To find the two-digit number based on the given conditions, let's go through a detailed step-by-step solution.
### Step 1: Representation of the Number
Represent the two-digit number as 10a + b, where:
- [tex]\(a\)[/tex] is the tens digit.
- [tex]\(b\)[/tex] is the unit digit.
### Step 2: Formulate the Given Conditions
Condition 1: The sum of the number and the number formed by interchanging its digits is 110.
This can be written as:
[tex]\[ (10a + b) + (10b + a) = 110 \][/tex]
Simplifying this equation:
[tex]\[ 10a + a + 10b + b = 110 \][/tex]
[tex]\[ 11a + 11b = 110 \][/tex]
[tex]\[ a + b = 10 \][/tex]
Condition 2: If ten is subtracted from the first number, the new number is 4 more than 5 times the sum of its digits.
This can be written as:
[tex]\[ (10a + b) - 10 = 5(a + b) + 4 \][/tex]
Simplifying this equation:
[tex]\[ 10a + b - 10 = 5(a + b) + 4 \][/tex]
[tex]\[ 10a + b - 10 = 5a + 5b + 4 \][/tex]
[tex]\[ 10a - 5a + b - 5b = 14 \][/tex]
[tex]\[ 5a - 4b = 14 \][/tex]
### Step 3: Solve the System of Equations
We now have a system of two equations:
1. [tex]\( a + b = 10 \)[/tex]
2. [tex]\( 5a - 4b = 14 \)[/tex]
Equation 1: [tex]\( a + b = 10 \)[/tex]
Solve Equation 1 for [tex]\(a\)[/tex]:
[tex]\[ a = 10 - b \][/tex]
Substitute into Equation 2:
[tex]\[ 5(10 - b) - 4b = 14 \][/tex]
[tex]\[ 50 - 5b - 4b = 14 \][/tex]
[tex]\[ 50 - 9b = 14 \][/tex]
[tex]\[ -9b = 14 - 50 \][/tex]
[tex]\[ -9b = -36 \][/tex]
[tex]\[ b = 4 \][/tex]
Substitute [tex]\( b = 4 \)[/tex] back into [tex]\( a + b = 10 \)[/tex]:
[tex]\[ a + 4 = 10 \][/tex]
[tex]\[ a = 6 \][/tex]
### Step 4: Determine the First Number
With [tex]\(a = 6\)[/tex] and [tex]\(b = 4\)[/tex], the two-digit number is:
[tex]\[ 10a + b = 10(6) + 4 = 60 + 4 = 64 \][/tex]
Therefore, the first number is:
[tex]\[ \boxed{64} \][/tex]
The correct answer is (C) 64.
