Let's analyze the given number analogy problem to find the pattern and determine the missing number. The analogy is presented as three pairs:

5 : 27 :: 7 : ? :: 8 : 39

We need to find the relationship between the first number and the second number in the first pair (5 and 27) and in the third pair (8 and 39). Once the relationship or rule is identified, we can apply it to the second pair (7 and ?) to find the missing number.

Analyzing the Number Relationships

Let's examine the relationship between the numbers in the given pairs:

• Pair 1: 5 and 27
• Pair 3: 8 and 39

We are looking for a mathematical operation or combination of operations that transforms the first number of a pair into the second number. Let's assume a common linear relationship exists, such as multiplying the first number by a constant factor (\(k\)) and then adding or subtracting another constant (\(c\)). The rule would be \(n \times k + c\), where \(n\) is the first number in the pair.

Finding the Pattern Rule

Using the assumed linear relationship \(n \times k + c\), we can set up equations based on the known pairs:

For the pair (5, 27):

\[ 5k + c = 27 \quad (Equation\ 1) \]

For the pair (8, 39):

\[ 8k + c = 39 \quad (Equation\ 2) \]

Now we can solve this system of linear equations for \(k\) and \(c\). Subtract Equation 1 from Equation 2:

\[ (8k + c) - (5k + c) = 39 - 27 \]

\[ 3k = 12 \]

Divide by 3:

\[ k = \frac{12}{3} \]

\[ k = 4 \]

Now substitute the value of \(k=4\) into Equation 1 to find \(c\):

\[ 5(4) + c = 27 \]

\[ 20 + c = 27 \]

Subtract 20 from both sides:

\[ c = 27 - 20 \]

\[ c = 7 \]

So, the pattern rule is \(n \times 4 + 7\).

Applying the Pattern to Find the Missing Number

Now we apply the discovered rule \(n \times 4 + 7\) to the second pair, which starts with the number 7. Here, \(n = 7\). The missing number will be:

\[ 7 \times 4 + 7 \]

\[ 28 + 7 \]

\[ 35 \]

Therefore, the missing number in the analogy 7 : ? is 35.

Verifying the Pattern

Let's quickly check if the rule works for all given pairs:

• For 5: \(5 \times 4 + 7 = 20 + 7 = 27\) (Correct)
• For 7: \(7 \times 4 + 7 = 28 + 7 = 35\) (This is our calculated missing number)
• For 8: \(8 \times 4 + 7 = 32 + 7 = 39\) (Correct)

The rule \(n \times 4 + 7\) consistently explains the relationship in the given number analogy.

Conclusion on the Missing Number

Based on the established pattern, the number that relates to 7 in the same way as 5 relates to 27 and 8 relates to 39 is 35.

The options provided were:

1. 38
2. 37
3. 35
4. 36

Our calculated missing number is 35, which matches option 3.

Revision Table: Number Analogy Solving

Solving number analogy problems involves identifying the mathematical relationship between the given numbers. Key steps include:

• Analyze the relationship in known pairs.
• Hypothesize a potential rule (addition, subtraction, multiplication, division, powers, combination of operations, or a linear relationship like \(nk+c\)).
• Test the hypothesis on all given pairs to ensure consistency.
• Apply the confirmed rule to the number with the missing counterpart.

Additional Information: Types of Number Analogies

Number analogies in logical reasoning and quantitative aptitude tests can involve various patterns. Some common types include:

• Arithmetic Operations: Simple addition, subtraction, multiplication, or division.
• Powers and Roots: Squaring, cubing, square roots, etc., possibly combined with other operations.
• Sequences: The relationship might depend on the position in a series (e.g., \(n^2+1\)).
• Digit Operations: Operations on the digits of the number (sum of digits, product of digits).
• Combinations: A mix of different operations.
• Linear Relationships: As seen in this problem, \(nk+c\).

Practicing different types helps in quickly identifying the underlying pattern.