We are tasked with dividing the total remuneration among three persons based on the given conditions. Let's analyze the problem step by step.

---

Let:

• The total reward = $R$.
• The shares of the first, second, and third persons = $x_1$, $x_2$, $x_3$ respectively.

Conditions given:

1. $30\%$ of the total reward $R$ belongs to the third person: $x_3 = 0.3R$

2. Half of the total reward is shared between the first and third persons: $x_1 + x_3 = 0.5R$

3. The second person's share is given as $x_2$, but its value is missing from the question. Let's denote it by $x_2$.

---

From condition 1, substitute $x_3 = 0.3R$ into condition 2:

$x_1 + 0.3R = 0.5R$ $x_1 = 0.5R - 0.3R$ $x_1 = 0.2R$

---

Total reward distribution:

The total reward is distributed as: $R = x_1 + x_2 + x_3$ Substituting the values of $x_1$ and $x_3$: $R = 0.2R + x_2 + 0.3R$ $R = 0.5R + x_2$ $x_2 = 0.5R$

---

Interpretation:

• The second person's share, $x_2$, is $0.5R$, meaning the second person gets half of the total reward.
• If the second person's share is Rs. $x_2$, we can express the total reward $R$ as: $R = 2 \cdot x_2$

---

Final Results:

If the second person's share is given (say Rs. $200$, for example): $R = 2 \cdot 200 = Rs. 400$ $x_1 = 0.2R = 0.2 \cdot 400 = Rs. 80$ $x_3 = 0.3R = 0.3 \cdot 400 = Rs. 120$

---

Do you want me to solve it with a specific value for $x_2$? 😊

---

Expanding the Discussion:

1. What happens if the conditions are altered, e.g., $40\%$ for the third person?
2. How would this work if we used ratios instead of percentages?
3. What if all three persons had equal shares?
4. How do we adjust if $x_2$ is expressed as a fraction instead of an exact percentage?
5. How can this problem be visualized using pie charts for clarity?

Tip:

Always verify your calculations by adding the parts to check if they sum to the total!