Finding the Number Analogy Relationship

This question asks us to identify the relationship between the numbers in the given pairs and then use that relationship to find the missing number in the third pair. The structure is a number analogy: 27 is to 841 as 18 is to 400, and the missing number is to 324.

Analyzing the First Number Pair: 27 and 841

Let's look at the first pair: 27 and 841. We need to find a mathematical operation or pattern that connects 27 to 841. Often in such problems, the relationship involves basic arithmetic operations, squares, cubes, or roots.

• Is 841 a multiple of 27? $841 \div 27$ is not a whole number.
• Let's check for squares or cubes. What is $\sqrt{841}$?

We can estimate or calculate the square root of 841:

• $20^2 = 400$
• $30^2 = 900$

So, $\sqrt{841}$ is between 20 and 30. The last digit of 841 is 1, so its square root must end in 1 or 9. Let's try a number ending in 9, like 29.

• $29 \times 29 = (30-1)(30-1) = 900 - 30 - 30 + 1 = 900 - 60 + 1 = 841$.

So, $\sqrt{841} = 29$.

Now, how is 29 related to 27? $27 + 2 = 29$.

This suggests a potential rule: The second number is the square of (the first number plus 2).

Let's express this as a formula: Second Number $= (\text{First Number} + 2)^2$.

Applying this to the first pair:

$(27 + 2)^2 = 29^2 = 841$. This matches the first pair.

Verifying the Relationship with the Second Pair: 18 and 400

Let's test this rule with the second pair: 18 and 400.

According to the rule, the second number should be $(\text{First Number} + 2)^2$.

$(\text{First Number} + 2)^2 = (18 + 2)^2 = 20^2 = 400$.

This also matches the second pair. The rule seems consistent.

Applying the Relationship to the Third Pair: ? and 324

Now, let the missing number in the third pair be $x$. The pair is $x$ and 324. We apply the same rule:

$(\text{First Number} + 2)^2 = \text{Second Number}$

$(x + 2)^2 = 324$

To find $x$, we need to take the square root of both sides of the equation.

$\sqrt{(x + 2)^2} = \sqrt{324}$

$x + 2 = \sqrt{324}$

Now, we need to find the square root of 324.

• $10^2 = 100$
• $20^2 = 400$

So, $\sqrt{324}$ is between 10 and 20. The last digit of 324 is 4, so its square root must end in 2 or 8. Let's try 18.

• $18 \times 18 = 324$.

So, $\sqrt{324} = 18$.

Substituting this back into our equation:

$x + 2 = 18$

Now, solve for $x$ by subtracting 2 from both sides:

$x = 18 - 2$

$x = 16$

The missing number is 16.

Conclusion

The relationship between the numbers in each pair is that the second number is the square of the first number plus 2. Applying this rule to the third pair, we found that the missing number is 16.

Let's check this third pair: 16 and 324.

$(16 + 2)^2 = 18^2 = 324$. This confirms our answer.

Answer Derivation Summary

The missing number that fits the pattern is 16.

Revision Table: Number Analogy Patterns

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

• Arithmetic operations (addition, subtraction, multiplication, division)
• Squares and cubes of numbers or related numbers
• Square roots and cube roots
• Operations on digits of the number
• Prime numbers, composite numbers
• Odd/Even numbers
• Series based on a rule (e.g., $n^2+1$, $n^3-n$)

Additional Information: Solving Number Puzzles

When tackling number analogy or pattern questions in competitive exams, follow a systematic approach:

1. Observe the Numbers: Look at the size of the numbers. Are they close, or is one much larger than the other? Large differences often suggest multiplication, squares, cubes, or exponents. Small differences might involve addition or subtraction.
2. Test Simple Relations First: Check for basic arithmetic (+, -, ×, ÷).
3. Look for Squares and Cubes: Be familiar with squares up to at least 30 and cubes up to at least 10. This helps quickly identify if a number is a perfect square or cube, or close to one.
4. Check for Relations with Adjacent Numbers: If a number is not a direct square/cube, check if it's related to the square/cube of the original number plus or minus a small value (like in this problem, adding 2 before squaring).
5. Consider Digit Operations: Sometimes the pattern involves the sum, product, or other manipulation of the digits of the number.
6. Test the Pattern: Once you find a potential pattern from the first pair, always test it rigorously with the second pair provided in the question. If it works for both, it's likely the correct pattern.
7. Apply to Find the Missing Number: Use the confirmed pattern to calculate the missing number in the final pair.
8. Verify the Answer: Plug the calculated missing number back into the pattern to ensure it holds true for the final pair.