Understanding the Number Analogy

The question asks us to find the relationship between the first two numbers and apply the same relationship to the third number to find the missing fourth number. The given analogy is:

343 ∶ 49 ∶∶ 729 ∶ ?

Analyzing the First Pair: 343 and 49

Let's examine the numbers 343 and 49 to find a pattern or relationship. Both numbers might seem familiar in terms of powers of integers.

• We know that 49 is the square of 7. Mathematically, this is \(7^2 = 49\).
• Let's check if 343 is related to the same base, 7. Calculating \(7^3\): \(7 \times 7 \times 7 = 49 \times 7 = 343\).

So, the relationship between 343 and 49 is that the first number (343) is the cube of a base number (7), and the second number (49) is the square of the same base number (7).

Relationship: First number is (Base)\(^3\), Second number is (Base)\(^2\).

Applying the Relationship to the Second Pair: 729 and ?

Now, we apply the same relationship to the third number, 729, to find the fourth number. We need to identify the base number for 729 in a similar way.

• We need to find a number whose cube is 729. Let's test small integers:
  • \(1^3 = 1\)
  • \(2^3 = 8\)
  • \(3^3 = 27\)
  • \(4^3 = 64\)
  • \(5^3 = 125\)
  • \(6^3 = 216\)
  • \(7^3 = 343\)
  • \(8^3 = 512\)
  • \(9^3 = 729\)

The third number, 729, is the cube of 9. So, the base number here is 9.

According to the relationship identified in the first pair, the fourth number should be the square of this base number (9).

Calculating the square of the base (9): \(9^2 = 9 \times 9 = 81\).

Therefore, the missing number in the analogy 729 ∶ ? is 81.

Summary of the Analogy

The relationship is \(n^3 : n^2\).

• For the first pair: \(7^3 : 7^2 \implies 343 : 49\).
• For the second pair: \(9^3 : 9^2 \implies 729 : 81\).

Final Answer Derivation

Based on the relationship found, the number that replaces the question mark is 81.

Revision Table: Number Analogy Concepts

Additional Information: Solving Number Series and Analogies

Number analogies and series questions test your ability to identify patterns in numbers. Common patterns include:

• Arithmetic progression (addition/subtraction of a constant difference).
• Geometric progression (multiplication/division by a constant ratio).
• Squares and cubes of numbers.
• Combinations of operations (e.g., multiply by 2 and add 1).
• Differences or sums between consecutive terms follow a pattern.
• Prime numbers or other special sequences.

To solve such problems, it's helpful to:

• Look for simple relationships first (addition, subtraction, multiplication, division).
• Check for squares or cubes of numbers.
• Calculate the differences between consecutive terms to see if there's a pattern.
• Consider combinations of operations.
• Practice recognizing common number patterns.

Understanding the relationship between cubes and squares of the same base, as seen in this problem, is a common pattern in number analogy and series questions.