Answer :
Certainly! Let's solve the problem of finding the missing number in the provided array:
[tex]$ \begin{array}{llll} 1 & 4 & 9 & 6 \\ 5 & 6 & 9 & 4 \\ 1 & ? & 1 & 4 \end{array} $[/tex]
We are given three rows of numbers, with a missing value represented by `?` in the third row. To solve for the missing number, we will follow a logical approach based on the numbers given.
### Step-by-Step Solution:
1. Identify the positions:
- Let's denote the columns as:
[tex]\[ \text{Column 1}, \text{Column 2}, \text{Column 3}, \text{Column 4} \][/tex]
- The missing number is in [tex]\(\text{Row 3, Column 2}\)[/tex].
2. First Observation:
- Row 1: [tex]\([1, 4, 9, 6]\)[/tex]
- Row 2: [tex]\([5, 6, 9, 4]\)[/tex]
- Row 3: [tex]\([1, ?, 1, 4]\)[/tex]
3. Formulate the hypothesis:
- Given the structure and values, we will compare sums or differences of specific columns within the rows to find a pattern. As a hypothesis, consider the thought that the missing number in [tex]\(\text{Row 3, Column 2}\)[/tex] could relate to the sum or difference of elements in columns of the first two rows.
4. Computational Step:
- Take into account the values surrounding the missing number. We hypothesize that the number in [tex]\(\text{Row 3, Column 2}\)[/tex] can be derived based on some arithmetic condition met by other elements in [tex]\(\text{Row 3}\)[/tex].
- Assuming that the sum of elements in [tex]\(\text{Row 3, Column 1}\)[/tex] and [tex]\(\text{Row 3, Column 3}\)[/tex] minus the element in [tex]\(\text{Row 3, Column 4}\)[/tex] gives the element in [tex]\(\text{Row 3, Column 2}\)[/tex].
Since:
[tex]\[ \text{Column 1: } 1 \][/tex]
[tex]\[ \text{Column 3: } 1 \][/tex]
[tex]\[ \text{Column 4: } 4 \][/tex]
Let's perform the arithmetic:
[tex]\[ \text{Looking for a number } x \text{ in \(\text{Row 3, Column 2}\)} \][/tex]
Sum the values and subtract the fourth column value from the sum of the other elements in [tex]\(\text{Row 3}\)[/tex]:
[tex]\[ x = 1 + 1 - 4 = 2 - 4 = -2 \][/tex]
5. Missing Number:
- Therefore, the missing number in [tex]\(\text{Row 3, Column 2}\)[/tex] is:
[tex]\[ -2 \][/tex]
So, the missing number is [tex]\(-2\)[/tex]. The complete array with the missing value filled in is:
[tex]$ \begin{array}{llll} 1 & 4 & 9 & 6 \\ 5 & 6 & 9 & 4 \\ 1 & -2 & 1 & 4 \end{array} $[/tex]
[tex]$ \begin{array}{llll} 1 & 4 & 9 & 6 \\ 5 & 6 & 9 & 4 \\ 1 & ? & 1 & 4 \end{array} $[/tex]
We are given three rows of numbers, with a missing value represented by `?` in the third row. To solve for the missing number, we will follow a logical approach based on the numbers given.
### Step-by-Step Solution:
1. Identify the positions:
- Let's denote the columns as:
[tex]\[ \text{Column 1}, \text{Column 2}, \text{Column 3}, \text{Column 4} \][/tex]
- The missing number is in [tex]\(\text{Row 3, Column 2}\)[/tex].
2. First Observation:
- Row 1: [tex]\([1, 4, 9, 6]\)[/tex]
- Row 2: [tex]\([5, 6, 9, 4]\)[/tex]
- Row 3: [tex]\([1, ?, 1, 4]\)[/tex]
3. Formulate the hypothesis:
- Given the structure and values, we will compare sums or differences of specific columns within the rows to find a pattern. As a hypothesis, consider the thought that the missing number in [tex]\(\text{Row 3, Column 2}\)[/tex] could relate to the sum or difference of elements in columns of the first two rows.
4. Computational Step:
- Take into account the values surrounding the missing number. We hypothesize that the number in [tex]\(\text{Row 3, Column 2}\)[/tex] can be derived based on some arithmetic condition met by other elements in [tex]\(\text{Row 3}\)[/tex].
- Assuming that the sum of elements in [tex]\(\text{Row 3, Column 1}\)[/tex] and [tex]\(\text{Row 3, Column 3}\)[/tex] minus the element in [tex]\(\text{Row 3, Column 4}\)[/tex] gives the element in [tex]\(\text{Row 3, Column 2}\)[/tex].
Since:
[tex]\[ \text{Column 1: } 1 \][/tex]
[tex]\[ \text{Column 3: } 1 \][/tex]
[tex]\[ \text{Column 4: } 4 \][/tex]
Let's perform the arithmetic:
[tex]\[ \text{Looking for a number } x \text{ in \(\text{Row 3, Column 2}\)} \][/tex]
Sum the values and subtract the fourth column value from the sum of the other elements in [tex]\(\text{Row 3}\)[/tex]:
[tex]\[ x = 1 + 1 - 4 = 2 - 4 = -2 \][/tex]
5. Missing Number:
- Therefore, the missing number in [tex]\(\text{Row 3, Column 2}\)[/tex] is:
[tex]\[ -2 \][/tex]
So, the missing number is [tex]\(-2\)[/tex]. The complete array with the missing value filled in is:
[tex]$ \begin{array}{llll} 1 & 4 & 9 & 6 \\ 5 & 6 & 9 & 4 \\ 1 & -2 & 1 & 4 \end{array} $[/tex]
