1/8 + 1/9}{4}
\]

### Step 3: Determine the median amount of cake.
The median is the middle value when the amounts are arranged in ascending order. Let's list them:
\[
\frac{1}{9}, \frac{1}{8}, \frac{1}{7}, \frac{1}{6}
\]
Since there are four values, the median will be the average of the two middle values \( \frac{1}{8} \) and \( \frac{1}{7} \).

### Step 4: Determine the mode.
Since each fraction occurs exactly once, there is no mode. In this case, we can say the mode does not exist or consider it as zero.

### Step 5: Calculate the mean of the mean, median, and mode.

Since the mode is zero, the mean of the three values will be:
\[
\text{Final Mean} = \frac{\text{Mean} + \text{Median} + \text{Mode}}{3}
\]

Now, let's calculate these values step by step.The mean value of the mean, median, and mode amounts of cake that each student received is approximately **0.0901**.

Do you have any questions or need more details on this solution?

Here are five related questions you might consider:

1. How would the result change if the mode was not zero but instead one of the cake portions?
2. What is the harmonic mean of the cake portions?
3. If the cakes were cut into different numbers of pieces (e.g., 5ths, 6ths, 7ths, and 8ths), how would the result change?
4. How would you calculate the standard deviation of the cake portions each student received?
5. What are other situations where the mean, median, and mode are used together in problem-solving?

**Tip:** When data is highly skewed, the median is often a better indicator of central tendency than the mean.

Explore the mean, median, and mode calculations for portions of cake cut into sixths, sevenths, eighths, and ninths. Understand how to calculate the arithmetic mean of these values. Suitable for students in Grades 6-8.

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