Let's denote Raj's age as $R$, his sister's age as $S$, and his father's age as $F$.

From the problem, we have the following information:

1. Raj is twice as old as his sister: $R = 2S$

2. Raj is half as old as his father: $R = \frac{1}{2}F$

3. After 50 years, Raj's sister will be half their father’s age: $S + 50 = \frac{1}{2}(F + 50)$

Step 1: Express everything in terms of $R$

From the first equation: $S = \frac{R}{2}$

From the second equation: $F = 2R$

Step 2: Substitute into the third equation

Substitute $S = \frac{R}{2}$ and $F = 2R$ into the third equation: $\frac{R}{2} + 50 = \frac{1}{2}(2R + 50)$

Step 3: Simplify the equation

Simplify both sides: $\frac{R}{2} + 50 = R + 25$

Subtract $\frac{R}{2}$ from both sides: $50 = \frac{R}{2} + 25$

Subtract 25 from both sides: $25 = \frac{R}{2}$

Multiply both sides by 2 to solve for $R$: $R = 50$

Conclusion

Raj is currently 50 years old.

Would you like more details or have any questions? Here are 5 related questions you might find helpful:

1. How old are Raj's sister and father now?
2. What would Raj's age be in 50 years?
3. How would the solution change if Raj was three times as old as his sister?
4. Can you solve a similar problem where Raj's sister is three years younger?
5. How can we verify our solution by checking each condition?

Tip: Expressing everything in terms of one variable can simplify solving age-related problems.