To solve this problem, we need to calculate how many provisions remain after the first 12 days and then determine how many additional days the remaining provisions can sustain the increased garrison at the new consumption rate.

Step-by-Step Solution

1. Initial Provision Calculation:

   • The initial number of men is 200.
   • Each man consumes 400 grams per day.
   • The provisions are sufficient for 30 days.

   The total amount of provisions available is: $\text{Total provisions} = 200 \times 400 \times 30 \text{ grams}$

2. Provisions Consumed in the First 12 Days:

   • For the first 12 days, the number of men remains 200, and each consumes 400 grams per day.

   The total provisions consumed in 12 days is: $\text{Provisions consumed in 12 days} = 200 \times 400 \times 12 \text{ grams}$

3. Remaining Provisions After 12 Days: $\text{Remaining provisions} = \text{Total provisions} - \text{Provisions consumed in 12 days}$

   Substituting the values: $\text{Remaining provisions} = (200 \times 400 \times 30) - (200 \times 400 \times 12)$

   Simplify by factoring out the common terms: $\text{Remaining provisions} = 200 \times 400 \times (30 - 12) = 200 \times 400 \times 18$

4. New Situation with Additional Men:

   • After 12 days, 50 more men join, making the total number of men 250.
   • The new daily consumption rate per man is 360 grams.

   The total daily consumption for 250 men is: $\text{New daily consumption} = 250 \times 360 \text{ grams}$

5. Finding $x$: Number of Additional Days Provisions Last:

   • Let $x$ be the number of additional days the remaining provisions can last.

   The total consumption for $x$ days by 250 men at the new rate is: $\text{Total consumption for } x \text{ days} = 250 \times 360 \times x \text{ grams}$

   Since the remaining provisions must equal this total consumption: $200 \times 400 \times 18 = 250 \times 360 \times x$

6. Solving for $x$: $x = \frac{200 \times 400 \times 18}{250 \times 360}$

   Simplify the fraction: $x = \frac{200 \times 400 \times 18}{250 \times 360} = \frac{200 \times 4 \times 18}{9 \times 25}$

   $x = \frac{200 \times 4 \times 2}{5} = \frac{1600}{5} = 16$

So, the value of $x$ is 16 days.

Conclusion

The remaining provisions will be sufficient for 16 more days after 50 men join the garrison and the consumption rate changes to 360 grams per day per man.

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Would you like more details on any part of this solution or have any questions?

Here are some related questions to consider:

1. What would the remaining days be if only 25 men had joined instead of 50?
2. How would the calculation change if the initial provisions were for 45 days instead of 30?
3. What would happen if the consumption rate did not change and remained at 400 grams per day per man?
4. How would you solve for the remaining days if instead of grams, the units were in kilograms?
5. How would an increase in the number of men and the consumption rate simultaneously affect the remaining provisions?

Tip: When solving word problems involving rates and amounts, carefully track each step by writing down what each number represents to avoid confusion.