Solving the Number Analogy Question

This question asks us to find the missing number in an analogy based on the relationship between the given pairs of numbers. We are given three pairs: 16 : 64, 36 : ?, and 20 : 100. We need to identify the rule or pattern that connects the first number to the second number in the given pairs (16:64 and 20:100) and then apply that same rule to the third pair (36:?) to find the missing number.

Analyzing the Given Number Pairs

Let's look closely at the two complete pairs:

• First pair: 16 and 64
• Third pair: 20 and 100

We need to find a relationship that works for both pairs. Let's consider possible relationships:

• Addition/Subtraction: \(64 - 16 = 48\), \(100 - 20 = 80\). The difference is not constant.
• Multiplication/Division: \(64 \div 16 = 4\), \(100 \div 20 = 5\). The multiplier is different (4 and 5).
• Squares/Cubes: \(16 = 4^2\), \(64 = 8^2\) or \(4^3\). \(20\) and \(100\) don't seem to fit simple squares or cubes related directly.

Let's reconsider the multiplication idea. The first number is multiplied by a certain value to get the second number. For the first pair (16:64), the multiplier is 4. For the third pair (20:100), the multiplier is 5.

Is there a connection between the first number and the multiplier? For 16, the multiplier is 4. Notice that \(16 \div 4 = 4\). For 20, the multiplier is 5. Notice that \(20 \div 4 = 5\).

This suggests a consistent pattern: the multiplier used to get the second number from the first number is the first number divided by 4. The relationship can be written as:

\[ \text{Second Number} = \text{First Number} \times \left( \frac{\text{First Number}}{4} \right) \]

Verifying the Pattern

Let's check if this pattern holds true for the given pairs:

• For 16 : 64: First Number = 16. Multiplier = \(16 \div 4 = 4\). Second Number = \(16 \times 4 = 64\). This matches.
• For 20 : 100: First Number = 20. Multiplier = \(20 \div 4 = 5\). Second Number = \(20 \times 5 = 100\). This also matches.

The pattern is confirmed.

Applying the Pattern to the Third Pair

Now we apply this pattern to the second pair, 36 : ?. First Number = 36. According to the pattern, the multiplier is the First Number divided by 4.

Multiplier \( = \frac{36}{4} = 9 \)

The missing number (the second number) is the First Number multiplied by the multiplier:

Missing Number \( = 36 \times 9 \)

\[ 36 \times 9 = 324 \]

So, the missing number in the analogy 36 : ? is 324.

Checking the Options

Let's compare our result with the given options:

• Option 1: 404
• Option 2: 256
• Option 3: 144
• Option 4: 324

Our calculated missing number, 324, matches Option 4.

Conclusion

The relationship in the analogy is that the second number is obtained by multiplying the first number by the result of dividing the first number by 4. Applying this rule to 36:?, we find the missing number is 324.

Revision Table: Key Concepts for Number Analogies

Additional Information: Types of Number Series Patterns

Number analogy questions often rely on patterns similar to those found in number series. Some common types of patterns include:

• Arithmetic Series: A constant difference between consecutive terms (e.g., 2, 5, 8, 11...).
• Geometric Series: A constant ratio between consecutive terms (e.g., 3, 6, 12, 24...).
• Square/Cube Patterns: Numbers are squares or cubes of sequential numbers or related to them (e.g., 1, 4, 9, 16...).
• Mixed Operations: A combination of addition, subtraction, multiplication, or division in a sequence.
• Step Patterns: The difference between terms forms another series, or the operation changes at each step.
• Digit Based Patterns: Patterns related to the digits of the numbers themselves.

Identifying these types of patterns can help solve number analogy and series questions more effectively.