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do a step by step explanation on how to find the other numbers

To solve this problem, we’ll break down the information given and work systematically to find the other five numbers based on the properties listed.

### Information provided:
1. **The lowest number is 37.**
2. **The range is 24**, which means the difference between the highest and lowest numbers is 24. So, the highest number is \(37 + 24 = 61\).
3. **The mode is 43.** This indicates that 43 appears more frequently than any other number, so we should include at least two occurrences of 43.
4. **The median is 46.** Since we have six numbers, the median will be the average of the third and fourth numbers when they are arranged in ascending order. Thus, the third and fourth numbers must average to 46.
5. **One number is a multiple of 11.** The multiples of 11 near our range are 33, 44, and 55. However, the lowest number is 37, so 33 is not possible. Therefore, the multiple of 11 must be either 44 or 55.

### Step-by-Step Solution:

#### Step 1: Determine the highest and lowest numbers
- The lowest number is **37** (given).
- Since the range is **24**, the highest number is \(37 + 24 = 61\).

#### Step 2: Use the median information
- Since the median of the six numbers is **46**, the third and fourth numbers must average to 46. Therefore:
  \[
  \frac{\text{Third number + Fourth number}}{2} = 46
  \]
  which implies:
  \[
  \text{Third number + Fourth number} = 92
  \]

#### Step 3: Apply the mode condition
- The mode is **43**, so **43** must appear at least twice in the list. We can tentatively place two instances of 43 in our list.

#### Step 4: Consider the multiple of 11
- The only possible values for the multiple of 11 within our range are **44** and **55**.
- Since 44 is close to 46, it’s plausible that **44** and **48** could be the third and fourth numbers (as they average to 46). We’ll try this arrangement.

#### Step 5: Arrange the tentative list and check
With the information we’ve organized, let’s try constructing a list:
- **37** (lowest number)
- **43** (mode, and appears at least twice)
- **43** (second instance to satisfy the mode condition)
- **44** (a multiple of 11 and part of the median calculation)
- **48** (to satisfy the median as 46 when averaged with 44)
- **61** (highest number, satisfying the range condition)

Our tentative list in ascending order is:
\[
37, 43, 43, 44, 48, 61
\]

#### Step 6: Verify the solution meets all conditions
1. **Lowest number is 37** — ✔️
2. **Range is 24** (61 - 37 = 24) — ✔️
3. **Mode is 43** (43 appears twice, more than any other number) — ✔️
4. **Median is 46** (\(\frac{44 + 48}{2} = 46\)) — ✔️
5. **One number is a multiple of 11** (44) — ✔️

All conditions are satisfied with the list:
\[
\boxed{37, 43, 43, 44, 48, 61}
\]

### Final Answer
The six numbers are **37, 43, 43, 44, 48, and 61**.

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### Related Questions
1. How would the solution change if the mode was given as a different number?
2. What if the median was an odd number, such as 45?
3. How does changing the range affect the possible values of the numbers?
4. What if there were seven numbers instead of six? How would the median be determined?
5. Can the highest number be a multiple of 11 and still meet all conditions?

### Tip
When solving problems with multiple conditions, it helps to list out each condition and solve them in the order that most limits the options, such as identifying the range or mode first.

Here is some information about six numbers: The lowest number is 37. The range is 24. The mode is 43. The median is 46. One number is a multiple of 11. Find the other five numbers.

Find the missing numbers in a set of six numbers given specific statistical conditions: lowest number is 37, range is 24, mode is 43, median is 46, and one number is a multiple of 11. Includes a step-by-step solution suitable for Grades 6-8.

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