Step 1: Analyze the given examples to identify the patternLet's examine how the result is obtained from the two numbers in each example. A common approach in such puzzles is to look at the product of the numbers and any additional value needed to reach the given result.
- For \(3 \times 4 = 25\):
The direct product of the numbers is \(3 \times 4 = 12\).
The difference between the given result and the product is \(25 - 12 = 13\). - For \(5 \times 2 = 27\):
The direct product is \(5 \times 2 = 10\).
The difference is \(27 - 10 = 17\). - For \(6 \times 3 = 39\):
The product is \(6 \times 3 = 18\).
The difference is \(39 - 18 = 21\).
Step 2: Identify the pattern in the "added" valuesThe values added in each case are: \(13, 17, 21\).
These form an arithmetic progression with a common difference of 4:
- \(17 - 13 = 4\)
- \(21 - 17 = 4\)
Step 3: Determine the rule for the "added" valueLet the first number be \(A\) and the second be \(B\). Assume the added value \(K\) is given by:\[K = 3A + B\]Verifying with the values:
- For \((A, B) = (3, 4)\): \(K = 3(3) + 4 = 9 + 4 = 13\)
- For \((A, B) = (5, 2)\): \(K = 3(5) + 2 = 15 + 2 = 17\)
- For \((A, B) = (6, 3)\): \(K = 3(6) + 3 = 18 + 3 = 21\)
These calculated values match the observed differences.
Step 4: Formulate the complete patternThe pattern is: \(A \times B \rightarrow (A \times B) + (3A + B)\)
Step 5: Verify the pattern with all given examples - \(3 \times 4 + (3 \times 3 + 4) = 12 + 13 = 25\) ✅
- \(5 \times 2 + (3 \times 5 + 2) = 10 + 17 = 27\) ✅
- \(6 \times 3 + (3 \times 6 + 3) = 18 + 21 = 39\) ✅
Step 6: Apply the pattern to solve the target problemFor \(A = 7\) and \(B = 5\):\[7 \times 5 + (3 \times 7 + 5) = 35 + (21 + 5) = 35 + 26 = 61\]
Final Answer: 61