Understanding the Analogy Question

The question asks us to find the missing term in an analogy based on the relationship between given pairs of numbers. We are given three pairs, with one term missing in the second pair: 8 : 496 :: ? : 204 :: 9 : 711. This means the relationship between 8 and 496 is the same as the relationship between the missing number and 204, and also the same as the relationship between 9 and 711.

Identifying the Pattern in the Number Pairs

Let's examine the relationship between the numbers in the first and third pairs to find a consistent pattern. The pairs are (8, 496) and (9, 711).

For the first pair (8, 496), let's try some common numerical relationships involving 8:

• Squaring: \(8^2 = 64\). Not close to 496.
• Cubing: \(8^3 = 512\). This is close to 496. The difference is \(512 - 496 = 16\). Notice that \(16 = 2 \times 8\). So, the relationship could be \(n^3 - 2n\), where \(n\) is the first term. Let's check if this works: \(8^3 - 2 \times 8 = 512 - 16 = 496\). This pattern fits the first pair.

Now, let's test this potential pattern \(n^3 - 2n\) on the third pair (9, 711). Here, \(n = 9\).

• Applying the pattern: \(9^3 - 2 \times 9 = 729 - 18 = 711\). This matches the second term of the third pair.

The pattern \(n^3 - 2n\) seems to be the correct relationship between the first term \(n\) and the second term \(n^3 - 2n\) in each pair.

Applying the Pattern to the Second Pair

The second pair is ? : 204. Let the missing term be \(x\). According to the pattern we found, the relationship is \(x : x^3 - 2x\). We are given that the second term is 204.

So, we need to solve the equation:

\(x^3 - 2x = 204\)

We can test the given options to find which value of \(x\) satisfies this equation.

When \(x = 6\), the expression \(x^3 - 2x\) evaluates to 204, which matches the second term in the second pair. Therefore, the missing term is 6.

Conclusion

The relationship between the terms in the analogy 8 : 496 :: ? : 204 :: 9 : 711 follows the pattern where the second term is obtained by calculating the cube of the first term and subtracting twice the first term (\(n^3 - 2n\)). By applying this pattern to the second pair and testing the options, we found that 6 satisfies the relationship with 204.

Revision Table: Key Analogy Pattern

Additional Information: Solving Analogy Questions

Analogy questions test your ability to identify relationships between pairs of items. In number analogies, these relationships are often mathematical operations or patterns. Strategies for solving number analogies include:

• Look for common operations: addition, subtraction, multiplication, division.
• Consider powers or roots: squares, cubes, square roots, cube roots.
• Check for sequences or series patterns.
• Look for relationships involving digits of the numbers.
• Combine operations: For example, squaring and adding, cubing and subtracting, etc.
• Once a potential pattern is found, test it thoroughly on all given pairs to ensure consistency.
• If testing options, substitute each option into the potential relationship to see which one fits.

Practicing with different types of number analogies helps in quickly recognizing patterns during exams.