Finding the Pattern in Number Sets

The question asks us to identify the mathematical relationship between the numbers in the given sets and then find the option set that follows the exact same relationship. We are told that operations should be performed on the whole numbers as they are, without breaking them down into individual digits.

Analyzing the Given Number Sets

Let's look at the first given set: (63, 81, 7).

• The numbers are 63, 81, and 7.
• We need to find a pattern connecting these three numbers.
• Notice that 81 is a perfect square. \(81 = 9^2\).
• Let's see if we can obtain 9 from the other two numbers, 63 and 7.
• If we divide the first number (63) by the third number (7), we get \(63 \div 7 = 9\).
• Now, if we square this result (9), we get \(9^2 = 81\).
• This matches the second number in the set.

So, a possible relationship is: (First number / Third number)$^2$ = Second number.

Let's test this relationship with the second given set: (84, 49, 12).

• The numbers are 84, 49, and 12.
• Let's apply the same relationship: (First number / Third number)$^2$ = Second number.
• Calculate (First number / Third number): \(84 \div 12 = 7\).
• Now, square the result: \(7^2 = 49\).
• This matches the second number (49) in the set.

The relationship (First number / Third number)$^2$ = Second number holds true for both given sets.

Applying the Number Relationship to Options

Now, we will check each option to see which one satisfies the relationship: \((\text{First number} / \text{Third number})^2 = \text{Second number}\).

Option 1: (99, 121, 9)

• First number = 99
• Second number = 121
• Third number = 9
• Let's calculate (First number / Third number): \(99 \div 9 = 11\).
• Now, square the result: \(11^2 = 121\).
• Does this equal the Second number? Yes, 121 = 121.

Option 1 follows the identified number relationship.

Option 2: (76, 234, 12)

• First number = 76
• Second number = 234
• Third number = 12
• Let's calculate (First number / Third number): \(76 \div 12\). This does not give a whole number. \(76 \div 12 = \frac{76}{12} = \frac{19}{3}\).
• Squaring this would be \((\frac{19}{3})^2 = \frac{361}{9}\), which is not equal to 234.

Option 2 does not follow the identified number relationship.

Option 3: (12, 125, 56)

• First number = 12
• Second number = 125
• Third number = 56
• Let's calculate (First number / Third number): \(12 \div 56 = \frac{12}{56} = \frac{3}{14}\).
• Squaring this would be \((\frac{3}{14})^2 = \frac{9}{196}\), which is not equal to 125.
• Also, 125 is not a perfect square, which is required for the second number based on the pattern.

Option 3 does not follow the identified number relationship.

Option 4: (78, 14, 56)

• First number = 78
• Second number = 14
• Third number = 56
• Let's calculate (First number / Third number): \(78 \div 56 = \frac{78}{56} = \frac{39}{28}\).
• Squaring this would be \((\frac{39}{28})^2 = \frac{1521}{784}\), which is not equal to 14.
• Also, 14 is not a perfect square, which is required for the second number based on the pattern.

Option 4 does not follow the identified number relationship.

Conclusion

Based on the analysis, only Option 1, (99, 121, 9), follows the same number relationship as the given sets (63, 81, 7) and (84, 49, 12). The relationship is \((\text{First number} / \text{Third number})^2 = \text{Second number}\).

Revision Table: Key Concepts Reviewed

This question involves understanding and identifying mathematical patterns within sets of numbers. Key concepts include:

• Basic arithmetic operations: Division and squaring.
• Identifying relationships: Looking for how numbers in a set are connected.
• Applying patterns: Using a discovered relationship to test other sets.
• Checking conditions: Ensuring operations are on whole numbers, not digits.

Additional Information: Number Pattern Recognition

Number pattern recognition is a common type of question in logical reasoning and quantitative aptitude tests. These questions assess your ability to observe, identify, and extend patterns in sequences or sets of numbers. The patterns can involve various mathematical operations or logical rules. Some common types of patterns include:

• Arithmetic progression (constant difference)
• Geometric progression (constant ratio)
• Squares or cubes of numbers
• Combinations of operations (like the pattern in this question)
• Fibonacci-like sequences

Practicing different types of number pattern problems helps in developing skills to quickly identify the underlying rule. It often requires trial and error, testing different operations or combinations of numbers within the set.