The ratio of the present ages of a mother and daughter is 7:1. Four years ago the ratio of their ages was 19:1. What will be the mother's age four years from now?

Options

A. 42 years
B. 38 years
C. 46 years
D. 36 years

Answer

To solve the problem, we will set up equations based on the information provided about the ages of the mother and daughter.

### Step 1: Define Variables
Let the present age of the mother be \( M \) and the present age of the daughter be \( D \).

### Step 2: Set Up the First Equation
According to the problem, the ratio of the present ages of the mother and daughter is 7:1. This can be expressed as:
\[ \frac{M}{D} = 7 \implies M = 7D \]

### Step 3: Set Up the Second Equation
The problem states that four years ago, the ratio of their ages was 19:1. Four years ago, the mother's age was \( M - 4 \) and the daughter's age was \( D - 4 \). This gives us the second equation:
\[ \frac{M - 4}{D - 4} = 19 \implies M - 4 = 19(D - 4) \]

### Step 4: Expand the Second Equation
Now, we will expand the second equation:
\[ M - 4 = 19D - 76 \]
Rearranging gives:
\[ M = 19D - 72 \]

### Step 5: Substitute the First Equation into the Second
Now, we can substitute \( M = 7D \) from the first equation into the second equation:
\[ 7D = 19D - 72 \]

### Step 6: Solve for \( D \)
Rearranging the equation to isolate \( D \):
\[ 7D - 19D = -72 \implies -12D = -72 \implies D = 6 \]

### Step 7: Find \( M \)
Now that we have \( D \), we can find \( M \):
\[ M = 7D = 7 \times 6 = 42 \]

### Step 8: Calculate Mother's Age Four Years from Now
To find the mother's age four years from now, we add 4 years to her current age:
\[ M + 4 = 42 + 4 = 46 \]

### Final Answer
The mother's age four years from now will be \( 46 \) years.

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