Number Analogy Reasoning: Solving the Pattern

This question asks us to identify the relationship between pairs of numbers in an analogy and apply that relationship to find a missing number. The given analogy is: 216 ∶ 3 ∶∶ ? ∶ 4 ∶∶ 1728 ∶ 6.

We have three pairs of numbers implied by the analogy structure:

• Pair 1: 216 and 3
• Pair 2: ? and 4
• Pair 3: 1728 and 6

The symbol '∶' means "is to", and '∶∶' means "as". So, the analogy reads: 216 is to 3, as ? is to 4, as 1728 is to 6.

Analyzing the Given Number Pairs to Find the Relationship

Let's examine the first and third pairs to find a consistent rule connecting the two numbers in each pair. The pairs are (216, 3) and (1728, 6).

We can look for various mathematical relationships, such as addition, subtraction, multiplication, division, powers, or roots.

Let's consider powers. We notice that 216 and 1728 are common cubic numbers:

• \(6^3 = 6 \times 6 \times 6 = 36 \times 6 = 216\)
• \(12^3 = 12 \times 12 \times 12 = 144 \times 12 = 1728\)

So, 216 is the cube of 6, and 1728 is the cube of 12.

Now let's see how the second number in each pair (3 and 6) relates to the base of the cube (6 and 12):

• For the first pair (216, 3), the base is 6, and the second number is 3. We can see that \(6 = 3 \times 2\).
• For the third pair (1728, 6), the base is 12, and the second number is 6. We can see that \(12 = 6 \times 2\).

The pattern appears to be that the first number is the cube of twice the second number. Let's express this as a formula:

\(N_1 = (N_2 \times 2)^3\)

Where \(N_1\) is the first number and \(N_2\) is the second number in a pair.

Applying the Pattern to Find the Missing Number

The missing number is in the second pair, which is (? : 4). Here, \(N_2 = 4\), and we need to find \(N_1\) (the missing number).

Using the pattern \(N_1 = (N_2 \times 2)^3\):

1. Substitute the value of \(N_2\): \(N_1 = (4 \times 2)^3\)
2. Perform the multiplication inside the parentheses: \(N_1 = (8)^3\)
3. Calculate the cube: \(N_1 = 8 \times 8 \times 8\)
4. \(N_1 = 64 \times 8\)
5. \(N_1 = 512\)

So, the missing number is 512.

Confirming the Answer

The calculated missing number is 512. Let's check the given options:

• Option 1: 512
• Option 2: 864
• Option 3: 728
• Option 4: 946

Our calculated value, 512, matches Option 1.

Revision Table: Number Analogy Patterns

Additional Information: Strategies for Solving Analogy Questions

Number analogy questions test your ability to find logical relationships between numbers. Here are some tips for tackling them:

• Look for common relationships: Start with simple arithmetic (add, subtract, multiply, divide), then consider powers and roots (squares, cubes).
• Check both pairs: The relationship you find must hold true for all given pairs in the analogy.
• Think about digits: Sometimes the pattern involves the digits of the numbers (sum of digits, product of digits, reversing digits).
• Consider position: In longer series, the position of the number might be relevant.
• Break down complex numbers: Large numbers might be products or powers of smaller numbers.
• Test your hypothesis: Once you think you've found the pattern, apply it to the known pairs to ensure it fits before calculating the missing number.
• Review options: The options can sometimes give you clues or help you verify your calculated answer.

Practice with different types of analogy questions will help you recognize common patterns more quickly.