To solve this problem, we need to carefully track the food consumption over time.

Step 1: Initial food supply

• Number of soldiers: 4000
• Daily food consumption per soldier: 1250 grams
• Total food supply for 40 days:
  $\text{Total food} = 4000 \times 1250 \times 40 = 200,000,000 \text{ grams}.$

Step 2: Food consumption for the first 22 days

For the first 22 days, all 4000 soldiers were consuming 1250 grams of food per day.
The total food consumed during this time is: $\text{Food consumed in 22 days} = 4000 \times 1250 \times 22 = 110,000,000 \text{ grams}.$

Step 3: Remaining food after 22 days

After 22 days, the remaining food is: $\text{Remaining food} = 200,000,000 - 110,000,000 = 90,000,000 \text{ grams}.$

Step 4: Change in number of soldiers

• After 22 days, 1600 soldiers leave, so the remaining soldiers are: $4000 - 1600 = 2400 \text{ soldiers}.$
• Let $x$ be the number of new soldiers who join the camp after 22 days, where each new soldier consumes 1000 grams of food daily.
• So, after 22 days, there are $2400 + x$ soldiers in the camp.

Step 5: Food consumption for the remaining 15 days

The total food consumed in the remaining 15 days is given by: $\text{Food consumption for 15 days} = (2400 \times 1250 + x \times 1000) \times 15.$ This food consumption must be equal to the remaining food supply, which is 90,000,000 grams: $(2400 \times 1250 + x \times 1000) \times 15 = 90,000,000.$

Step 6: Solving the equation

First, simplify the equation: $(2400 \times 1250 + 1000x) \times 15 = 90,000,000,$ $2400 \times 1250 = 3,000,000,$ $(3,000,000 + 1000x) \times 15 = 90,000,000.$ Dividing both sides by 15: $3,000,000 + 1000x = 6,000,000.$ Now solve for $x$: $1000x = 6,000,000 - 3,000,000,$ $1000x = 3,000,000,$ $x = 3000.$

Thus, 3000 new soldiers joined the camp.

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Would you like more details or clarifications on this solution? Here are 5 related questions:

1. How would the solution change if the food supply increased?
2. What if the daily consumption of the original soldiers changed after 22 days?
3. How do different food consumption rates affect total supply?
4. Could we solve this problem graphically by plotting food consumption?
5. How would the solution change if soldiers were added gradually over time?

Tip: When solving word problems, organize the information step-by-step to avoid missing any key details.