Select the number that is related to the third number in the same way as the second number is related to the first number. 108 ∶ 6 ∶∶ 256 ∶ ?
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Solving the Number Analogy: 108 ∶ 6 ∶∶ 256 ∶ ?
The question asks us to find the number that shares the same relationship with 256 as 6 shares with 108. This is a classic number analogy problem where we need to identify the rule connecting the first pair of numbers and apply it to the third number to find the fourth one.
Identifying the Relationship between 108 and 6
Let's analyze the numbers 108 and 6. We need to find a mathematical operation or pattern that links 6 to 108. Let's consider different possibilities involving the number 6:
- Addition/Subtraction: \(6 + X = 108\) (X=102); \(108 - X = 6\) (X=102). Not a simple addition or subtraction.
- Multiplication/Division: \(6 \times X = 108\). \(X = 108 / 6 = 18\). The relationship could be multiplying the second number by 18 to get the first.
- Powers: \(6^2 = 36\), \(6^3 = 216\). Neither is 108.
- Roots: \(\sqrt{108}\) is not an integer.
- Combination of operations: Let's look at the powers again. We saw \(6^3 = 216\). Interestingly, 108 is exactly half of 216. So, the relationship could be: the first number is half of the cube of the second number.
Let's test this proposed relationship: If the second number is \(B\), the first number \(A\) is related by \(A = \frac{B^3}{2}\).
For the first pair, \(A = 108\) and \(B = 6\). Let's check if \(108 = \frac{6^3}{2}\):
\[ \frac{6^3}{2} = \frac{6 \times 6 \times 6}{2} = \frac{36 \times 6}{2} = \frac{216}{2} = 108 \]This relationship holds true for the first pair (108 ∶ 6).
Applying the Relationship to 256
Now we apply the same relationship to the third number, 256, to find the fourth number. Let the third number be \(C = 256\), and the unknown fourth number be \(D\). The relationship is \(C = \frac{D^3}{2}\).
We need to solve for \(D\):
\[ 256 = \frac{D^3}{2} \]Multiply both sides by 2:
\[ 256 \times 2 = D^3 \] \[ 512 = D^3 \]To find \(D\), we need to calculate the cube root of 512:
\[ D = \sqrt[3]{512} \]We need to find a number that, when multiplied by itself three times, equals 512.
- \(7^3 = 7 \times 7 \times 7 = 49 \times 7 = 343\)
- \(8^3 = 8 \times 8 \times 8 = 64 \times 8 = 512\)
So, \(D = 8\).
Verifying with the Options
The calculated fourth number is 8. Let's check the given options:
| Option | Value | Does it match? |
|---|---|---|
| 1 | 8 | Yes |
| 2 | 16 | No |
| 3 | 12 | No |
| 4 | 9 | No |
The value 8 matches Option 1.
Conclusion
The relationship between the numbers is that the first number is half of the cube of the second number. Applying this rule to 256, we find that it is related to 8 because 256 is half of the cube of 8 (\(8^3/2 = 512/2 = 256\)).
Revision Table: Key Concepts
| Concept | Description | Application in this problem |
|---|---|---|
| Number Analogy | Finding a relationship between a pair of numbers and applying it to another. | Used to determine the missing number based on the pattern 108:6. |
| Cube of a Number | A number multiplied by itself three times (\(n^3\)). | Finding \(6^3\) and \(8^3\) was crucial to identify and use the relationship. |
| Cube Root | The inverse operation of cubing; finding the number that, when cubed, gives the original number (\(\sqrt[3]{x}\)). | Used to find the fourth number \(D\) from the equation \(D^3 = 512\). |
Additional Information: Solving Reasoning Questions
Number analogy and reasoning questions often require identifying patterns based on basic arithmetic operations, powers, roots, prime numbers, or combinations of these. Here are some tips for solving such questions:
- Look for simple arithmetic first: addition, subtraction, multiplication, division.
- Check for squares and cubes: Is one number the square or cube of the other, or related to it by squaring/cubing?
- Consider roots: Is one number the square root or cube root of the other?
- Examine digit properties: Sum of digits, product of digits, reverse digits, etc.
- Look for combined operations: Often, the relationship involves more than one step (e.g., squaring and then adding a number, cubing and dividing).
- Check prime or composite numbers: Are the numbers prime, composite, or related based on these properties?
- Test your hypothesis: Once you find a potential relationship, apply it to the first pair to ensure it holds true before applying it to the second pair.
- Consider common patterns: Look up common number series and analogy patterns encountered in exams.
Practice is key to quickly identifying patterns in numerical reasoning problems.
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In the following question, select the related number from the given alternatives.
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