Solving the Number Analogy Pattern

The question presents a number analogy in the format A : B :: C : D :: E : F. This means that the relationship between A and B is the same as the relationship between C and D, and also the same as the relationship between E and F.

We are given the pairs: 7 : 330, ? : 203, and 9 : 716. We need to find the missing number in the second pair that maintains the same relationship found in the other two pairs.

Identifying the Number Pattern

Let's analyze the relationship between the numbers in the given pairs. We have the pairs (7, 330) and (9, 716). We need to find a mathematical operation or combination of operations that connects the first number (let's call it 'n') to the second number in each pair.

Let's consider common patterns involving powers:

• For the pair (7, 330):
• Try squaring the first number: \(7^2 = 49\). This is not close to 330.
• Try cubing the first number: \(7^3 = 7 \times 7 \times 7 = 49 \times 7 = 343\).
• The number 343 is very close to 330. The difference is \(343 - 330 = 13\).
• So, the relationship could be \(n^3 - 13\). Let's test this pattern with the other known pair.

• For the pair (9, 716):
• Using the proposed pattern \(n^3 - 13\), let's substitute n=9:
• Calculate \(9^3 = 9 \times 9 \times 9 = 81 \times 9 = 729\).
• Subtract 13: \(729 - 13 = 716\).
• This matches the second number in the pair (9, 716).

The pattern appears to be consistently \(n^3 - 13\), where 'n' is the first number in the pair.

Finding the Missing Term

Now we apply this established pattern to the middle pair: ? : 203. Let the missing number be \(x\). The relationship is \(x : 203\). Based on the pattern, we have:

\(x^3 - 13 = 203\)

To find \(x\), we need to solve this equation:

• Add 13 to both sides of the equation:
• \(x^3 = 203 + 13\)
• \(x^3 = 216\)
• To find \(x\), we need to find the cube root of 216:
• \(x = \sqrt[3]{216}\)

We need to find a number that, when multiplied by itself three times, equals 216. Let's try small integer cubes:

• \(1^3 = 1\)
• \(2^3 = 8\)
• \(3^3 = 27\)
• \(4^3 = 64\)
• \(5^3 = 125\)
• \(6^3 = 216\)

So, the missing number \(x\) is 6.

Checking the Options

Let's check the given options to see which one matches our calculated missing term:

1. 14
2. 16
3. 6
4. 4

Our calculated missing term is 6, which corresponds to Option 3.

The relationship \(n^3 - 13\) holds true for all pairs when the missing term is 6.

Revision Table: Number Pattern Summary

Additional Information: Logical Reasoning Analogies

Number analogies are common types of questions in logical reasoning tests. They require you to find the relationship between a pair of numbers and apply that same relationship to another pair or find a missing number in a pair. Common relationships include:

• Addition, Subtraction, Multiplication, Division
• Squares and Cubes (as seen in this problem)
• Operations involving squares or cubes (e.g., \(n^2 + k\), \(n^3 - k\), \(n^2 \times k\), etc.)
• Prime numbers, composite numbers
• Digits sum or product
• Sequential patterns

Solving number analogies involves careful observation, hypothesis testing, and applying basic mathematical operations. It's important to test the assumed pattern across all given pairs to ensure consistency before applying it to find the missing term.