Let's break down the problem step-by-step.

1. Initial Situation:
  - 1200 soldiers
  - Food stock lasts for 30 days

  The total amount of food stock can be represented in terms of soldier-days. So, the total food stock is enough for:

  \[
  1200 \text{ soldiers} \times 30 \text{ days} = 36,000 \text{ soldier-days}
  \]

2. After 10 Days:
  - Food consumption for 10 days with 1200 soldiers:

  \[
  1200 \text{ soldiers} \times 10 \text{ days} = 12,000 \text{ soldier-days}
  \]

  Remaining food stock:

  \[
  36,000 \text{ soldier-days} - 12,000 \text{ soldier-days} = 24,000 \text{ soldier-days}
  \]

3. New Situation:
  - After 10 days, 300 more soldiers join, making the total:

  \[
  1200 + 300 = 1500 \text{ soldiers}
  \]

4. Calculate How Many More Days the Remaining Food Will Last:

  The remaining food stock is 24,000 soldier-days. With 1500 soldiers consuming food, we can calculate the number of days the food will last:

  \[
  \text{Number of days} = \frac{24,000 \text{ soldier-days}}{1500 \text{ soldiers}}
  \]
  \[
  \text{Number of days} = 16
  \]

So, the remaining food stock will last for 16 days.

Remaining days = 30 - 10 = 20 days

For 1200 soldiers, the stock will last for 20 more days

For 1 soldier, the stock will last for 20 x 1200 more days

So, for (1200 + 300 = 1500) soldiers, the stock will last = 20 x 1200 / 1500 = 16 more days