Select the number that is related to the third number in the same way as the second number is related to the first number. 14 : 238 : : 18 : ?
342
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:
- 342
- 651
- 301
- 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 \(18 \times 19 = 342\).
| First Number | Next Prime Number | Operation | Result |
|---|---|---|---|
| 14 | 17 (Next prime > 14) | \(14 \times 17\) | 238 |
| 18 | 19 (Next prime > 18) | \(18 \times 19\) | 342 |
Revision Table: Essential Concepts for Analogies
| Concept | Description | Relevance to this Problem |
|---|---|---|
| Logical Reasoning | Using critical thinking to identify patterns and relationships. | Needed to spot the \(n \times (\text{next prime})\) pattern. |
| Number Patterns | Mathematical rules connecting numbers in a sequence or pair. | The specific pattern discovered was multiplication by the next prime. |
| Prime Numbers | Numbers greater than 1 divisible only by 1 and themselves. | Crucial for defining the multiplier in this specific analogy. |
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.
Similar Questions
Select the set in which the numbers are related in the same way as are the numbers of the following set.
(NOTE: Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13 – Operations on 13 such as adding /subtracting /multiplying etc. to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is NOT allowed)
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Select the set in which the numbers are related in the same way as are the numbers of the following sets.
(NOTE : Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13- Operations on 13 such as adding/subtracting/multiplying etc. to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed.)
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(10, 13, 12)
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Select the option that is related to the third term in the same way as the second term is related to the first term and the sixth term is related to the fifth term.
16 : 50 :: 49 : ? :: 144 : 338
Important Questions from Letter and Number Based
Select the related number from the given alternatives that will complete the series:
Y 2 : 4 : : V 2 : ?Select the related number from the given alternatives:
F : 216 : : L : ?Select the option in which the numbers are related in the same way as are the numbers of the following set.
(12, 60, 84)
Three of the following four number-pairs ale alike in a certain way and one is different. Find the odd one out.
In the following question, select the related number from the given alternatives.
52 : 57 ∷ 46 : ?
