In a triangle ABC, there are three point P, Q & R on side BC, such that `BP:PQ:QR:RC=1:2:3:4` . If G is the centroid, then find the ratio of area of `Delta PGR` to area `Delta ABC`.

Answer

To solve the problem, we need to find the ratio of the area of triangle PGR to the area of triangle ABC given the ratio of segments on side BC.

### Step-by-Step Solution:

1. **Identify the segments on BC**:
Given the ratio \( BP : PQ : QR : RC = 1 : 2 : 3 : 4 \), we can assign lengths to these segments. Let:
- \( BP = 1x \)
- \( PQ = 2x \)
- \( QR = 3x \)
- \( RC = 4x \)

Therefore, the total length of \( BC \) is:
\[ BC = BP + PQ + QR + RC = 1x + 2x + 3x + 4x = 10x \]

2. **Locate points P, Q, and R**:
- Point \( P \) divides \( BC \) into \( BP \) and \( PQ \).
- Point \( Q \) divides \( PQ \) and \( QR \).
- Point \( R \) divides \( QR \) and \( RC \).

3. **Find the coordinates of points P, Q, and R**:
Assuming \( B \) is at \( (0, 0) \) and \( C \) is at \( (10x, 0) \):
- \( P \) is at \( (1x, 0) \)
- \( Q \) is at \( (3x, 0) \)
- \( R \) is at \( (6x, 0) \)

4. **Determine the centroid G of triangle ABC**:
The centroid \( G \) of triangle \( ABC \) can be calculated using the formula:
\[ G = \left( \frac{x_A + x_B + x_C}{3}, \frac{y_A + y_B + y_C}{3} \right) \]
Let \( A \) be at \( (a, h) \), then:
\[ G = \left( \frac{a + 0 + 10x}{3}, \frac{h + 0 + 0}{3} \right) = \left( \frac{a + 10x}{3}, \frac{h}{3} \right) \]

5. **Calculate the areas of triangles**:
The area of triangle \( ABC \) can be calculated as:
\[ \text{Area}_{ABC} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 10x \times h = 5xh \]

The area of triangle \( PGR \) can be calculated using the base \( PR \) and the height from \( G \) to line \( PR \):
- The length \( PR = QR - PQ = 6x - 1x = 5x \)
- The height from \( G \) to line \( PR \) is the same as the height from \( A \) to line \( BC \) since \( G \) is vertically aligned with \( A \).

Therefore, the area of triangle \( PGR \) is:
\[ \text{Area}_{PGR} = \frac{1}{2} \times PR \times \text{height from G} = \frac{1}{2} \times 5x \times \frac{h}{3} = \frac{5xh}{6} \]

6. **Find the ratio of the areas**:
Now, we can find the ratio of the areas:
\[ \text{Ratio} = \frac{\text{Area}_{PGR}}{\text{Area}_{ABC}} = \frac{\frac{5xh}{6}}{5xh} = \frac{1}{6} \]

### Final Answer:
The ratio of the area of triangle PGR to the area of triangle ABC is \( \frac{1}{6} \).