Finding the Logic in Number Relationships

This question asks us to identify a specific relationship or pattern that connects two numbers in given pairs and then apply that same logic to a third number to find its related counterpart. We are given two example pairs: 24 relates to 75, and 36 relates to 111. We need to find the number related to 43 using the same pattern.

Analyzing the Given Number Pairs

Let's examine the two pairs provided:

1. Pair 1: 24 and 75
2. Pair 2: 36 and 111

We need to find a mathematical operation or a series of operations that transforms the first number in each pair into the second number.

Discovering the Pattern or Logic

Let's try to find a relationship between the numbers. Often, these types of problems involve multiplication and addition/subtraction. Let's assume the relationship is linear, in the form \(y = ax + b\), where \(x\) is the first number and \(y\) is the second number in the pair.

Using the first pair (24, 75):

\[ 75 = a \times 24 + b \quad \text{(Equation 1)} \]

Using the second pair (36, 111):

\[ 111 = a \times 36 + b \quad \text{(Equation 2)} \]

Now we can solve these two linear equations simultaneously to find the values of \(a\) and \(b\).

Subtract Equation 1 from Equation 2:

\[ (111 - 75) = (a \times 36 + b) - (a \times 24 + b) \]

\[ 36 = a \times 36 - a \times 24 \]

\[ 36 = a \times (36 - 24) \]

\[ 36 = a \times 12 \]

To find \(a\), divide both sides by 12:

\[ a = \frac{36}{12} = 3 \]

Now substitute the value of \(a = 3\) back into Equation 1:

\[ 75 = 3 \times 24 + b \]

\[ 75 = 72 + b \]

To find \(b\), subtract 72 from both sides:

\[ b = 75 - 72 = 3 \]

So, the relationship or logic is \(y = 3x + 3\). Let's verify this logic with the second pair (36, 111):

For \(x = 36\), \(y = 3 \times 36 + 3 = 108 + 3 = 111\). This matches the given second pair.

The established logic is: Multiply the first number by 3 and then add 3 to the result.

Applying the Logic to the Third Number (43)

Now we apply the same logic to the number 43 to find the related number. Let \(x = 43\).

According to the logic \(y = 3x + 3\):

\[ y = 3 \times 43 + 3 \]

First, calculate \(3 \times 43\):

\[ 3 \times 43 = 129 \]

Then, add 3 to the result:

\[ y = 129 + 3 \]

\[ y = 132 \]

So, following the same logic, 43 is related to 132.

Comparing with Options

Let's compare our result, 132, with the given options:

1. 130
2. 138
3. 135
4. 132

Our calculated number, 132, matches option 4.

Revision Table: Number Analogy Logic

Additional Information on Number Pattern Recognition

Number pattern recognition questions, like this number analogy, are common in logical reasoning and quantitative aptitude tests. They assess your ability to observe numerical sequences or pairs and identify the underlying rule. The rules can vary widely, including:

• Arithmetic operations (addition, subtraction, multiplication, division).
• Multiple operations combined (like multiplication and addition, as seen here).
• Squaring or cubing numbers.
• Operations involving the digits of the number.
• Sequences based on prime numbers, Fibonacci numbers, etc.

To solve these problems, it's helpful to systematically test common relationships. Start with simple operations and move to more complex ones. Comparing the differences or ratios between paired numbers is often a good starting point to find the potential multipliers or constants.