Question

If \( 3 \times 4 = 25 \), \( 5 \times 2 = 27 \), \( 6 \times 3 = 39 \), then what is \( 7 \times 5 = ? \)

• A. 61
• B. 65
• C. 71
• D. 73

Hint:

In number puzzles, check for patterns beyond standard arithmetic—try differences, sequences in the added values, or relations like square, cube, or increasing arithmetic sequences.

Answer 1

The Correct Option is C

Step 1: Analyze the given examples to identify the pattern
Let'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.

1. 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 \(25 - 12 = 13\).
2. For \(5 \times 2 = 27\):
   The direct product is \(5 \times 2 = 10\).
   The difference is \(27 - 10 = 17\).
3. 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" values
The values added in each case are: \(13, 17, 21\).
These form an arithmetic progression with common difference:

• \(17 - 13 = 4\)
• \(21 - 17 = 4\)

Step 3: Determine the rule for the "added" value
Let the first number be \(A\) and the second be \(B\). Suppose the added value \(K\) is given by: \[ K = 3A + B \] Using 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 \((A, B) = (6, 3)\): \(K = 3(6) + 3 = 18 + 3 = 21\)

All match the differences observed earlier. Step 4: Formulate the complete pattern
\[ A \times B \rightarrow (A \times B) + (3A + B) \] 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 problem
Let \(A = 7\), \(B = 5\)
\[ 7 \times 5 + (3 \times 7 + 5) = 35 + (21 + 5) = 35 + 26 = \boxed{61} \] Final Answer: 61