Understanding the Number Analogy Pattern

The question presents a number analogy in the format: 6 : 646 :: 5 : ? :: 4 : 190. This means that the first term is related to the second term in the same way that the third term is related to the missing term, and the fifth term is related to the sixth term. Our goal is to discover the specific mathematical pattern or rule that connects the numbers in the given pairs (6, 646) and (4, 190), and then apply that same rule to find the missing number related to 5.

Analyzing the Given Number Pairs

Let's look at the two complete pairs provided:

• Pair 1: 6 and 646
• Pair 2: 4 and 190

We need to find a consistent relationship between the first number (let's call it $N$) and the second number (let's call it $R$) in each pair. The numbers 646 and 190 are significantly larger than 6 and 4, respectively, which suggests that the pattern might involve powers of the first number, such as squares ($N^2$) or cubes ($N^3$).

Exploring Potential Mathematical Relationships

Let's consider the cube of the first number $N$:

• For $N=6$, $N^3 = 6^3 = 6 \times 6 \times 6 = 216$.
• For $N=4$, $N^3 = 4^3 = 4 \times 4 \times 4 = 64$.

Now, let's compare these cubic values to the given second numbers:

• For $N=6$, the second number is 646. The difference is $646 - 216 = 430$.
• For $N=4$, the second number is 190. The difference is $190 - 64 = 126$.

The differences (430 and 126) still don't immediately show a simple pattern based on $N$. Let's consider if the pattern involves multiplying the cube by a constant and adding or subtracting another constant. Let's assume the pattern is of the form $R = A \times N^3 + B$, where $A$ and $B$ are constants.

Deriving the Pattern Formula

Using the two given pairs, we can set up a system of linear equations to solve for the constants $A$ and $B$ in the potential pattern $R = A \times N^3 + B$.

For the pair (6, 646), with $N=6$ and $R=646$:

\( A \times 6^3 + B = 646 \)

\( 216A + B = 646 \) (Equation 1)

For the pair (4, 190), with $N=4$ and $R=190$:

\( A \times 4^3 + B = 190 \)

\( 64A + B = 190 \) (Equation 2)

Now, we solve this system of two equations. Subtract Equation 2 from Equation 1:

\( (216A + B) - (64A + B) = 646 - 190 \)

\( 216A - 64A = 456 \)

\( 152A = 456 \)

Divide both sides by 152:

\( A = \frac{456}{152} \)

\( A = 3 \)

Now that we have the value of $A$, substitute it back into either Equation 1 or Equation 2 to find $B$. Using Equation 2:

\( 64A + B = 190 \)

\( 64 \times 3 + B = 190 \)

\( 192 + B = 190 \)

Subtract 192 from both sides:

\( B = 190 - 192 \)

\( B = -2 \)

So, the derived pattern rule is \( R = 3N^3 - 2 \).

Verifying the Pattern Rule

Let's quickly check if this rule holds for the given pairs:

• For $N=6$: \( R = 3 \times 6^3 - 2 = 3 \times 216 - 2 = 648 - 2 = 646 \). Correct.
• For $N=4$: \( R = 3 \times 4^3 - 2 = 3 \times 64 - 2 = 192 - 2 = 190 \). Correct.

The pattern \( R = 3N^3 - 2 \) consistently relates the first number to the second number in both given pairs of the analogy.

Applying the Pattern to Find the Missing Number

The question asks for the number related to 5 in the analogy 5 : ?. Using the established pattern \( R = 3N^3 - 2 \), we substitute $N=5$ to find the missing term:

\( R = 3 \times 5^3 - 2 \)

\( R = 3 \times 125 - 2 \)

\( R = 375 - 2 \)

\( R = 373 \)

Therefore, the missing number in the analogy 5 : ? is 373.

Conclusion

The number that completes the analogy 6 : 646 :: 5 : ? :: 4 : 190 is 373, based on the pattern \( R = 3N^3 - 2 \).

Revision Table: Key Concepts in Number Patterns

Additional Information: Solving Analogy Questions

Solving number analogy questions often involves systematic exploration of potential mathematical relationships between the given terms. Here are some common types of patterns to look for:

• Basic Arithmetic: Addition, subtraction, multiplication, division.
• Powers: Squares ($N^2$), cubes ($N^3$), or higher powers.
• Roots: Square root, cube root.
• Combinations: Combinations of basic operations and powers (e.g., $N^2 + k$, $N^3 - k$, $k \times N + m$, $A \times N^2 + B \times N + C$).
• Digit Manipulation: Sum of digits, product of digits, reversing digits (less common unless explicitly suggested).
• Special Numbers: Prime numbers, composite numbers, factorials.

A good strategy is to start with simple patterns and move towards more complex ones if the initial hypotheses don't fit. For patterns involving polynomial relationships like \( AN^3 + B \), using the given pairs to set up and solve equations is a standard approach when simpler methods fail.