Analyzing the Number Pattern

The question asks us to identify a missing number in a given pattern. We are provided with three rows of numbers, and we need to find the number that should replace the question mark (?) in the third row. The key constraint is that all operations must be performed on the given whole numbers, not on their individual digits.

Let's denote the numbers in each row as:

• Row 1: First Number = 21, Second Number = 12, Third Number = 19
• Row 2: First Number = 27, Second Number = 10, Third Number = 19
• Row 3: First Number = 24, Second Number = 10, Third Number = ?

Identifying the Pattern Rule

We need to find a consistent mathematical relationship between the First Number, Second Number, and the Third Number in each row. Let's explore potential patterns. A common approach is to see how the first two numbers relate to the third.

Let's consider the operation: Sum = First Number + Second Number

• Row 1 Sum: $21 + 12 = 33$
• Row 2 Sum: $27 + 10 = 37$
• Row 3 Sum: $24 + 10 = 34$

Now, let's see how this Sum relates to the Third Number. We can look at the difference: SubtractValue (SV) = Sum - Third Number

• Row 1: $SV = 33 - 19 = 14$
• Row 2: $SV = 37 - 19 = 18$
• Row 3: $SV = 34 - ?$. If the answer is 18, then $SV = 34 - 18 = 16$.

So, the sequence of Subtract Values (SV) for the three rows is 14, 18, 16.

Finding the Pattern for SubtractValue (SV)

We need to find how SV (14, 18, 16) is derived from the numbers in each row. Let's test if SV relates to the Second Number.

Pattern: $SV = \text{Second Number} + \text{Adjustment}$

• Row 1: $14 = 12 + \text{Adjustment}_1 \implies \text{Adjustment}_1 = 14 - 12 = 2$
• Row 2: $18 = 10 + \text{Adjustment}_2 \implies \text{Adjustment}_2 = 18 - 10 = 8$
• Row 3: $16 = 10 + \text{Adjustment}_3 \implies \text{Adjustment}_3 = 16 - 10 = 6$

The sequence of Adjustments is 2, 8, 6.

Now, let's see how this Adjustment relates to the First Number in each row.

Pattern: $\text{Adjustment} = \text{First Number} - \text{Constant}$

• Row 1: $2 = 21 - \text{Constant}_1 \implies \text{Constant}_1 = 21 - 2 = 19$
• Row 2: $8 = 27 - \text{Constant}_2 \implies \text{Constant}_2 = 27 - 8 = 19$
• Row 3: $6 = 24 - \text{Constant}_3 \implies \text{Constant}_3 = 24 - 6 = 18$

The sequence of Constants is 19, 19, 18. This sequence appears to be the defining rule for the adjustment calculation.

Applying the Pattern to Find the Missing Number

The complete pattern derived is:

1. Calculate the Sum: $\text{Sum} = \text{First Number} + \text{Second Number}$
2. Determine the Constant (C) for the row: C is 19 for rows 1 and 2, and 18 for row 3.
3. Calculate the Adjustment: $\text{Adjustment} = \text{First Number} - C$
4. Calculate the Subtract Value: $SV = \text{Second Number} + \text{Adjustment}$
5. Calculate the Third Number: $\text{Third Number} = \text{Sum} - SV$

Let's apply this to Row 3:

• First Number = 24, Second Number = 10
• Sum = $24 + 10 = 34$
• The Constant (C) for Row 3 is 18.
• Adjustment = $24 - 18 = 6$
• SV = $10 + 6 = 16$
• Third Number = $34 - 16 = 18$

Therefore, the number that can replace the question mark (?) is 18.