For soldiers in a desert there are food provisions for 60 days. If after 15 days 500 soldiers joined them and the food lasts 40 days longer, how many soldiers are there in the desert?

Let's solve the problem step by step.

### Step 1: Define Variables
- Let \( N \) be the number of soldiers initially in the desert.
- Let the total amount of food provisions be \( F \). 

### Step 2: Calculate Food Consumption
The initial number of soldiers \( N \) have food for 60 days. Therefore, the daily consumption of food is:

\[
\text{Daily consumption} = \frac{F}{60}
\]

### Step 3: Calculate Food Consumed in the First 15 Days
In the first 15 days, the soldiers consume:

\[
\text{Food consumed in 15 days} = 15 \times \frac{F}{60} = \frac{15F}{60} = \frac{F}{4}
\]

### Step 4: Food Remaining After 15 Days
After 15 days, the remaining food is:

\[
\text{Remaining food} = F - \frac{F}{4} = \frac{3F}{4}
\]

### Step 5: New Total Number of Soldiers
After 15 days, 500 more soldiers join. Therefore, the new total number of soldiers is \( N + 500 \).

### Step 6: Calculate Food Consumption for the Remaining 40 Days
The remaining food \( \frac{3F}{4} \) lasts 40 more days for \( N + 500 \) soldiers. So, the daily consumption for the new group of soldiers is:

\[
\text{Daily consumption for } N + 500 \text{ soldiers} = \frac{3F}{4 \times 40} = \frac{3F}{160}
\]

However, we also know that the daily consumption for \( N + 500 \) soldiers is:

\[
\frac{F}{60} \times (N + 500)
\]

### Step 7: Set Up the Equation
We equate the two expressions for daily consumption:

\[
\frac{F}{60} \times (N + 500) = \frac{3F}{160}
\]

### Step 8: Solve the Equation
Cancel \( F \) from both sides:

\[
\frac{N + 500}{60} = \frac{3}{160}
\]

Multiply both sides by 60:

\[
N + 500 = \frac{180}{160}
\]

Simplify further:

\[
N + 500 = 1.125N
\]

Now, solve for \( N \):

\[
160(N + 500) = 180N
\]

Expanding and solving the equation:

\[
160N + 80000 = 180N
\]

\[
80000 = 20N
\]

\[
N = 4000
\]

So, the initial number of soldiers is \( N = 4000 \). The total number of soldiers after 500 joined is:

\[
N + 500 = 4500
\]

### Final Answer:
There are 4500 soldiers in the desert after 500 joined.

Would you like more details or have any other questions? 

Here are 5 related questions:

1. How would the solution change if 1000 soldiers joined instead of 500?
2. What if the food lasted 50 days longer instead of 40 days?
3. How would the problem be different if the number of soldiers decreased instead of increased?
4. What if the soldiers had a varied rate of food consumption?
5. How would the scenario change if the initial food supply was unknown?

**Tip:** When solving word problems, always define your variables clearly and build equations step by step.

Learn how to solve the problem involving soldiers in a desert with food provisions. Understand the application of algebraic equations and rates in determining the number of soldiers. Suitable for Grades 7-9 students studying algebra and word problems.

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