Understanding Number Analogies and Finding Patterns

Number analogies are a type of logical reasoning question where you are given a pair of numbers related by a certain rule or pattern. You then need to apply the same rule to a third number to find a missing fourth number. The analogy given is:

\(14 : 238 :: 18 : ?\)

Our goal is to figure out the relationship between 14 and 238 and use that to find the number related to 18.

Analyzing the Relationship in the First Pair (14 : 238)

Let's explore different ways 14 could be related to 238:

• Multiplication: Can we multiply 14 by a number to get 238? Let's divide 238 by 14: \(238 \div 14 = 17\). So, \(14 \times 17 = 238\). The multiplier is 17.
• Squaring: \(14^2 = 196\). The difference between 238 and 196 is \(238 - 196 = 42\). Notice that \(42 = 3 \times 14\). This suggests a pattern like \(n^2 + 3n\). Let's check: \(n^2 + 3n = n(n+3)\). For \(n=14\), this is \(14(14+3) = 14 \times 17 = 238\). This also works and confirms the multiplier is 17, which is \(14+3\).

We have found that \(14 \times 17 = 238\), and the multiplier 17 can be seen as \(14+3\).

Identifying the Pattern for the Multiplier

Let's think about the number 17. Is it special in relation to 14 in another way? The prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23, ... Looking at this sequence, 17 is the very next prime number that comes after 14.

This leads to a potential pattern: The second number is the first number multiplied by the next prime number greater than the first number.

• For 14: The next prime after 14 is 17. \(14 \times 17 = 238\). This matches the given pair.

This pattern seems promising.

Applying the Pattern to the Second Pair (18 : ?)

Now, let's apply the pattern "multiply by the next prime number greater than the current number" to 18.

• Identify the third number: It is 18.
• Find the next prime number after 18: The prime numbers sequence continues ... 17, 19, 23, ... The next prime number immediately following 18 is 19.
• Apply the multiplication: Multiply 18 by the next prime, which is 19.

Let's calculate \(18 \times 19\):

\(18 \times 19 = 18 \times (20 - 1)\)

\(= (18 \times 20) - (18 \times 1)\)

\(= 360 - 18\)

\(= 342\)

So, the missing number is 342.

Verifying with the Options

The calculated missing number is 342. Let's compare this with the provided options:

1. 342
2. 651
3. 301
4. 405

Our calculated number, 342, matches option 1.

Summary of the Number Analogy Solution

The relationship in the analogy \(14 : 238 :: 18 : ?\) is that the second number is the product of the first number and the next prime number greater than the first number. Applying this rule to 18 gives \(18 \times 19 = 342\).

Revision Table: Essential Concepts for Analogies

Additional Information: Strategies for Solving Number Analogies

When tackling number analogies and number series problems, consider these common strategies:

• Look for basic arithmetic relationships (addition, subtraction, multiplication, division).
• Check for squares or cubes (\(n^2\), \(n^3\)) and variations (\(n^2 \pm k\), \(n^3 \pm k\)).
• Investigate relationships involving digits (sum of digits, product of digits).
• Consider sequences based on types of numbers (prime, composite, odd, even).
• Look for combined operations (e.g., \(n \times (n+1)\), \(n^2 - n\)).
• Always test the potential pattern on the first pair before applying it to the second.
• If your first hypothesis doesn't yield an option, try a different pattern.

Practice with various types of number patterns will help you quickly recognize relationships during exams.