Select the option that is related to the fourth term in the same way as the first term is related to the second term and the fifth term is related to the sixth term. 8 : 96 :: ? : 54 :: 12 : 216
6
Logical Reasoning: Number Analogy Solution
This question is based on the concept of number analogy, where we need to find the relationship between the numbers in the given pairs and apply the same relationship to find the missing term.
Analyzing the Given Number Pairs
We are given three pairs related by the '::' symbol:
- 8 : 96
- ? : 54
- 12 : 216
We need to find the pattern connecting the first number to the second number in the known pairs (8:96 and 12:216).
Finding the Relationship Pattern
Let's examine the first pair, 8 and 96. How can 8 be related to 96? We can try various operations:
- Multiplication: $8 \times 12 = 96$. The multiplier is 12.
- Squaring: $8^2 = 64$. $96 - 64 = 32$. $32 = 4 \times 8$. So, $8^2 + 4 \times 8 = 96$. Notice that 4 is half of 8. This suggests a potential pattern: $N^2 + (N/2) \times N$.
- Multiplication based on the number itself: $8 \times (8 + \text{something})$. $96 / 8 = 12$. So, $8 \times (8 + 4) = 96$. Again, 4 is half of 8. This suggests the pattern: $N \times (N + N/2)$.
Let's check the second known pair, 12 and 216, with the pattern $N \times (N + N/2)$:
- For $N=12$: $12 \times (12 + 12/2) = 12 \times (12 + 6) = 12 \times 18 = 216$. This matches the given pair!
So, the established pattern is that the second term is obtained by multiplying the first term (N) by the sum of the first term (N) and half of the first term (N/2). Mathematically, this is $N \times (N + N/2)$ or $N \times (3N/2)$ or $1.5 \times N^2$. Let's use $N \times (N + N/2)$.
Applying the Pattern to the Missing Term
Now, we apply this pattern to the second pair: ? : 54. Let the missing term be N.
According to the pattern:
$N \times (N + N/2) = 54$
Let's solve for N:
$N \times (3N/2) = 54$
$\frac{3N^2}{2} = 54$
$3N^2 = 54 \times 2$
$3N^2 = 108$
$N^2 = \frac{108}{3}$
$N^2 = 36$
$N = \sqrt{36}$
$N = 6$ (Assuming the positive value as is standard in these types of problems)
Verifying the Answer
Let's check if N=6 fits the pattern:
$6 \times (6 + 6/2) = 6 \times (6 + 3) = 6 \times 9 = 54$. This matches the given second term!
Therefore, the missing term is 6.
| Pair | First Term (N) | Relationship Applied: $N \times (N + N/2)$ | Calculated Second Term | Given Second Term |
|---|---|---|---|---|
| 1st | 8 | $8 \times (8 + 8/2) = 8 \times 12$ | 96 | 96 |
| 3rd | 12 | $12 \times (12 + 12/2) = 12 \times 18$ | 216 | 216 |
| 2nd | ? (N) | $N \times (N + N/2) = 54$ | 54 | 54 |
Conclusion
The relationship between the terms in each pair is $N : N \times (N + N/2)$. Applying this relationship to the second pair, we found that the missing term is 6.
Revision Table: Number Analogy Patterns
| Common Pattern Type | Description | Example |
|---|---|---|
| Arithmetic Operations | Adding, subtracting, multiplying, or dividing by a constant or variable number. | N : N+k, N : N*k |
| Square/Cube Relations | Relating to the square or cube of the number, often with additions/subtractions. | N : N$^2$+k, N : N$^3$-k |
| Combinations | Combining multiple operations or involving relations with N/2, N+1, N-1, etc. | N : N$^2$+N, N : N*(N+1), N : N*(N/2) |
Additional Information: Solving Logical Reasoning Problems
Logical reasoning questions, especially number analogies, require careful observation to identify the underlying pattern or rule. Here are some tips:
- Look for simple arithmetic relations first.
- Consider squares, cubes, square roots, or cube roots.
- Check for patterns involving the number itself, its half, or its neighbors (N+1, N-1).
- Sometimes the pattern involves the sum or product of digits.
- Practice with various types of problems to become familiar with common patterns.
Identifying the relationship accurately is key to solving number analogy questions correctly.
Similar Questions
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Important Questions from Letter and Number Based
Select the related number from the given alternatives that will complete the series:
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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 : ?
