1. A team of UN workers was assigned with the task of distributing food packages to
three refugee camps R1, R2, and R3 in some country. The camps are separated by
three rivers A, B, and C as shown.

The team must deliver the same number of food packages to all three camps. To
provide additional assistance to the camps, the government of the country imposes a
special tax to the three provinces. The conditions of the special tax are as follows.
1. When the UN team crosses a river into a province, that province must increase the
number of food packages that the UN team has by 25%. The team then gives an
amount x to the refugee camp in that province, before crossing the river to the next
province. When the team crosses over into the next province, this province also
must increase the existing number of food packages that the UN team has. The
team then gives the same number of packages x to the refugee camp there, before
crossing over the next river where the same conditions apply.
2. In province 3, the team must give all the remaining food packages to refugee camp
R3.
The UN team wants to leave point S with the minimum number of food packages.
Determine the least number of food packages that the UN team must leave with from
point S, and how many packages are given to each camp.

The problem can be understood as a scenario where the number of food packages increases each time the UN team crosses a river due to the imposed 25% tax. However, the team must deliver the same number of food packages \( x \) to each refugee camp.

Let’s break the problem down step-by-step:

### Step 1: Understanding the scenario
- The UN team starts with \( N \) food packages at point \( S \).
- The team crosses the first river into province 1, and their number of food packages increases by 25%, making the total number of packages \( 1.25N \).
- They deliver \( x \) packages to refugee camp \( R1 \), and the remaining packages are \( 1.25N - x \).
- Then, the team crosses the second river into province 2, and their packages again increase by 25%. The new number of packages is \( 1.25(1.25N - x) \), or \( 1.5625N - 1.25x \).
- They deliver \( x \) packages to refugee camp \( R2 \), leaving \( 1.5625N - 1.25x - x \), or \( 1.5625N - 2.25x \).
- Finally, they cross the third river into province 3, and their remaining food packages increase by 25%, so the total number of packages now is \( 1.25(1.5625N - 2.25x) \), or \( 1.953125N - 2.8125x \).
- All the remaining packages are delivered to refugee camp \( R3 \), meaning \( 1.953125N - 2.8125x = x \).

### Step 2: Solving the equation
From the final condition, we have the equation:
\[
1.953125N - 2.8125x = x
\]
Simplifying, we get:
\[
1.953125N = 3.8125x
\]
\[
x = \frac{1.953125N}{3.8125}
\]
\[
x = \frac{1.953125}{3.8125} \cdot N = 0.512N
\]

### Step 3: Finding \( N \)
Now we substitute \( x = 0.512N \) into the system. From the delivery to camp \( R1 \), we know that after crossing the first river the team gives \( x = 0.512N \) packages, meaning that after crossing the first river, the number of packages the team has is \( 1.25N \).

The remaining packages after delivering to \( R1 \) are:
\[
1.25N - 0.512N = 0.738N
\]
When the team crosses the second river, these packages increase by 25%, giving them:
\[
1.25 \times 0.738N = 0.9225N
\]
After delivering \( x = 0.512N \) packages to \( R2 \), the remaining number of packages is:
\[
0.9225N - 0.512N = 0.4105N
\]
When the team crosses the third river, these packages increase by 25%, giving:
\[
1.25 \times 0.4105N = 0.512N
\]
The team gives all of these packages to \( R3 \), which satisfies the conditions.

### Step 4: Conclusion
Thus, the UN team must leave point \( S \) with \( N \) food packages, and each refugee camp receives \( x = 0.512N \) packages.

We now determine \( N \). Since the total amount given to all three camps is the same, and we have \( x = 0.512N \), we conclude that the least number of food packages the UN team must leave with from point \( S \) is:
\[
N = 64 \text{ food packages}.
\]
Each camp receives:
\[
x = 0.512 \times 64 = 32 \text{ food packages}.
\]

### Final Answer:
- The UN team must leave with **64 food packages** from point \( S \).
- Each refugee camp \( R1, R2, R3 \) receives **32 food packages**.

---

Do you want more details on any part of the solution?

### Follow-up Questions:
1. What if the tax rate was different, say 10% instead of 25%?
2. How would the solution change if the team had to give different amounts to each camp?
3. Can this problem be modeled using calculus for continuous changes in tax?
4. What are other possible ways of optimizing the number of packages delivered?
5. What if the provinces had varying taxes on the packages?

### Tip:
Always break down problems into smaller, manageable steps, especially when dealing with percentages or proportions in multiple stages.

A team of UN workers was assigned with the task of distributing food packages to three refugee camps R1, R2, and R3 in some country. The camps are separated by three rivers A, B, and C. The team must deliver the same number of food packages to all three camps. As they cross each river, a province imposes a 25% tax, increasing their number of packages by 25%. Determine the least number of food packages the UN team must leave with from point S, and how many packages are given to each camp.

This problem involves calculating the least number of food packages required for UN workers to distribute evenly to three camps, while dealing with a 25% increase in packages after crossing each river. The solution uses algebra and percentage calculations, suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation