Four years ago, a mother was four times as old as her daughter. Six years later, the mother will be two and a half times as old as her daughter at that time. Find the present ages of mother and her daughter.

Answer

To solve the problem of finding the present ages of the mother and daughter, we can follow these steps:

### Step 1: Define the Variables
Let:
- \( x \) = present age of the mother
- \( y \) = present age of the daughter

### Step 2: Set Up the First Equation
According to the problem, four years ago, the mother was four times as old as her daughter. This can be expressed as:
\[ x - 4 = 4(y - 4) \]
Simplifying this equation:
\[ x - 4 = 4y - 16 \]
Rearranging gives:
\[ x - 4y = -12 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
The problem also states that six years later, the mother will be two and a half times as old as her daughter. This can be expressed as:
\[ x + 6 = \frac{5}{2}(y + 6) \]
Multiplying both sides by 2 to eliminate the fraction:
\[ 2(x + 6) = 5(y + 6) \]
Expanding this gives:
\[ 2x + 12 = 5y + 30 \]
Rearranging gives:
\[ 2x - 5y = 18 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
Now we have two equations:
1. \( x - 4y = -12 \) (Equation 1)
2. \( 2x - 5y = 18 \) (Equation 2)

To eliminate \( x \), we can multiply Equation 1 by 2:
\[ 2(x - 4y) = 2(-12) \]
This simplifies to:
\[ 2x - 8y = -24 \quad \text{(Equation 3)} \]

### Step 5: Subtract Equation 2 from Equation 3
Now we can subtract Equation 2 from Equation 3:
\[ (2x - 8y) - (2x - 5y) = -24 - 18 \]
This simplifies to:
\[ -3y = -42 \]
Dividing both sides by -3 gives:
\[ y = 14 \]

### Step 6: Substitute Back to Find \( x \)
Now that we have \( y \), we can substitute it back into Equation 1 to find \( x \):
\[ x - 4(14) = -12 \]
This simplifies to:
\[ x - 56 = -12 \]
Adding 56 to both sides gives:
\[ x = 44 \]

### Final Answer
Thus, the present ages are:
- Mother's age \( x = 44 \) years
- Daughter's age \( y = 14 \) years