Understanding Number Analogies and Patterns

Number analogy questions test your ability to find the relationship or pattern between a pair of numbers and apply that same relationship to another number or pair to find the missing element. In this question, we are given two complete pairs and one incomplete pair in the format A : B :: C : ? :: D : E.

We need to find the relationship between the first pair (14 and 400) and verify it with the second complete pair (42 and 2304). Once the relationship is confirmed, we will apply it to the third term (37) to find the missing fourth term.

Analyzing the Given Number Pairs

Let's look at the given pairs:

• First pair: 14 : 400
• Second complete pair: 42 : 2304
• Incomplete pair: 37 : ?

Finding the Relationship Pattern

Let's examine the first pair, 14 and 400. We need to find an operation or a series of operations that transforms 14 into 400.

• Addition: $14 + 386 = 400$. This is a large difference, but possible.
• Multiplication: $14 \times X = 400$. $X = 400/14$, which is not a whole number.
• Squaring: $14^2 = 196$. Not 400.
• Let's consider squaring a number related to 14. What if we add a constant to 14 and then square it?
• Consider $(14 + k)^2 = 400$. Since $\sqrt{400} = 20$, we can say $14 + k = 20$. This gives $k = 20 - 14 = 6$.
• So, the potential relationship is $(Number + 6)^2 = Result$.

Verifying the Relationship with the Second Complete Pair

Let's test if the relationship $(Number + 6)^2 = Result$ holds true for the pair 42 : 2304.

• Apply the rule to the first number, 42: $(42 + 6)^2$
• $(42 + 6)^2 = 48^2$
• Calculate $48^2$: $48 \times 48$.

Calculation of $48^2$:

Alternatively, $48^2 = (50-2)^2 = 50^2 - 2 \times 50 \times 2 + 2^2 = 2500 - 200 + 4 = 2304$.

Since $(42 + 6)^2 = 2304$, the relationship $(Number + 6)^2 = Result$ is confirmed for both given pairs.

Applying the Pattern to Find the Missing Term

Now, we apply the same relationship to the third term, 37, to find the missing fourth term.

• The third term is 37.
• The rule is $(Number + 6)^2 = Result$.
• So, the missing term is $(37 + 6)^2$.
• $(37 + 6)^2 = 43^2$.

Calculate $43^2$:

Alternatively, $43^2 = (40+3)^2 = 40^2 + 2 \times 40 \times 3 + 3^2 = 1600 + 240 + 9 = 1849$.

The missing term is 1849.

Conclusion

The relationship in the analogy 14 : 400 :: 37 : ? :: 42 : 2304 is that the second number is the square of the first number plus 6, i.e., $(First\ Number + 6)^2 = Second\ Number$. Applying this rule to 37, we get $(37 + 6)^2 = 43^2 = 1849$.

The correct option is the one containing 1849.

Revision Table: Analogy Pattern Summary

Additional Information: Tips for Solving Analogies

When solving number analogies or pattern recognition questions, consider these common relationships:

• Basic arithmetic operations: Addition, subtraction, multiplication, division.
• Powers and roots: Squares, cubes, square roots, cube roots.
• Sequences: Arithmetic progressions, geometric progressions.
• Combinations of operations: Adding/subtracting a constant before/after multiplication/division or squaring/cubing.
• Digit operations: Sum of digits, product of digits, reversing digits.

Always test your assumed relationship on all given pairs to confirm its validity before applying it to find the missing term.