Understanding the Number Analogy Problem

This question is a numerical analogy problem where you need to find the relationship between the first pair of numbers and apply that same relationship to the third number to find the fourth one. The given analogy is 7 : 343 : : 9 : ?

Identifying the Relationship

Let's look at the first pair: 7 and 343. We need to figure out how 343 is related to 7. Let's try common mathematical operations:

• Is it addition? $7 + x = 343 \implies x = 343 - 7 = 336$. No obvious simple addition.
• Is it multiplication? $7 \times x = 343 \implies x = 343 / 7$. Let's perform the division: $343 \div 7 = 49$. So, $7 \times 49 = 343$. The relationship could be multiplying the first number by 49.
• Is it a power? Let's check powers of 7.
  • $7^1 = 7$
  • $7^2 = 7 \times 7 = 49$
  • $7^3 = 7 \times 7 \times 7 = 49 \times 7 = 343$

We see that 343 is the cube of 7 ($7^3$). This is a very common relationship in number analogy problems. Multiplying by 49 is equivalent to multiplying by $7^2$, so $7 \times 49 = 7 \times 7^2 = 7^3$. The most direct relationship seems to be cubing the first number.

Applying the Relationship to Find the Missing Number

Now, we apply the same relationship (cubing) to the third number, which is 9. We need to find the cube of 9 ($9^3$).

Calculation:

$\qquad 9^3 = 9 \times 9 \times 9$

First, $9 \times 9 = 81$.

Then, $81 \times 9$:

$\qquad 81 \times 9 = (80 + 1) \times 9 = (80 \times 9) + (1 \times 9) = 720 + 9 = 729$.

So, $9^3 = 729$.

Checking the Options

The missing number is 729. Let's check the given options:

1. 529
2. 629
3. 729
4. 1008

Our calculated number, 729, matches option 3.

Conclusion

The relationship in the analogy 7 : 343 : : 9 : ? is that the second number is the cube of the first number. Since $7^3 = 343$, applying the same rule to 9 gives $9^3 = 729$. Therefore, the related number is 729.

Revision Table: Cubes of Small Numbers

Knowing the cubes of small integers can be helpful in solving number analogy problems quickly.

Additional Information: Powers and Roots in Reasoning

Number reasoning problems often involve relationships based on powers and roots. It's useful to be familiar with squares, cubes, square roots, and cube roots of common numbers.

• Squares: $n^2 = n \times n$. E.g., $8^2 = 64$.
• Cubes: $n^3 = n \times n \times n$. E.g., $4^3 = 64$.
• Square Roots: The number that when squared gives the original number. $\sqrt{n}$. E.g., $\sqrt{64} = 8$.
• Cube Roots: The number that when cubed gives the original number. $\sqrt[3]{n}$. E.g., $\sqrt[3]{64} = 4$.

Recognizing these patterns quickly can significantly improve your speed in solving numerical reasoning questions in competitive exams and aptitude tests.