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
What comes next?
AJ, BL, CN, ?
In the word "FUNDAMENTAL", if we assign even numbers to vowels (A=2, E=4, I=6, O=8, U=10), what is the sum of all vowel values?
Select the term from among the given options that can replace the question mark (?) in the following series.
YL 93, XK 88, WJ 83, VI 78, ?
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)
(49, 63, 441)
(7, 14, 14)
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.)
(15, 20, 4)
(30, 30, 3)
Select the set in which the numbers are related in the same way as the numbers of the given 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.)
(4, 6, 8)
(6, 8, 10)
Select the option that is related to the fifth term in the same way as the second term is related to the first term and the fourth term is related to the third term.
132 : 10 :: 42 : 5 :: 272 : ?
Study the given pattern carefully and select the number that can replace the question mark (?) in it.
(6, 2, 34)
(1, 5, 13)
(9, 1, ?)
(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)
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)
(2, 114, 19)
(12, 216, 6)
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.
6 : 646 :: 5 : ? :: 4 : 190
Important Questions from Letter and Number Based
Select the option that is related to the third number in the same way as the second number is related to the first number.
22 : 441 :: 13 : ?Select the option that is related to the third number in the same way as the second number is related to the first number.
31 : 90 :: 43 : ?
Select the option which is related to the third number in the same way as the second number is related to the first number.
7 : 56 :: 13 : ?
Select the option that is related to the third number in the same way as the second number is related to the first number.
77 : 11 :: 259 : ?
Select the option that is related to the third number in the same way as the second number is related to the first number.
15 : 270 :: 13 : ?
