A and B are friends and their ages differ by 2 years. As father D is twice as old as A and B is twice as old as his sister C. The age of D and C differ by 40 years. Find the ages of A and B.

Answer

To solve the problem step by step, we will define the variables and set up the equations based on the information given:

1. **Define the Variables:**
Let the age of A be \( A \) and the age of B be \( B \).

2. **Set Up the First Equation:**
According to the problem, the ages of A and B differ by 2 years. This can be expressed as:
\[ A - B = \pm 2 \]
This means either \( A - B = 2 \) or \( B - A = 2 \).

3. **Set Up the Second Equation:**
It is given that father D is twice as old as A. Therefore, we can express D's age as:
\[ D = 2A \]

4. **Set Up the Third Equation:**
B is twice as old as his sister C. This gives us:
\[ B = 2C \quad \Rightarrow \quad C = \frac{B}{2} \]

5. **Set Up the Fourth Equation:**
The ages of D and C differ by 40 years. Since D is older than C, we can write:
\[ D - C = 40 \]
Substituting the expressions for D and C gives:
\[ 2A - \frac{B}{2} = 40 \]

6. **Multiply the Fourth Equation by 2 to Eliminate the Fraction:**
\[ 4A - B = 80 \quad \text{(Equation 1)} \]

7. **Consider the Two Cases for A - B:**
- **Case 1:** \( A - B = 2 \)
\[ A = B + 2 \quad \text{(Equation 2)} \]
- **Case 2:** \( A - B = -2 \)
\[ B = A + 2 \quad \text{(Equation 3)} \]

8. **Substituting Equation 2 into Equation 1:**
Substitute \( A = B + 2 \) into \( 4A - B = 80 \):
\[ 4(B + 2) - B = 80 \]
Simplifying this:
\[ 4B + 8 - B = 80 \]
\[ 3B + 8 = 80 \]
\[ 3B = 72 \quad \Rightarrow \quad B = 24 \]
Now substitute \( B = 24 \) back into Equation 2:
\[ A = 24 + 2 = 26 \]

9. **Substituting Equation 3 into Equation 1:**
Substitute \( B = A + 2 \) into \( 4A - B = 80 \):
\[ 4A - (A + 2) = 80 \]
Simplifying this:
\[ 4A - A - 2 = 80 \]
\[ 3A - 2 = 80 \]
\[ 3A = 82 \quad \Rightarrow \quad A = \frac{82}{3} \]
Now substitute \( A = \frac{82}{3} \) back into Equation 3:
\[ B = \frac{82}{3} + 2 = \frac{82}{3} + \frac{6}{3} = \frac{88}{3} \]

10. **Final Results:**
We have two possible solutions:
- From Case 1: \( A = 26 \) and \( B = 24 \)
- From Case 2: \( A = \frac{82}{3} \) and \( B = \frac{88}{3} \)

### Summary of Ages:
- One solution is \( A = 26 \) years and \( B = 24 \) years.
- The other solution is \( A = \frac{82}{3} \) years and \( B = \frac{88}{3} \) years.