To solve the problem, let’s go through the details step-by-step and find the two-digit number Carmen is thinking of.
1. Understand the given conditions:
- The sum of the digits is 12.
- The product of the digits is 35.
- The digit in the tens place is greater than the digit in the ones place.
2. Set up variables:
- Let \(\(T\)\) be the tens-place digit.
- Let \(\(O\)\) be the ones-place digit.
3. Form equations based on the conditions:
- \(\(T + O = 12\)\)
- \(\(T \times O = 35\)\)
- \(\(T > O\)\)
4. Find pairs of digits satisfying the sum condition:
- We need to find pairs \(\((T, O)\)\) such that the sum \(\(T + O = 12\)\).
5. Check each pair to see if they fulfill the product condition:
- Let’s pair up the digits and check:
- \(\(1 + 11 = 12\)\) but 11 is not a valid single digit.
- \(\(2 + 10 = 12\)\) but 10 is not a valid single digit.
- \(\(3 + 9 = 12\)\), and the product \(\(3 \times 9 = 27\)\) (not 35).
- \(\(4 + 8 = 12\)\), and the product \(\(4 \times 8 = 32\)\) (not 35).
- \(\(5 + 7 = 12\)\), and the product \(\(5 \times 7 = 35\)\).
6. Verify if the digits satisfy the third condition:
- For \(\(T = 7\)\) and \(\(O = 5\)\):
- \(\(T + O = 12\)\), which is true.
- \(\(T \times O = 35\)\), which is true.
- \(\(T > O\)\), which is true (7 > 5).
Since \(\(T = 7\)\) and \(\(O = 5\)\) satisfy all three conditions, the two-digit number Carmen is thinking of is 75.
