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