Solving Number Manipulation and Average Calculation Problem

This problem requires us to apply a set of given rules to two separate rows of numbers to find a resultant number for each row. After finding the resultant numbers, we need to calculate their average.

Understanding the Rules for Number Rows

Let's first list down the rules provided for manipulating the numbers in each row:

1. If an even number is followed by an even number, the first is to be subtracted from the second.
2. If an odd number is followed by an odd number, both are to be added.
3. If an odd number is followed by an even number, the even number is to be subtracted from the square of the odd number.
4. If an even number is followed by an odd number, the odd number is to be subtracted from the even number.

We need to apply these rules sequentially from left to right for each row.

Calculating the Resultant for Row I

Row I is: 28, 19, 12

Let's process the numbers step-by-step:

• Consider the first two numbers: 28 and 19.
  • 28 is an even number.
  • 19 is an odd number.
  • This combination is an even number followed by an odd number. According to Rule iv, the odd number (19) is subtracted from the even number (28).
  • Calculation: \(28 - 19 = 9\)
• Now, consider the result from the first step (9) and the third number in the row (12).
  • 9 is an odd number.
  • 12 is an even number.
  • This combination is an odd number followed by an even number. According to Rule iii, the even number (12) is subtracted from the square of the odd number (9).
  • Calculation: \(9^2 - 12 = 81 - 12 = 69\)

The resultant number for Row I is 69.

Calculating the Resultant for Row II

Row II is: 13, 7, 32

Let's process the numbers step-by-step:

• Consider the first two numbers: 13 and 7.
  • 13 is an odd number.
  • 7 is an odd number.
  • This combination is an odd number followed by an odd number. According to Rule ii, both numbers are added.
  • Calculation: \(13 + 7 = 20\)
• Now, consider the result from the first step (20) and the third number in the row (32).
  • 20 is an even number.
  • 32 is an even number.
  • This combination is an even number followed by an even number. According to Rule i, the first number (20) is subtracted from the second number (32).
  • Calculation: \(32 - 20 = 12\)

The resultant number for Row II is 12.

Calculating the Average of Resultant Numbers

We have the resultant number for Row I as 69 and for Row II as 12.

The average is calculated by adding the two resultant numbers and dividing by 2.

Average \( = \frac{\text{Resultant (Row I)} + \text{Resultant (Row II)}}{2} \)

Average \( = \frac{69 + 12}{2} \)

Average \( = \frac{81}{2} \)

Average \( = 40.5 \)

The average of the resultant numbers of the first row and the second row is 40.5.

Revision Table: Summary of Calculations

Average Calculation:

\( \frac{69 + 12}{2} = \frac{81}{2} = 40.5 \)

Additional Information: Understanding Parity and Arithmetic Operations

This problem heavily relies on the concept of parity, which means whether a number is even or odd. Understanding how arithmetic operations behave with even and odd numbers is crucial for solving such problems quickly.

• Even Numbers: Integers divisible by 2 (\(\dots, -4, -2, 0, 2, 4, \dots\)).
• Odd Numbers: Integers not divisible by 2 (\(\dots, -3, -1, 1, 3, 5, \dots\)).

Properties used in the rules:

• Even - Odd = Odd
• Odd + Odd = Even
• Odd\(^2\) = Odd
• Odd\(^2\) - Even = Odd - Even = Odd
• Even - Even = Even

In our calculations:

• Row I, Step 1: Even (28) - Odd (19) = Odd (9). This matches the property.
• Row I, Step 2: Odd (9\(^2\)) - Even (12) = Odd (81) - Even (12) = Odd (69). This matches the property.
• Row II, Step 1: Odd (13) + Odd (7) = Even (20). This matches the property.
• Row II, Step 2: Even (32) - Even (20) = Even (12). This matches the property.

While checking the parity of the intermediate results isn't strictly necessary to solve this specific problem using the given rules, understanding these properties can sometimes help in verifying steps or solving related problems.