Question

Two years ago, father was thrice as old as his daughter, and 6 years later he will be 4 years older than twice her age. How old are they now?

Hint:

To solve age-related problems, set up equations based on the given conditions, and solve the system of equations to find the ages.

Answer 1

Step 1: Let the current age of the father be \( f \) and the current age of the daughter be \( d \).
Two years ago, the father's age was \( f - 2 \), and the daughter's age was \( d - 2 \). According to the given information, two years ago the father was thrice as old as his daughter: \[ f - 2 = 3(d - 2). \]
Step 2: Six years later, the father's age will be \( f + 6 \), and the daughter's age will be \( d + 6 \). According to the problem, six years later, the father will be 4 years older than twice the daughter's age. So, we can write:} \[ f + 6 = 2(d + 6) + 4. \]
Step 3: Solve the system of equations.
From the first equation: \[ f - 2 = 3(d - 2) \quad \Rightarrow \quad f - 2 = 3d - 6 \quad \Rightarrow \quad f = 3d - 4 \quad \text{(Equation 1)}. \] From the second equation: \[ f + 6 = 2(d + 6) + 4 \quad \Rightarrow \quad f + 6 = 2d + 12 + 4 \quad \Rightarrow \quad f + 6 = 2d + 16 \quad \Rightarrow \quad f = 2d + 10 \quad \text{(Equation 2)}. \]
Step 4: Equate the two expressions for \( f \).
From Equation 1 and Equation 2, we have: \[ 3d - 4 = 2d + 10. \] Solving for \( d \): \[ 3d - 2d = 10 + 4 \quad \Rightarrow \quad d = 14. \]

Step 5: Find \( f \).
Substitute \( d = 14 \) into Equation 1: \[ f = 3(14) - 4 = 42 - 4 = 38. \]
Step 6: Conclusion.
The father is 38 years old, and the daughter is 14 years old.