Understanding Number Relations and Patterns

This question asks us to identify a specific mathematical relationship or pattern between pairs of numbers and apply that pattern to a third number to find its corresponding pair from the given options. We are given two examples of this relationship:

• 125 is related to 50
• 230 is related to 92

We need to find out what 315 is related to, following the same logic.

Identifying the Number Pattern

Let's examine the relationship between the first number and the second number in the given pairs. We can try to find a common operation or proportion that connects them.

Consider the first pair: 125 and 50.

Possible relationships could involve addition, subtraction, multiplication, division, or a combination of these operations.

• Is there a constant difference? $125 - 50 = 75$. Let's check the second pair: $230 - 92 = 138$. The difference is not constant.
• Is there a constant ratio? Let's try dividing the first number by the second number, or vice versa.

Ratio of the first number to the second number for the first pair:

$$ \frac{125}{50} = \frac{125 \div 25}{50 \div 25} = \frac{5}{2} = 2.5 $$

Ratio of the first number to the second number for the second pair:

$$ \frac{230}{92} $$

To simplify this ratio, we can look for common factors. Both are even numbers. Let's divide by 2:

$$ \frac{230 \div 2}{92 \div 2} = \frac{115}{46} $$

115 ends in 5, so it's divisible by 5. $115 = 5 \times 23$. 46 is $2 \times 23$. So 23 is a common factor.

$$ \frac{115 \div 23}{46 \div 23} = \frac{5}{2} = 2.5 $$

The ratio of the first number to the second number is indeed constant for both pairs! The first number is 2.5 times the second number.

Alternatively, we can say that the second number is $\frac{1}{2.5}$ times the first number.

$$ \frac{1}{2.5} = \frac{1}{5/2} = \frac{2}{5} = 0.4 $$

So, the relationship is: Second number = First number $\times$ 0.4.

Verifying the Relationship

Let's verify this relationship with the given pairs:

• For the pair 125 and 50: $125 \times 0.4 = 125 \times \frac{4}{10} = 125 \times \frac{2}{5} = \frac{125}{5} \times 2 = 25 \times 2 = 50$. This is correct.
• For the pair 230 and 92: $230 \times 0.4 = 230 \times \frac{4}{10} = 230 \times \frac{2}{5} = \frac{230}{5} \times 2 = 46 \times 2 = 92$. This is also correct.

The established pattern is consistent: the second number is 0.4 times the first number.

Applying the Pattern to 315

Now we apply the same relationship to 315 to find the number it is related to. Let the missing number be $X$.

According to the pattern:

$$ X = 315 \times 0.4 $$

Let's calculate the value of $X$:

$$ X = 315 \times \frac{4}{10} = 315 \times \frac{2}{5} $$

First, divide 315 by 5:

$$ \frac{315}{5} = 63 $$

Now, multiply the result by 2:

$$ X = 63 \times 2 = 126 $$

So, 315 is related to 126 based on the same logic.

Comparing with Options

Let's compare our result, 126, with the given options:

• Option 1: 145
• Option 2: 135
• Option 3: 120
• Option 4: 126

Our calculated value, 126, matches Option 4.

Therefore, following the same logic, 315 is related to 126.

Revision Table: Number Relation Pattern

Let's quickly review the pattern and the calculations involved in this number relation problem.

Additional Information: Solving Number Analogy Problems

Number analogy problems, like the one we just solved, require you to find a hidden rule or relationship connecting numbers in a given pair or series. Here are some common types of relationships to look for:

• Arithmetic Operations: Addition, subtraction, multiplication, division, or a combination of these.
• Powers and Roots: Squares, cubes, square roots, cube roots.
• Digit Manipulation: Sum of digits, product of digits, reversing digits, or operations based on the digits themselves.
• Prime or Composite Numbers: The numbers might follow a pattern related to prime or composite numbers.
• Sequences: Fibonacci sequence, arithmetic progression, geometric progression, etc., if there are multiple numbers in a sequence.
• Proportions: As seen in this problem, the ratio between numbers might be constant.
• Algebraic Relationships: The relationship might follow a simple algebraic formula like $y = ax + b$ or $y = ax^2$.

Strategy for Solving:

1. Look at the given pairs carefully.
2. Try simple arithmetic operations first (addition, subtraction, multiplication, division) to see if a constant value or ratio exists.
3. If simple operations don't work, consider powers, roots, or digit manipulation.
4. Test the potential rule with all the given pairs to ensure consistency.
5. Once the rule is confirmed, apply it to the new number to find the missing term.
6. Compare your result with the provided options.

Practice with different types of number relation problems helps in quickly identifying common patterns.