The average age of two boys and their father is greater than the average age of those two boys and their mother by 3 yr . The average age of the four is 19yr. If the average age of the two boys be 11/2 yr , then the age of the father and mother.

Options

A. 37yr and 28 yr
B. 47 yr and 38 yr
C. 50 yr and 41 yr
D. 35 yr and 32 yr

Answer

To solve the problem step by step, we will use the information provided in the question.

### Step 1: Understand the given information
We are given:
1. The average age of two boys and their father is greater than the average age of those two boys and their mother by 3 years.
2. The average age of the four (the two boys, their father, and their mother) is 19 years.
3. The average age of the two boys is \( \frac{11}{2} \) years (which is 5.5 years).

### Step 2: Calculate the total age of the four individuals
The average age of the four individuals is 19 years. Therefore, the total age of the four individuals can be calculated as follows:

\[ \text{Total age of four} = \text{Average age} \times \text{Number of individuals} = 19 \times 4 = 76 \text{ years} \]

### Step 3: Calculate the total age of the two boys
The average age of the two boys is \( \frac{11}{2} \) years. Therefore, the total age of the two boys can be calculated as follows:

\[ \text{Total age of two boys} = \text{Average age} \times \text{Number of boys} = \frac{11}{2} \times 2 = 11 \text{ years} \]

### Step 4: Calculate the total age of the father and mother
We know the total age of the four individuals is 76 years, and the total age of the two boys is 11 years. Therefore, the total age of the father and mother can be calculated as follows:

\[ \text{Total age of father and mother} = \text{Total age of four} - \text{Total age of two boys} = 76 - 11 = 65 \text{ years} \]

### Step 5: Set up the equation based on the age difference
According to the problem, the average age of the two boys and their father is greater than the average age of the two boys and their mother by 3 years. Let \( F \) be the age of the father and \( M \) be the age of the mother. We can express this as:

\[ \frac{B1 + B2 + F}{3} = \frac{B1 + B2 + M}{3} + 3 \]

Substituting the total age of the two boys (11 years):

\[ \frac{11 + F}{3} = \frac{11 + M}{3} + 3 \]

### Step 6: Simplify the equation
Multiplying through by 3 to eliminate the fraction:

\[ 11 + F = 11 + M + 9 \]

This simplifies to:

\[ F = M + 9 \]

### Step 7: Substitute into the total age equation
Now we have two equations:
1. \( F + M = 65 \)
2. \( F = M + 9 \)

Substituting the second equation into the first:

\[ (M + 9) + M = 65 \]

### Step 8: Solve for the age of the mother
Combining like terms:

\[ 2M + 9 = 65 \]

Subtracting 9 from both sides:

\[ 2M = 56 \]

Dividing by 2:

\[ M = 28 \text{ years} \]

### Step 9: Solve for the age of the father
Using the value of \( M \) to find \( F \):

\[ F = M + 9 = 28 + 9 = 37 \text{ years} \]

### Final Answer
The ages are:
- Age of the mother: 28 years
- Age of the father: 37 years

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