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The image appears to show a problem related to a math or reasoning challenge that involves calculating the total amount of money raised by children for a cause. Here's the content and how you can approach solving it:

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**Problem Text:**
*"Tania and John collected some money for a charity. Tania will be charged $2 for each amount and John $3. Each child collected only full amounts contributing to the final total amount collected in pounds. Determine the number of pounds with the least number of points that Tania and John collected."*

**Example:**
\[ \_\_a\_\_\] equals \[ \_\_b\_\_ \] pounds.

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### Interpretation and Solution Approach:

1. **Understanding the Problem:**
   - Tania collects $ x $ pounds at a rate of $2 per pound.
   - John collects $ y $ pounds at a rate of $3 per pound.
   - The task is to find the smallest possible total amount of money that can be collected using these contributions.

2. **Formulation:**
   You need to find the smallest positive integer $ z $ such that:
   \[
   2x + 3y = z
   \]
   where $ x $ and $ y $ are non-negative integers.

3. **Finding the Least Total:**
   This is a classic linear Diophantine equation problem where the smallest solution can be determined by testing different values of $ x $ and $ y $.

   To solve it, you can start with small values of $ x $ and determine $ y $ such that $ 2x + 3y = z $ is minimized.

   - Start with $ x = 0 $: $ z = 3y $, smallest $ y = 1 $ gives $ z = 3 $.
   - Next, try $ x = 1 $: $ 2(1) + 3y = z $, so $ z = 2 + 3y $, smallest $ y = 1 $ gives $ z = 5 $.
   - If you continue, the smallest $ z $ that fits is $ z = 1 $, but this cannot be obtained with positive integers for $ x $ and $ y $.

4. **Finding the Least Number of Pounds:**
   Typically, the answer would be derived from the smallest linear combination of 2 and 3 that produces an integer solution. The smallest amount that fits is $ z = 5 $ pounds when $ x = 1 $ and $ y = 1 $.

   Therefore, the answer is likely **5 pounds**.

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Would you like to go into further details or do you have any other questions?

### Relative Questions:
1. What is a Diophantine equation and how is it used in number theory?
2. How do you solve linear combinations for integer solutions?
3. What are the properties of linear Diophantine equations with two variables?
4. How can you generalize the method to solve similar problems with different coefficients?
5. What is the significance of finding the least possible value in this context?

### Tip:
When solving problems involving linear combinations with integer coefficients, the smallest solution can often be found by systematically testing small integer values and verifying the resulting sums.

Learn how to solve the problem of money collection involving Tania and John for a charity. This problem focuses on linear Diophantine equations, calculating total amounts collected by children at different rates. Suitable for high school students.

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