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Let's solve the problem step by step.

### Given:
- The printed rectangular surface has dimensions: \(100 \, \text{cm} \times 140 \, \text{cm}\).
- There is a uniform gray border of width \(x\) around the edges.
- The area of the gray border is \(\frac{3}{10}\) of the printed area.

### (a) Calculate the total area of the paper.

1. **Area of the printed surface:**
   \[
   A_{\text{printed}} = 100 \, \text{cm} \times 140 \, \text{cm} = 14000 \, \text{cm}^2
   \]

2. **Area of the total paper (including the gray border):**
   The dimensions of the total paper are \((100 + 2x)\) and \((140 + 2x)\), where \(2x\) accounts for the gray border on both sides.
   \[
   A_{\text{total}} = (100 + 2x) \times (140 + 2x)
   \]

3. **Area of the gray border:**
   The area of the gray border is the difference between the total area and the printed area:
   \[
   A_{\text{gray}} = A_{\text{total}} - A_{\text{printed}}
   \]

4. **Relationship between the gray border area and the printed area:**
   Given that the gray area is \(\frac{3}{10}\) of the printed area:
   \[
   A_{\text{gray}} = \frac{3}{10} \times A_{\text{printed}} = \frac{3}{10} \times 14000 = 4200 \, \text{cm}^2
   \]

5. **Set up the equation:**
   \[
   A_{\text{total}} - 14000 = 4200
   \]
   \[
   A_{\text{total}} = 18200 \, \text{cm}^2
   \]

So, the total area of the paper is \(18200 \, \text{cm}^2\).

### (b) What fraction of the total paper does the printed image represent?

The fraction is simply the ratio of the printed area to the total area:
\[
\text{Fraction} = \frac{A_{\text{printed}}}{A_{\text{total}}} = \frac{14000}{18200} = \frac{70}{91} \approx 0.7692
\]

So, the printed image represents approximately \( \frac{70}{91} \) or about 76.92% of the total paper.

---

Would you like a deeper explanation of any step, or do you have further questions?

Here are 5 related questions to explore:

1. How would the dimensions of the border \(x\) change if the area of the border were \(\frac{1}{2}\) of the printed area?
2. How would the total area change if the dimensions of the printed surface were doubled?
3. What would happen to the fraction if the gray border's area was reduced to \(\frac{1}{5}\) of the printed area?
4. What is the perimeter of the total paper including the gray border?
5. If the printed area was a square instead of a rectangle, how would the equation change?

**Tip:** Always check units and dimensions when calculating areas and perimeters to avoid errors.

Un cartel tiene una superficie rectangular impresa de 100 cm por 140 cm y una franja gris de ancho uniforme x alrededor de los bordes cuya área equivale a las tres décimas partes del área impresa. (a) Calcule el área total del papel. (b) ¿Qué fracción representa la imagen impresa del total del papel?

Learn how to calculate the total area of a poster including a gray border, given that the border's area is three-tenths of the printed area. This problem involves key geometry and algebra concepts, suitable for students in Grades 10-12.

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