Understanding the Number Analogy Problem

The question asks us to identify the relationship between pairs of numbers and apply that same relationship to a third number to find a missing term. The given analogy is:

\(7 : 84 :: 5 : ? :: 9 : 144\)

This format means "7 is to 84 as 5 is to ? as 9 is to 144". We need to find the pattern connecting 7 and 84, and 9 and 144, and then use that pattern to find the number related to 5.

Identifying the Pattern in the Given Number Pairs

Let's examine the relationships in the two complete pairs provided.

Analyzing the Pair 7 and 84

How is 84 related to 7? We can test common mathematical operations.

• Is it addition? \(7 + N = 84 \implies N = 77\).
• Is it multiplication? \(7 \times N = 84\). Dividing 84 by 7, we get \(N = \frac{84}{7} = 12\). So, \(84 = 7 \times 12\).

Multiplication by 12 seems like a possible relationship.

Analyzing the Pair 9 and 144

Let's apply a similar approach to the second pair.

• Is it multiplication? \(9 \times M = 144\). Dividing 144 by 9, we get \(M = \frac{144}{9} = 16\). So, \(144 = 9 \times 16\).

Multiplication by 16 is the relationship here. We have different multipliers (12 and 16) for different starting numbers (7 and 9).

Deducing the General Pattern Rule

The multiplier is not constant. It depends on the first number in the pair. Let the first number be \(n\) and the multiplier be \(P\).

• When \(n = 7\), the multiplier \(P = 12\).
• When \(n = 9\), the multiplier \(P = 16\).

We need to find a rule that relates \(P\) to \(n\). Let's look at the relationship between \(n\) and \(P\):

• For \(n=7\), \(P=12\). Notice that \(12 = 7 + 5\).
• For \(n=9\), \(P=16\). Notice that \(16 = 9 + 7\).

The number added to \(n\) to get the multiplier \(P\) is increasing (5, then 7). Let's see if there's a relationship between this added number and \(n\).

• When \(n=7\), the added number is 5, which is \(7 - 2\).
• When \(n=9\), the added number is 7, which is \(9 - 2\).

This suggests the pattern for the multiplier \(P\) might be \(P = n + (n - 2)\). Let's test this:

• For \(n=7\), \(P = 7 + (7 - 2) = 7 + 5 = 12\). Correct.
• For \(n=9\), \(P = 9 + (9 - 2) = 9 + 7 = 16\). Correct.

So the multiplier is \(n + (n - 2) = 2n - 2\).

The general rule is: Second term = First term \(\times\) (2 \(\times\) First term - 2).

In mathematical terms, if the first term is \(n\), the second term is \(n \times (2n - 2)\) or \(2n(n-1)\).

Applying the Pattern to Find the Missing Number

The analogy asks for the term related to 5. Here, the first term is \(n = 5\).

Using the rule: Missing Term = \(5 \times (2 \times 5 - 2)\).

First, calculate the value inside the parenthesis:

\(2 \times 5 - 2 = 10 - 2 = 8\).

Now, multiply by the first term, 5:

Missing Term = \(5 \times 8 = 40\).

Verifying the Answer with Options

The calculated missing term is 40. Let's compare this with the provided options:

1. 30
2. 40
3. 36
4. 45

The value 40 matches Option 2.

Conclusion

By analyzing the relationship between the given pairs (7:84 and 9:144), we found that the second term is calculated by multiplying the first term (\(n\)) by \(2n-2\). Applying this pattern to the number 5, we found the missing term to be 40. Therefore, the correct option is 40.

Revision Table: Solving Number Analogy Problems

Additional Information: Tips for Reasoning Pattern Questions

Number analogy and pattern recognition questions are common in reasoning sections of exams. Here are some general tips:

• Start Simple: Always check for basic arithmetic operations (addition, subtraction, multiplication, division) first.
• Consider Squares and Cubes: Many patterns involve squares or cubes of the number, or numbers close to them.
• Look at Differences/Ratios: Calculate the difference or ratio between the numbers in each pair. See if there is a pattern in these differences or ratios.
• Check for Prime Numbers or Series: Sometimes the numbers follow a sequence of prime numbers or other specific series.
• Examine Digits: In some cases, the pattern depends on the sum or product of the digits, or rearranging digits.
• Combine Operations: The pattern might involve multiple steps, like multiplying and then adding, or squaring and then subtracting.
• Systematic Testing: If one type of pattern doesn't work, move on to the next systematically. Don't get stuck on a single idea.
• Practice: The more you practice different types of patterns, the quicker you will become at recognizing them.