This question is a type of number analogy problem where we need to find the relationship between the first two numbers and the relationship between the third and fourth numbers. Once we identify this relationship or pattern, we apply it to the fifth number to find the missing sixth number.

Understanding the Number Analogy Pattern

Let's look at the given pairs:

• First pair: 7 : 21
• Second pair: 14 : 42
• Third pair: 18 : ?

We need to find the rule that connects the numbers in the first two pairs. Let's examine the relationship between the first number and the second number in each complete pair.

Analyzing the First Pair (7 : 21)

How is 21 related to 7? Let's think about basic arithmetic operations:

• Addition: \(7 + \text{something} = 21\). That something is \(21 - 7 = 14\). So, \(7 + 14 = 21\).
• Multiplication: \(7 \times \text{something} = 21\). That something is \(21 \div 7 = 3\). So, \(7 \times 3 = 21\).

Both addition by 14 and multiplication by 3 are possible patterns for the first pair.

Analyzing the Second Pair (14 : 42)

Now let's check which of these patterns holds true for the second pair (14 : 42).

• Applying the addition pattern (\(+ 14\)): \(14 + 14 = 28\). But the second number is 42. So, addition by 14 is not the correct pattern.
• Applying the multiplication pattern (\(\times 3\)): \(14 \times 3 = 42\). This matches the second number in the pair.

Therefore, the consistent pattern in this number analogy is multiplying the first number by 3 to get the second number.

Applying the Pattern to Find the Missing Number

The identified pattern is: Second Number = First Number \(\times 3\).

Now we apply this pattern to the third pair, which is 18 : ?.

Missing Number = 18 \(\times 3\).

Let's calculate:

\(18 \times 3 = 54\)

So, the missing number is 54.

Step-by-Step Solution for Number Analogy

Here are the steps taken to solve this number analogy question:

1. Examine the first given pair of numbers (7 : 21) and identify a possible relationship (e.g., addition, subtraction, multiplication, division, squares, cubes).
2. Identify potential patterns: \(7 + 14 = 21\) or \(7 \times 3 = 21\).
3. Examine the second given pair of numbers (14 : 42) and test the potential patterns found in step 2.
4. Test pattern 1 (\(+ 14\)): \(14 + 14 = 28\). This does not equal 42.
5. Test pattern 2 (\(\times 3\)): \(14 \times 3 = 42\). This matches the second number.
6. Confirm the pattern: The pattern is multiplying the first number by 3.
7. Apply the confirmed pattern to the third pair (18 : ?).
8. Calculate the missing number: \(18 \times 3 = 54\).
9. The missing number is 54.

Checking the Options

Let's look at the provided options:

• Option 1: 57
• Option 2: 54
• Option 3: 63
• Option 4: 72

Our calculated missing number is 54, which matches Option 2.

Revision Table: Key Concepts in Number Analogy

Additional Information: Tips for Solving Number Analogy Questions

Solving number analogy and number pattern questions often involves looking for common mathematical relationships. Here are some tips:

• Check for simple arithmetic operations: Addition, subtraction, multiplication, or division between the numbers in a pair.
• Look for patterns involving squares or cubes: Is the second number the square or cube of the first? Or is the relationship related to squares or cubes?
• Consider combined operations: Sometimes the pattern might involve two steps, like multiplying by a number and then adding another number (\(a \times n + m = b\)).
• Look for differences or ratios: Calculate the difference (\(b-a\)) or the ratio (\(b/a\)) between the numbers in the known pairs. See if it's constant or follows another pattern.
• Check for patterns in digits: Less common, but sometimes the pattern involves manipulating the digits of the numbers.
• Systematically test potential patterns: Don't stop at the first pattern you find; verify it using the other complete pairs provided in the analogy.