Six years before, the age of mother was numerically equal to the square of son's age. Three years hence, her age will be thrice the age of her son then. Find the present ages of the mother and son.

Explanation

To solve the problem, we need to set up equations based on the information provided. Let's denote the present age of the mother as \( x \) and the present age of the son as \( y \).

### Step 1: Formulate the first equation
According to the problem, six years ago, the age of the mother was numerically equal to the square of the son's age. This can be expressed as:
\[ x - 6 = (y - 6)^2 \]

### Step 2: Formulate the second equation
The problem also states that three years from now, the mother's age will be three times the son's age. This can be expressed as:
\[ x + 3 = 3(y + 3) \]

### Step 3: Simplify the second equation
Let's simplify the second equation:
\[ x + 3 = 3y + 9 \]
Rearranging this gives:
\[ x = 3y + 6 \]

### Step 4: Substitute the second equation into the first equation
Now, we will substitute \( x \) from the second equation into the first equation:
\[ (3y + 6) - 6 = (y - 6)^2 \]
This simplifies to:
\[ 3y = (y - 6)^2 \]

### Step 5: Expand and rearrange the equation
Expanding the right side:
\[ 3y = y^2 - 12y + 36 \]
Rearranging gives:
\[ y^2 - 15y + 36 = 0 \]

### Step 6: Factor the quadratic equation
Now we will factor the quadratic equation:
\[ y^2 - 15y + 36 = (y - 12)(y - 3) = 0 \]
This gives us two possible solutions for \( y \):
\[ y - 12 = 0 \quad \Rightarrow \quad y = 12 \]
\[ y - 3 = 0 \quad \Rightarrow \quad y = 3 \]

### Step 7: Find the corresponding values of \( x \)
Now we will find the corresponding values of \( x \) for both values of \( y \):
1. If \( y = 12 \):
\[ x = 3(12) + 6 = 36 + 6 = 42 \]
2. If \( y = 3 \):
\[ x = 3(3) + 6 = 9 + 6 = 15 \]

### Step 8: Verify the solutions
We need to check which pair of ages satisfies both conditions:
- For \( (x, y) = (42, 12) \):
- Six years ago: Mother = 36, Son = 6 → \( 36 = 6^2 \) (True)
- Three years hence: Mother = 45, Son = 15 → \( 45 = 3 \times 15 \) (True)

- For \( (x, y) = (15, 3) \):
- Six years ago: Mother = 9, Son = -3 → \( 9 = (-3)^2 \) (True, but age cannot be negative)
- Three years hence: Mother = 18, Son = 6 → \( 18 = 3 \times 6 \) (True)

Thus, the only valid solution is:
- Present age of mother = 42 years
- Present age of son = 12 years

### Final Answer
- Present age of mother: 42 years
- Present age of son: 12 years