Two years ago a mother was 4 times as old as her son. Six years from now her age will become more than double her son's age by 10 years. What is the present ratio of their ages ?

Options

A. A) `2:1`
B. B) `38:11`
C. `C) 19:7`
D. D) `3:1`

Explanation

To solve the problem step by step, let's denote the present age of the mother as \( X \) and the present age of the son as \( Y \).

### Step 1: Set up the equations based on the information given.
1. **Two years ago**, the mother's age was \( X - 2 \) and the son's age was \( Y - 2 \).
2. According to the problem, two years ago the mother was 4 times as old as her son:
\[ X - 2 = 4(Y - 2) \]

### Step 2: Simplify the first equation.
Expanding the equation:
\[ X - 2 = 4Y - 8 \]
Rearranging gives us:
\[ X - 4Y = -6 \quad \text{(Equation 1)} \]

### Step 3: Set up the second equation based on future ages.
1. **Six years from now**, the mother's age will be \( X + 6 \) and the son's age will be \( Y + 6 \).
2. The problem states that six years from now, the mother's age will be more than double her son's age by 10 years:
\[ X + 6 = 2(Y + 6) + 10 \]

### Step 4: Simplify the second equation.
Expanding the equation:
\[ X + 6 = 2Y + 12 + 10 \]
This simplifies to:
\[ X + 6 = 2Y + 22 \]
Rearranging gives us:
\[ X - 2Y = 16 \quad \text{(Equation 2)} \]

### Step 5: Solve the system of equations.
Now we have the following two equations:
1. \( X - 4Y = -6 \) (Equation 1)
2. \( X - 2Y = 16 \) (Equation 2)

We can subtract Equation 1 from Equation 2:
\[ (X - 2Y) - (X - 4Y) = 16 - (-6) \]
This simplifies to:
\[ 2Y = 22 \]
Dividing both sides by 2 gives:
\[ Y = 11 \]

### Step 6: Substitute back to find \( X \).
Now, substitute \( Y = 11 \) into Equation 2:
\[ X - 2(11) = 16 \]
This simplifies to:
\[ X - 22 = 16 \]
Adding 22 to both sides gives:
\[ X = 38 \]

### Step 7: Find the present ratio of their ages.
Now we have:
- Mother's age \( X = 38 \)
- Son's age \( Y = 11 \)

The present ratio of their ages is:
\[ \text{Ratio} = \frac{X}{Y} = \frac{38}{11} \]

### Final Answer:
The present ratio of their ages is \( 38:11 \).

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