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Let's tackle the questions step by step.

### Question 27:
This is based on the given triangle diagram. Let's summarize the given dataLet's tackle the questions step by step.

### Question 27:
This is based on the given triangle diagram. Let's summarize the given 
- \( CD \parallel EF \) and \( BC \parallel DE \)
- \( AD = 10 \) cm, \( AF = 4 \) cm.

We are asked to find:

#### (a) \( FD \)
Since \( CD \parallel EF \), triangles \( AFD \) and \( CEF \) are similar by AA similarity criterion. The side ratios of similar triangles are proportional.

- Given \( AD = 10 \) cm and \( AF = 4 \) cm, we have:
  \[
  FD = AD - AF = 10 - 4 = 6 \text{ cm}.
  \]
Thus, \( FD = 6 \) cm.

#### (b) \( AE : EC \)
Again, since \( CD \parallel EF \), the corresponding sides of the similar triangles are proportional. Therefore, the ratio of \( AE \) to \( EC \) can be determined using similar triangles \( AFD \) and \( CEF \). The exact length of \( EC \) is not directly given, but the ratio should follow the triangle properties.

#### (c) Length of \( AB \)
To find \( AB \), we would need to apply the full proportionality and similarity rules based on the parallel lines and the geometry of the triangle. We will use triangle similarity, but we need more information about the proportions or side lengths for further calculation.

#### (d) Ratio of areas of triangles \( ADC \) and \( BDC \)
The ratio of the areas of two triangles with the same height is proportional to the base lengths. Since \( AD \) and \( DB \) form the base of triangles \( ADC \) and \( BDC \), the area ratio is:
  \[
  \text{Ratio of areas} = \frac{AD}{DB}.
  \]
But \( DB \) isn't provided, so we'd need that length to give an exact ratio.

### Question 28:
This involves solving a fractional equation problem.

The problem states that a fraction simplifies after subtracting 1 from the numerator to become \( \frac{1}{2} \), and when 7 is added to the denominator, it simplifies to \( \frac{1}{3} \).

Let the fraction be \( \frac{x}{y} \).

#### (a) Forming equations:
From the given conditions:
1. If 1 is subtracted from the numerator, the fraction becomes \( \frac{1}{2} \):
   \[
   \frac{x - 1}{y} = \frac{1}{2}.
   \]
   This gives the equation:
   \[
   2(x - 1) = y \quad \text{or} \quad y = 2x - 2.
   \]

2. If 7 is added to the denominator, the fraction becomes \( \frac{1}{3} \):
   \[
   \frac{x}{y + 7} = \frac{1}{3}.
   \]
   This gives the equation:
   \[
   3x = y + 7 \quad \text{or} \quad y = 3x - 7.
   \]

#### (b) Solving these equations:
Now, solve the system of equations:
1. \( y = 2x - 2 \),
2. \( y = 3x - 7 \).

Equating both expressions for \( y \):
\[
2x - 2 = 3x - 7.
\]
Solving for \( x \):
\[
-2 + 7 = x \quad \text{or} \quad x = 5.
\]

Substitute \( x = 5 \) into either equation for \( y \), e.g., \( y = 2x - 2 \):
\[
y = 2(5) - 2 = 10 - 2 = 8.
\]

Thus, the fraction is \( \frac{5}{8} \).

#### (c) The fraction is \( \frac{5}{8} \).

---

Would you like further details on any part? Here are five questions related to the concepts discussed:

1. How do we prove the similarity of triangles using the AA criterion?
2. How can we find the lengths of sides in similar triangles?
3. How do the ratios of areas of triangles relate to their sides?
4. Can you explore other methods to solve a system of linear equations?
5. How are proportions used to solve geometry problems involving parallel lines?

**Tip:** In similar triangles, corresponding sides are always proportional, and this can simplify the process of finding unknown lengths.

27. In the figure, CD is parallel to EF and BC is parallel to DE. Given: AD = 10 cm, AF = 4 cm. Find FD, the ratio AE:EC, the length of AB, and the ratio of the areas of triangles ADC and BDC. 28. A fraction becomes 1/2 after subtracting 1 from the numerator and 1/3 after adding 7 to the denominator. Solve for the original fraction.

Solve geometric and algebraic problems including finding side lengths and ratios in triangles using parallel lines, as well as determining a fraction based on given conditions. Suitable for students in Grades 8-10.

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