6 years ago, a mother's age was 6 times her daughter's age. Three years hence the daughter would be one-third her mother's age. The current age in years) of the mother is
A)45
B)39
C)36
D)42

Options

A. 45
B. 39
C. 36
D. 42

Answer

To solve the problem step by step, let's define the variables and set up the equations based on the information given.

### Step 1: Define the Variables
Let:
- \( x \) = Mother's current age
- \( y \) = Daughter's current age

### Step 2: Set Up the First Equation
According to the problem, 6 years ago, the mother's age was 6 times her daughter's age. Therefore, we can write the equation as:
\[ x - 6 = 6(y - 6) \]
Expanding this gives:
\[ x - 6 = 6y - 36 \]
Rearranging it, we get:
\[ x - 6y = -30 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
The problem also states that three years from now, the daughter will be one-third of her mother's age. This gives us the second equation:
\[ y + 3 = \frac{1}{3}(x + 3) \]
Multiplying both sides by 3 to eliminate the fraction:
\[ 3(y + 3) = x + 3 \]
Expanding this gives:
\[ 3y + 9 = x + 3 \]
Rearranging it, we get:
\[ x - 3y = 6 \quad \text{(Equation 2)} \]

### Step 4: Solve the System of Equations
Now we have a system of two equations:
1. \( x - 6y = -30 \)
2. \( x - 3y = 6 \)

We can solve these equations by elimination or substitution. Let's use substitution.

From Equation 2, we can express \( x \) in terms of \( y \):
\[ x = 3y + 6 \]

Now substitute \( x \) in Equation 1:
\[ (3y + 6) - 6y = -30 \]
Simplifying this gives:
\[ 3y + 6 - 6y = -30 \]
\[ -3y + 6 = -30 \]
Subtracting 6 from both sides:
\[ -3y = -36 \]
Dividing by -3:
\[ y = 12 \]

### Step 5: Find the Mother's Age
Now that we have the daughter's age, we can find the mother's age using Equation 2:
\[ x = 3y + 6 = 3(12) + 6 = 36 + 6 = 42 \]

### Final Answer
The current age of the mother is:
\[ \boxed{42} \]