Solving Number Analogy and Pattern Recognition

The question asks us to find the number that shares the same relationship with the third term (31) as the second term (264) shares with the first term (25), and the sixth term (456) shares with the fifth term (49).

This type of problem requires identifying a pattern or relationship between the pairs of numbers provided. The given pairs are (25, 264) and (49, 456). We need to find the pattern that transforms 25 into 264 and 49 into 456, and then apply that same pattern to 31 to find the missing number.

Analyzing the Number Pairs to Find the Pattern

Let's look closely at the given pairs:

• Pair 1: 25 and 264
• Pair 2: 49 and 456

We need to find a mathematical operation or a combination of operations that connects the first number in each pair to the second number.

Let's explore some possible relationships, such as multiplication, division, addition, subtraction, squares, cubes, or a combination of these. A common pattern in such analogies is a linear relationship of the form $n \times k + c$, where $n$ is the first number in the pair, $k$ and $c$ are constants, and the result is the second number.

Let's assume the relationship is $n \times k + c$. Applying this to the given pairs:

1. For the pair (25, 264):

   $$25k + c = 264 \quad (*)$$

2. For the pair (49, 456):

   $$49k + c = 456 \quad (**)$$

We now have a system of two linear equations with two variables, $k$ and $c$. We can solve this system to find the values of $k$ and $c$. A simple method is to subtract equation $(*)$ from equation $(**)$:

$$(49k + c) - (25k + c) = 456 - 264$$ $$(49 - 25)k + (c - c) = 192$$ $$24k = 192$$

Now, solve for $k$:

$$k = \frac{192}{24}$$ $$k = 8$$

Now that we have the value of $k$, we can substitute it back into either equation $(*)$ or $(**)$ to find the value of $c$. Let's use equation $(*)$:

$$25k + c = 264$$ $$25(8) + c = 264$$ $$200 + c = 264$$

Now, solve for $c$:

$$c = 264 - 200$$ $$c = 64$$

So, the pattern or relationship is $n \times 8 + 64$. Let's verify this with the second pair (49, 456):

$$49 \times 8 + 64 = 392 + 64 = 456$$

The pattern holds true for both given pairs.

Applying the Pattern to Find the Missing Term

Now we need to apply the same pattern, $n \times 8 + 64$, to the third term, which is 31. Here, $n = 31$.

$$31 \times 8 + 64 = 248 + 64$$ $$248 + 64 = 312$$

Therefore, the missing term in the analogy 31 : ? is 312.

Conclusion

The relationship between the terms in the analogy is defined by the formula $n \times 8 + 64$. Applying this formula to the third term (31) gives us the missing fourth term.

$$31 \times 8 + 64 = 312$$

Comparing this result with the given options, we find that 312 is one of the choices.

Revision Table: Key Concepts in Number Analogy

Additional Information: Tips for Solving Number Reasoning Questions

Number reasoning questions and analogies often involve finding patterns. Here are some common types of patterns to look for:

• Arithmetic Operations: Addition, subtraction, multiplication, or division by a constant number.
• Sequences: Patterns involving consecutive numbers, prime numbers, squares, cubes, etc.
• Differences/Ratios: A constant difference or ratio between consecutive terms.
• Squares or Cubes: The second term is the square or cube of the first term, or related to the square/cube ($n^2 \pm k$, $n^3 \pm k$).
• Digit Manipulation: Operations performed on the digits of the number.
• Combinations: A combination of multiple operations, like the linear pattern found in this problem ($n \times k + c$).

When approaching such questions, try applying these common patterns systematically to the given pairs to identify the underlying rule. Once the rule is found, apply it to the term with the missing value.