Identifying the Odd Number-Pair Pattern

The question asks us to identify the number-pair among the given options where the relationship between the first and second numbers is different from the others. We are told that the same mathematical operation(s) connect the numbers in all pairs except one.

Let's look at the given number-pairs:

• (7, 89)
• (8, 102)
• (5, 60)
• (6, 76)

We need to find a rule or pattern that applies to most of these pairs. Let the first number in a pair be \(n\) and the second number be \(m\). We can try to find a linear relationship of the form \(m = an + b\), where \(a\) and \(b\) are constants.

Let's use two of the number-pairs to find the potential values of \(a\) and \(b\). We can use (7, 89) and (8, 102).

For the pair (7, 89):

\(89 = a(7) + b \quad (Equation\ 1)\)

For the pair (8, 102):

\(102 = a(8) + b \quad (Equation\ 2)\)

Now, we can subtract Equation 1 from Equation 2:

\((102) - (89) = (8a + b) - (7a + b)\)

\(13 = 8a - 7a + b - b\)

\(13 = a\)

Now that we have the value of \(a\), we can substitute it back into either Equation 1 or Equation 2 to find \(b\). Let's use Equation 1:

\(89 = 7(13) + b\)

\(89 = 91 + b\)

\(b = 89 - 91\)

\(b = -2\)

So, the potential pattern connecting the number-pairs is \(m = 13n - 2\).

Now, let's test this pattern with all the given number-pairs to see which one does not follow this rule.

• Number-pair (7, 89):

  First number \(n = 7\).

  Using the pattern \(m = 13n - 2\), the expected second number is \(13 \times 7 - 2 = 91 - 2 = 89\). This matches the given second number.

• Number-pair (8, 102):

  First number \(n = 8\).

  Using the pattern \(m = 13n - 2\), the expected second number is \(13 \times 8 - 2 = 104 - 2 = 102\). This matches the given second number.

• Number-pair (5, 60):

  First number \(n = 5\).

  Using the pattern \(m = 13n - 2\), the expected second number is \(13 \times 5 - 2 = 65 - 2 = 63\). The given second number is 60. This does not match the expected number.

• Number-pair (6, 76):

  First number \(n = 6\).

  Using the pattern \(m = 13n - 2\), the expected second number is \(13 \times 6 - 2 = 78 - 2 = 76\). This matches the given second number.

From the testing, we can see that the number-pairs (7, 89), (8, 102), and (6, 76) all follow the pattern \(m = 13n - 2\). The number-pair (5, 60) is the only one that does not follow this pattern, as \(13 \times 5 - 2 = 63\), not 60.

Therefore, the odd number-pair is (5, 60).

Revision Table: Analyzing Number-Pair Patterns

This type of question requires careful observation and testing different possible mathematical operations or relationships between the numbers in a pair. Common patterns include linear relations (\(m = an + b\)), quadratic relations (\(m = an^2 + bn + c\) or \(m = an^2 + b\)), multiplicative relations (\(m = an\) or \(m = an + b\)), or relationships involving squares or cubes of the numbers.

Steps to solve such problems:

1. Examine the given number-pairs.
2. Look for simple relationships (addition, subtraction, multiplication, division).
3. If simple relations don't work, consider linear relationships (\(m = an + b\)). Use two pairs to set up equations and solve for \(a\) and \(b\).
4. Test the derived pattern on all other pairs.
5. The pair that does not fit the pattern is the odd one out.
6. If a linear pattern doesn't work, consider quadratic or other polynomial relationships, or relations involving squares/cubes and constants.

Additional Information on Number Series and Patterns

Pattern identification is a key skill in logical reasoning and quantitative aptitude tests. Questions involving number-pairs or number series assess your ability to find underlying rules. Besides linear patterns like the one found here, other common patterns include:

• Arithmetic Progression: A constant difference between consecutive terms (if the pairs were consecutive). Not applicable directly here, but the difference between the second numbers for consecutive first numbers can reveal linear patterns.
• Geometric Progression: A constant ratio between consecutive terms.
• Difference Patterns: The difference between consecutive terms forms its own pattern (e.g., arithmetic or geometric).
• Perfect Squares/Cubes: Numbers might be related to squares or cubes of the first number, possibly with additions or subtractions.
• Alternating Patterns: Sometimes two different patterns alternate.
• Fibonacci Sequence: Each term is the sum of the two preceding ones (not common for number-pairs unless specifically mentioned).

Practicing with various types of number pattern problems helps in quickly recognizing potential relationships and applying the correct method to find the odd one out or the next term in a series.