Understanding the Number Analogy Pattern

The question asks us to find the missing term in an analogy based on the relationship between the given pairs of numbers. The analogy is presented as:

$$ 8 : 512 :: ? : 1728 :: 18 : 5832 $$

This format means that the first term is related to the second term in the same way the fourth term is related to the third term (which is unknown), and the fifth term is related to the sixth term. We need to identify the pattern or rule connecting the numbers in each pair.

Analyzing the Given Pairs

Let's look at the two complete pairs provided:

1. The first pair is 8 and 512.
2. The third pair is 18 and 5832.

Discovering the Relationship

We need to find a mathematical relationship that connects 8 to 512 and 18 to 5832. Let's consider common relationships like addition, subtraction, multiplication, division, or powers.

• Addition/Subtraction: $512 - 8 = 504$. $5832 - 18 = 5814$. The difference is not constant.
• Multiplication: $512 \div 8 = 64$. $5832 \div 18 = 324$. The multiplier is not constant.
• Powers: Let's consider squaring or cubing the first term.
  • $8^2 = 8 \times 8 = 64$. This is not 512.
  • $8^3 = 8 \times 8 \times 8 = 64 \times 8 = 512$. This matches the first pair!

  Let's check if the same relationship holds for the third pair (18 and 5832).

  • $18^2 = 18 \times 18 = 324$. This is not 5832.
  • $18^3 = 18 \times 18 \times 18 = 324 \times 18 = 5832$. This matches the third pair!

The pattern established is that the second number in each pair is the cube of the first number.

The general rule for this analogy is: $$ \text{First Term} : (\text{First Term})^3 $$

Finding the Missing Term

The second pair in the analogy is ? : 1728. Let the missing term be represented by $x$. According to the pattern, the second term (1728) must be the cube of the first term ($x$).

So, we have the equation:

$$ x^3 = 1728 $$

To find $x$, we need to calculate the cube root of 1728. $$ x = \sqrt[3]{1728} $$

We can find the cube root by testing possible values, or by considering the properties of cubes. For example, the cube root of a number ending in 8 will end in 2 (since $2^3 = 8$ and $12^3 = 1728$).

Let's test the options provided:

• Option 1: $8^3 = 512$
• Option 2: $14^3 = 14 \times 14 \times 14 = 196 \times 14 = 2744$
• Option 3: $9^3 = 9 \times 9 \times 9 = 81 \times 9 = 729$
• Option 4: $12^3 = 12 \times 12 \times 12 = 144 \times 12 = 1728$

The calculation $12^3 = 1728$ matches the second term in the pair. Therefore, the missing term is 12.

Conclusion

The relationship throughout the analogy is that the second term is the cube of the first term. Since $12^3 = 1728$, the missing term is 12.

The complete analogy is: $$ 8 : 512 :: 12 : 1728 :: 18 : 5832 $$

Revision Table: Key Concepts in Number Analogies

Additional Information on Cube Relationships and Cube Roots

Understanding cubes and cube roots is essential for solving this type of number analogy.

• Cube of a Number: The cube of a number is the result of multiplying the number by itself three times. It is denoted by a superscript 3, like $x^3$. For example, $5^3 = 5 \times 5 \times 5 = 125$.
• Cube Root of a Number: The cube root of a number is the value that, when cubed, gives the original number. It is denoted by the symbol $\sqrt[3]{ }$. For example, $\sqrt[3]{125} = 5$ because $5^3 = 125$.
• Finding Cube Roots: For smaller perfect cubes like 1728, you can sometimes find the cube root by trial and error, testing integers. For larger numbers, prime factorization can be used. For example, to find $\sqrt[3]{1728}$:
  • Prime factorize 1728: $1728 = 2^6 \times 3^3 = (2^2)^3 \times 3^3 = 4^3 \times 3^3 = (4 \times 3)^3 = 12^3$.
  • So, $\sqrt[3]{1728} = 12$.

Practicing with different powers and roots helps in quickly identifying patterns in number analogy questions.