Understanding the Mathematical Pattern

The question asks us to find the value of an expression using a newly defined operation, represented by the symbol '#'. We are given three examples that show how this operation works between two numbers to produce a result. Our task is to identify the rule or pattern behind this operation and then apply it to find the value of 10 # 2.

Analyzing the Given Examples

Let's look at the examples provided:

• \(5 # 11 = 4\)
• \(13 # 7 = 5\)
• \(10 # 22 = 8\)

We need to find a relationship between the two numbers on the left of the equals sign (e.g., 5 and 11) and the number on the right (e.g., 4).

Identifying the Pattern for the Operation #

Let's try some simple arithmetic operations on the pairs of numbers:

• For 5 and 11:
  • Sum: \(5 + 11 = 16\)
  • Difference: \(|11 - 5| = 6\)
  • Product: \(5 \times 11 = 55\)
• For 13 and 7:
  • Sum: \(13 + 7 = 20\)
  • Difference: \(|13 - 7| = 6\)
  • Product: \(13 \times 7 = 91\)
• For 10 and 22:
  • Sum: \(10 + 22 = 32\)
  • Difference: \(|22 - 10| = 12\)
  • Product: \(10 \times 22 = 220\)

Now let's compare these results with the given output values (4, 5, 8).

Consider the sum of the two numbers:

• \(5 + 11 = 16\). The result is 4. How can we get 4 from 16? Maybe divide by 4? \(16 \div 4 = 4\).
• Let's check this idea with the second example. \(13 + 7 = 20\). If we divide by 4, we get \(20 \div 4 = 5\). This matches the given result 5.
• Let's check with the third example. \(10 + 22 = 32\). If we divide by 4, we get \(32 \div 4 = 8\). This matches the given result 8.

The pattern seems consistent across all three examples. The operation \(a # b\) appears to be defined as \((a + b) \div 4\).

Applying the Pattern to Find 10 # 2

Now we apply the discovered pattern to the expression 10 # 2.

Using the formula \(a # b = (a + b) \div 4\), where \(a = 10\) and \(b = 2\):

\(10 # 2 = (10 + 2) \div 4\)

First, calculate the sum inside the parentheses:

\(10 + 2 = 12\)

Next, perform the division:

\(12 \div 4 = 3\)

So, the value of 10 # 2 is 3.

Verification with Options

The calculated value is 3, which corresponds to Option 1.

The pattern \((a + b) \div 4\) consistently explains the given examples, and applying it to 10 # 2 yields 3.

Revision Table: Mathematical Pattern Solving

Additional Information: Logical Reasoning in Math

Problems like this fall under the category of logical reasoning or quantitative aptitude. They test your ability to observe patterns, form hypotheses, and test them systematically. These types of questions are common in competitive exams and aim to assess analytical skills rather than just rote learning of formulas.

• Types of Patterns: Patterns can be based on arithmetic operations (addition, subtraction, multiplication, division), powers, roots, sequences, spatial arrangements, or a combination of these.
• Problem-Solving Steps:
  1. Understand the problem and the given information.
  2. Analyze the examples provided.
  3. Look for simple relationships between the numbers. Try basic operations first.
  4. Formulate a hypothesis about the underlying rule or pattern.
  5. Test the hypothesis using all the given examples.
  6. If the hypothesis is consistent, apply it to the question asked.
  7. If the hypothesis is not consistent, go back to step 3 and try a different approach.
• Tips: Pay close attention to the numbers. Sometimes the pattern involves digits, number properties (like even/odd), or sequential logic. Don't assume the first pattern you see is correct; always test it with all examples.