Solving Number Analogy: Finding the Pattern in 643:9 :: 441:7 :: 324:?

This question asks us to find the relationship or pattern between the numbers in the given pairs and then apply that pattern to the third pair to find the missing number. The pairs are 643 : 9, 441 : 7, and 324 : ?

Let's analyze the first two pairs to discover the logic connecting the numbers on the left side of the colon (:) to the numbers on the right side.

Analyzing the First Pair: 643 : 9

Consider the digits of the first number, 643. The digits are 6, 4, and 3.

Let's try different operations involving these digits to get the second number, 9.

• Sum of digits: $6 + 4 + 3 = 13$ (Not 9)
• Product of digits: $6 \times 4 \times 3 = 72$

The product of the digits is 72. Now, let's look at the digits of the product (72), which are 7 and 2.

What if we sum the digits of the product?

• Sum of the digits of the product: $7 + 2 = 9$.

This matches the second number in the first pair (9). This seems like a promising pattern.

Analyzing the Second Pair: 441 : 7

Let's apply the same pattern to the second pair, 441 : 7.

Consider the digits of the first number, 441. The digits are 4, 4, and 1.

Calculate the product of these digits:

• Product of digits: $4 \times 4 \times 1 = 16$.

The product is 16. Now, sum the digits of the product (16), which are 1 and 6.

• Sum of the digits of the product: $1 + 6 = 7$.

This matches the second number in the second pair (7). The pattern holds for the second pair as well.

Applying the Pattern to the Third Pair: 324 : ?

Now, we apply the established pattern to the third pair, 324 : ?.

Consider the digits of the first number, 324. The digits are 3, 2, and 4.

Calculate the product of these digits:

• Product of digits: $3 \times 2 \times 4 = 24$.

The product is 24. Finally, sum the digits of the product (24), which are 2 and 4.

• Sum of the digits of the product: $2 + 4 = 6$.

So, the missing number in the third pair is 6.

Summary of the Logic

The pattern followed in this analogy is:

For a number composed of digits a, b, and c, the related number is the sum of the digits of the product $(a \times b \times c)$.

Following this pattern, the number related to 324 is 6.

Revision Table: Number Analogy Pattern

Let's quickly review the steps taken to solve this number analogy question:

1. Identify the relationship between the first number and the second number in the given pairs.
2. In this case, the pattern was found to be the sum of the digits of the product of the first number's digits.
3. Verify this pattern using the other given pairs.
4. Apply the confirmed pattern to the final pair to find the missing number.

Additional Information: Types of Number Analogies

Number analogies are a common type of logical reasoning question. They test your ability to identify patterns and relationships between numbers. Common patterns include:

• Arithmetic Operations: Adding, subtracting, multiplying, or dividing the number or its digits.
• Digit Manipulation: Sum of digits, product of digits, difference of digits, reversing digits, arranging digits, etc. (as seen in this problem).
• Squares, Cubes, and Roots: The numbers might be perfect squares, cubes, or related to their roots.
• Prime Numbers or Other Sequences: The numbers might follow a sequence like prime numbers, odd/even numbers, etc.
• Combinations of Operations: Sometimes, a combination of the above methods is used.

Practicing different types helps in quickly identifying the pattern during an exam.