Final answer:
The ages of the two sisters can be determined by setting up a system of equations and solving the quadratic equation. The younger sister is 12 years old, and the older sister is 17 years old.
Step-by-step explanation:
To calculate the ages of the two sisters based on the information given, we can set up a system of equations. Let's call the age of the younger sister y years, and the age of the older sister y + 5 years, since we know the difference in their ages is 5 years. We also know the product of their ages is 204. Hence, our system of equations is:
- The difference of ages: y + 5 - y = 5
- The product of ages: y(y + 5) = 204
Now we solve the second equation:
- y^2 + 5y = 204
- y^2 + 5y - 204 = 0
After factoring the quadratic equation, we find the possible values for y:
This means y could be -17 or 12, but since we cannot have a negative age, we take y = 12 years. Therefore, the younger sister is 12 years old, and the older sister is 12 years + 5 years = 17 years old.