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Organize numbers into a two-way frequency table based on divisibility by 3 and 4.

Check the final answer first, then review the worked steps.

Problem

Organize numbers into a two-way frequency table based on divisibility by 3 and 4.

Answer

[[9, 1], [1, 9]]

Step-by-step solution

  1. Identify the total number of items: The problem states there is a set of 20 numbers. This will be the grand total for our table.
  1. Determine the number of multiples of 3 but not multiples of 4: The problem explicitly states that 9 numbers are multiples of 3 but not multiples of 4. This value goes into the cell for 'Multiple of 3' and 'Not a multiple of 4'.
  1. Determine the total number of multiples of 3: The problem states that 10 numbers in total are a multiple of 3. This is the row total for 'Multiple of 3'.
  1. Calculate the number of multiples of 3 that are also multiples of 4: Since 10 numbers are multiples of 3 in total, and 9 of them are not multiples of 4, the remaining numbers in this category must be multiples of 4. So, $10 - 9 = 1$. This value goes into the cell for 'Multiple of 3' and 'Multiple of 4'.
  1. Determine the total number of multiples of 4: The problem states that 10 numbers in total are a multiple of 4. This is the column total for 'Multiple of 4'.
  1. Calculate the number of numbers that are not multiples of 3 and not multiples of 4: We know the total number of items is 20. We have accounted for 10 numbers that are multiples of 3. Therefore, the remaining numbers are not multiples of 3. So, $20 - 10 = 10$. This is the row total for 'Not a multiple of 3'.
  1. Calculate the number of numbers that are not multiples of 3 but are multiples of 4: We know that 10 numbers are multiples of 4 in total. We've already identified 1 number that is a multiple of 3 and a multiple of 4. Therefore, the remaining numbers in the 'Multiple of 4' column must not be multiples of 3. So, $10 - 1 = 9$. This value goes into the cell for 'Not a multiple of 3' and 'Multiple of 4'.
  1. Calculate the number of numbers that are not multiples of 3 and not multiples of 4: We know that 10 numbers are not multiples of 3 in total (from step 6). We've identified 9 of these that are multiples of 4 (from step 7). Therefore, the remaining numbers in this category must not be multiples of 4. So, $10 - 9 = 1$. This value goes into the cell for 'Not a multiple of 3' and 'Not a multiple of 4'.
  1. Construct the two-way frequency table: Based on the calculations above, the table is filled as follows:

| | Multiple of 4 | Not a multiple of 4 |
|-------------|---------------|---------------------|
| Multiple of 3 | 1 | 9 |
| Not a multiple of 3 | 9 | 1 |