Let:

• $x$ be Raju's current age,
• $f$ be Raju's father's current age,
• $m$ be Raju's mother's current age.

Step 1: Setting up equations based on the information given

1. Five years from now:

   • Raju's father's age will be $f + 5$,
   • Raju's age will be $x + 5$,
   • It is given that $f + 5 = 3(x + 5)$.

   So, we can write: $f + 5 = 3(x + 5)$ Expanding this, we get: $f + 5 = 3x + 15$ $f = 3x + 10$

2. Mother's age in terms of Raju's age:

   • Five years from now, Raju's mother's age will be $m + 5$,
   • We are given that $m + 5 = k(x + 5)$, where $k$ is an integer.

   This gives: $m + 5 = k(x + 5)$ Expanding this, we get: $m = kx + 5k - 5$

3. Present age ratio of Raju's father and mother:

   • The ratio of the father's age to the mother's age is given as $\frac{f}{m} = \frac{8}{5}$.

   Thus: $\frac{f}{m} = \frac{8}{5}$ which implies: $5f = 8m$

Step 2: Substitute $f = 3x + 10$ and $m = kx + 5k - 5$ into the ratio equation

Substituting $f = 3x + 10$ and $m = kx + 5k - 5$ into $5f = 8m$: $5(3x + 10) = 8(kx + 5k - 5)$ Expanding both sides, we get: $15x + 50 = 8kx + 40k - 40$ Bringing terms involving $x$ to one side and constants to the other: $15x - 8kx = 40k - 90$ Factoring out $x$: $x(15 - 8k) = 40k - 90$ Thus: $x = \frac{40k - 90}{15 - 8k}$

Step 3: Solve for integer values of $k$

Since $x$ must be a positive integer, $\frac{40k - 90}{15 - 8k}$ should yield a positive integer. After testing values of $k$, we find that $k = 2$ satisfies the conditions.

1. If $k = 2$: $x = \frac{40(2) - 90}{15 - 8(2)} = \frac{80 - 90}{15 - 16} = \frac{-10}{-1} = 10$ So, $x = 10$.

Step 4: Calculate $f$, $m$, and the sum of their ages

• Raju’s age $x = 10$,
• Father’s age $f = 3x + 10 = 3(10) + 10 = 30 + 10 = 40$,
• Mother’s age $m = kx + 5k - 5 = 2(10) + 5(2) - 5 = 20 + 10 - 5 = 25$.

The sum of their ages is: $x + f + m = 10 + 40 + 25 = 75$

Final Answer

The sum of the current ages of Raju and his parents is 75.

Would you like a further breakdown or more examples?

Here are some related questions you may find useful:

1. What if the ratio of the father's and mother's age had been different, like 7:4?
2. How would the solution change if 'k' had been a non-integer?
3. What other integer values of 'k' might lead to feasible solutions for different initial conditions?
4. How can you verify if the ages satisfy the ratio condition?
5. Could this approach be extended to more complex family age problems?

Tip: When working with age problems, setting clear variables for each person’s age can help keep equations organized and make substitution simpler.