Understanding Number Triads and Relationships

This question asks us to find a triad of numbers that shares the same relationship as two given example triads. A triad is simply a set of three numbers. We need to discover the rule or pattern connecting the three numbers in the example triads and then apply that rule to the given options to find the one that fits.

Analyzing the Given Triads to Find the Pattern

We are given two example triads:

• 6 - 7 - 42
• 8 - 13 - 104

Let's look at the relationship between the numbers in the first triad, 6, 7, and 42. We need to figure out how 42 is related to 6 and 7. Common operations include addition, subtraction, multiplication, or division.

• Is it addition? $6 + 7 = 13$. This is not 42.
• Is it subtraction? $7 - 6 = 1$ or $6 - 7 = -1$. This is not 42.
• Is it multiplication? $6 \times 7 = 42$. Yes, this works!

Let's check if this multiplication pattern holds for the second triad, 8, 13, and 104. According to our potential rule, the third number should be the product of the first two numbers.

• Multiply the first two numbers: $8 \times 13$.
• Let's calculate $8 \times 13$: $8 \times 10 = 80$, and $8 \times 3 = 24$. So, $80 + 24 = 104$.
• The third number in the triad is 104. $8 \times 13 = 104$. Yes, the pattern holds!

The relationship between the numbers in the given triads is that the third number is the product of the first and second numbers.

Mathematically, if the triad is represented as \(a - b - c\), the relationship is \(a \times b = c\).

Testing the Options Against the Discovered Relationship

Now we will examine each option to see which one follows the rule \(a \times b = c\), where \(a\) is the first number, \(b\) is the second number, and \(c\) is the third number in the triad.

Option 1: 5 - 15 - 125

• First number \(a = 5\), Second number \(b = 15\).
• Product \(a \times b = 5 \times 15\).
• $5 \times 15 = 75$.
• The third number in the triad is 125.
• Does \(75 = 125\)? No.
• This option does not follow the pattern.

Option 2: 6 - 4 - 72

• First number \(a = 6\), Second number \(b = 4\).
• Product \(a \times b = 6 \times 4\).
• $6 \times 4 = 24$.
• The third number in the triad is 72.
• Does \(24 = 72\)? No.
• This option does not follow the pattern.

Option 3: 12 - 11 - 132

• First number \(a = 12\), Second number \(b = 11\).
• Product \(a \times b = 12 \times 11\).
• $12 \times 11 = 132$.
• The third number in the triad is 132.
• Does \(132 = 132\)? Yes.
• This option follows the pattern.

Option 4: 9 - 5 - 24

• First number \(a = 9\), Second number \(b = 5\).
• Product \(a \times b = 9 \times 5\).
• $9 \times 5 = 45$.
• The third number in the triad is 24.
• Does \(45 = 24\)? No.
• This option does not follow the pattern.

Conclusion

Only Option 3, 12 - 11 - 132, follows the same relationship as the given triads, where the third number is the product of the first two numbers.

Revision Table: Number Triad Analysis

Additional Information: Reasoning with Number Patterns

Questions involving number triads or series often test your ability to identify mathematical relationships between numbers. These relationships can be simple arithmetic operations like addition, subtraction, multiplication, division, or more complex patterns involving squares, cubes, prime numbers, or sequences.

When approaching such problems:

• Look for patterns in the given examples first.
• Consider basic operations (add, subtract, multiply, divide).
• Check if the pattern involves the position of the numbers (e.g., first + second = third).
• Once a potential pattern is found, test it rigorously on all provided examples.
• Apply the confirmed pattern to the options one by one.
• Make sure to follow any specific instructions, such as the rule about not breaking down numbers into individual digits, as mentioned in this question. This rule reinforces that operations should be on the whole numbers themselves.

Practicing with different types of number pattern problems helps develop the skill to quickly identify the underlying rule.