Solving Number Analogy Reasoning Questions

The question asks us to find the missing term in a number analogy. We are given three pairs related in the same way: \(16 : 50 :: 49 : ? :: 144 : 338\). We need to identify the relationship between the first and second numbers in the known pairs and apply that relationship to the third pair to find the missing number.

Analyzing the Given Number Pairs

Let's look at the first numbers in each pair:

• First pair: 16
• Second pair: 49
• Third pair: 144

We can see that these numbers are perfect squares:

• \(16 = 4^2\)
• \(49 = 7^2\)
• \(144 = 12^2\)

Let's call the base of the square 'Base'. So, the first number in each pair is \(Base^2\).

Identifying the Relationship Pattern

Now let's examine how the second number in each pair is related to the first number or its base.

• Pair 1: \(16 (4^2) \rightarrow 50\)
• Pair 3: \(144 (12^2) \rightarrow 338\)

Let's try to find a pattern involving the Base (4 and 12) that results in the second number (50 and 338).

Consider a pattern involving \(Base^2\) and Base. Let's test the relationship \(a \times Base^2 + b \times Base + c\).

Let's try a simpler pattern. How about \(2 \times Base^2 + \text{something related to Base}\)?

• For Base = 4: \(2 \times 4^2 = 2 \times 16 = 32\). We need 50. The difference is \(50 - 32 = 18\).
• For Base = 12: \(2 \times 12^2 = 2 \times 144 = 288\). We need 338. The difference is \(338 - 288 = 50\).

Now let's look for a pattern in the differences (18 and 50) based on the Base (4 and 12).

How is 18 related to 4? How is 50 related to 12? Let's try a linear relationship for the difference: \(a \times Base + b\).

• For Base = 4: \(4a + b = 18\)
• For Base = 12: \(12a + b = 50\)

Subtracting the first equation from the second:

\[ (12a + b) - (4a + b) = 50 - 18 \] \[ 8a = 32 \] \[ a = \frac{32}{8} = 4 \]

Substitute the value of \(a\) into the first equation:

\[ 4(4) + b = 18 \] \[ 16 + b = 18 \] \[ b = 18 - 16 = 2 \]

So, the pattern for the difference is \(4 \times Base + 2\).

Verifying the Discovered Pattern

The proposed complete pattern is: Second Term = \(2 \times Base^2 + (4 \times Base + 2)\).

• Let's check this for the first pair (Base = 4):
  Second Term = \(2 \times 4^2 + (4 \times 4 + 2) = 2 \times 16 + (16 + 2) = 32 + 18 = 50\). This matches the given second term.
• Let's check this for the third pair (Base = 12):
  Second Term = \(2 \times 12^2 + (4 \times 12 + 2) = 2 \times 144 + (48 + 2) = 288 + 50 = 338\). This matches the given second term.

The pattern is consistent for both given pairs.

Applying the Pattern to Find the Missing Term

The second pair is \(49 : ?\). Here, the first term is \(49 = 7^2\), so the Base is 7.

Using the discovered pattern with Base = 7:

\[ \text{Missing Term} = 2 \times 7^2 + (4 \times 7 + 2) \] \[ \text{Missing Term} = 2 \times 49 + (28 + 2) \] \[ \text{Missing Term} = 98 + 30 \] \[ \text{Missing Term} = 128 \]

So, the missing term is 128.

The missing term that completes the analogy is 128.

Conclusion

Based on the identified pattern, the number related to 49 in the same way as 16 is related to 50 and 144 is related to 338 is 128.

The correct option is 128.

Revision Table: Number Analogy Pattern

Let's summarize the key steps for solving this type of number analogy problem:

• Identify if the first terms in the pairs follow a simple pattern, like being perfect squares or cubes.
• If they are perfect squares, find the base of the square (\(\sqrt{\text{First Term}}\)).
• Look for a relationship between the first term (or its base) and the second term that is consistent across all given pairs.
• Express this relationship as a formula involving the base or the first term.
• Verify the formula using the known pairs.
• Apply the verified formula to the pair with the missing term to find the answer.

Additional Information: Types of Number Analogies

Number analogy questions test your ability to identify relationships between numbers. Common relationships include:

• Arithmetic Operations: Addition, subtraction, multiplication, division, or a combination of these.
• Squares and Cubes: The numbers might be squares or cubes, or related to squares/cubes plus/minus a constant or the base number.
• Series Patterns: The relationship might involve squaring the number, multiplying by a factor, and adding/subtracting a sequence (like consecutive numbers, even/odd numbers, etc.).
• Digit Manipulation: Sum of digits, product of digits, reversing digits, etc.
• Prime or Composite Numbers: Relationships based on number properties.

Solving these questions requires careful observation and systematic testing of possible patterns.