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This image presents a mathematics problem involving proportional division of a lottery prize between two people, Paula and Sara. They spent a total of 80€ on lottery tickets, with Paula contributing two-fifths and Sara the remaining amount. When they won a prize of 48,000€, it was distributed based on their initial contributions. 

### Key Steps and Solution:

#### 1. **Initial Contributions (Gasto Total):**
- Paula contributed \( \frac{2}{5} \) of 80€, which is \( 32€ \).
- Sara contributed the remaining \( 80€ - 32€ = 48€ \).

#### 2. **Proportional Prize Distribution (Ingreso Total):**
- The total prize is 48,000€.
- Paula gets \( \frac{2}{5} \) of the prize, which is:
  \[
  \frac{2}{5} \times 48,000 = 19,200€
  \]
- Sara gets the remaining amount:
  \[
  48,000 - 19,200 = 28,800€
  \]

#### 3. **Re-division of Sara’s Winnings:**
Sara splits her winnings as follows:
- For the family: 3 tenths (\( \frac{3}{10} \)).
- For friends: 1 quarter (\( \frac{1}{4} \)).
- The rest she keeps for herself.

We need to compute these values for Sara:
- \( \frac{3}{10} \times 28,800 = 8,640€ \) for her family.
- \( \frac{1}{4} \times 28,800 = 7,200€ \) for friends.
- The remaining \( 28,800 - 8,640 - 7,200 = 12,960€ \) for herself.

#### 4. **Re-division of Paula’s Winnings:**
Paula splits her winnings as follows:
- For the family: 2 sixths (\( \frac{2}{6} = \frac{1}{3} \)).
- For friends: 1 fifth (\( \frac{1}{5} \)).
- The rest she keeps for herself.

We need to compute these values for Paula:
- \( \frac{1}{3} \times 19,200 = 6,400€ \) for her family.
- \( \frac{1}{5} \times 19,200 = 3,840€ \) for friends.
- The remaining \( 19,200 - 6,400 - 3,840 = 8,960€ \) for herself.

#### Answers:
1. **What fraction of the prize did Sara get?**  
   Sara received \( \frac{28,800}{48,000} = \frac{3}{5} \) of the total prize.
  
2. **How much money did Paula get?**  
   Paula received 19,200€.

3. **What fraction did Paula keep for herself?**  
   Paula kept \( \frac{8,960}{19,200} = \frac{7}{15} \).

4. **What fraction did Sara keep for herself?**  
   Sara kept \( \frac{12,960}{28,800} = \frac{9}{20} \).

Would you like more details or explanations on any of these points? Here are a few related questions:

1. How would the results change if Paula had contributed half the total cost?
2. What is the combined amount that both gave to their families?
3. Who gave a larger fraction of their winnings to friends?
4. What is the total amount Paula and Sara kept for themselves?
5. How much money did Paula’s friends receive?

**Tip:** Always double-check fractional distributions when dealing with proportional problems to avoid calculation errors.

Paula and Sara buy a lottery ticket for Christmas, spending a total of 80€. Paula contributed two-fifths of the money, and Sara paid the rest. They won a prize of 48,000€, which was distributed proportionally based on their initial contribution. What fraction of the prize did Sara get? How much did Paula receive? How much did each give to their family and friends, and how much did they keep for themselves?

Learn how to solve a problem involving proportional division of a lottery prize. Paula and Sara spent 80€, and won 48,000€, which was divided based on their initial contribution. Find out how much each received, how much they gave to family and friends, and what fraction they kept.

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