Let's begin by defining the variables and understanding the relationships between the ages of each family member based on the problem statement.
1. Let [tex]\( \text{mom\_age} \)[/tex] be the age of Kaitlyn's mom.
2. Kaitlyn is 26 years younger than her mom. Therefore, Kaitlyn's age = [tex]\( \text{mom\_age} - 26 \)[/tex].
3. Kaitlyn's dad is 6 years older than her mom. Therefore, Kaitlyn's dad's age = [tex]\( \text{mom\_age} + 6 \)[/tex].
4. The sum of their ages is 6 more than the number of years in a century. A century has 100 years, so the sum of their ages = [tex]\( 100 + 6 = 106 \)[/tex].
Now, let's set up an equation to represent the total sum of their ages:
[tex]\[ \text{mom\_age} + (\text{mom\_age} - 26) + (\text{mom\_age} + 6) = 106 \][/tex]
Combine like terms:
[tex]\[ \text{mom\_age} + \text{mom\_age} - 26 + \text{mom\_age} + 6 = 106 \][/tex]
Simplify the equation:
[tex]\[ 3 \times \text{mom\_age} - 20 = 106 \][/tex]
Add 20 to both sides to isolate the term involving [tex]\(\text{mom\_age}\)[/tex]:
[tex]\[ 3 \times \text{mom\_age} = 126 \][/tex]
Divide both sides by 3 to solve for [tex]\(\text{mom\_age}\)[/tex]:
[tex]\[ \text{mom\_age} = \frac{126}{3} = 42 \][/tex]
Thus, Kaitlyn's mom is 42 years old.
Next, find Kaitlyn's age:
[tex]\[ \text{Kaitlyn's age} = \text{mom\_age} - 26 = 42 - 26 = 16 \][/tex]
Finally, find Kaitlyn's dad's age:
[tex]\[ \text{Dad's age} = \text{mom\_age} + 6 = 42 + 6 = 48 \][/tex]
Therefore, the ages of the family members are:
- Kaitlyn is 16 years old.
- Mom is 42 years old.
- Dad is 48 years old.
