A rectangular courtyard is 3m 78 cm long and 5m 25 cm broad. It is desired to prove it with square tiles of the same size. What is the largest size of the tile that can be used ? Also find the number of such tiles.

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Convert the dimensions to the same unit
The dimensions of the courtyard are given as:
- Length = 3 m 78 cm
- Breadth = 5 m 25 cm

First, we convert these measurements to centimeters:
- Length = 3 m × 100 cm/m + 78 cm = 300 cm + 78 cm = 378 cm
- Breadth = 5 m × 100 cm/m + 25 cm = 500 cm + 25 cm = 525 cm

### Step 2: Find the HCF (Highest Common Factor)
Next, we need to find the HCF of the two dimensions (378 cm and 525 cm) to determine the largest size of the square tile that can be used.

**Factorization of 378:**
- 378 = 2 × 3 × 3 × 3 × 7 = 2 × 3^3 × 7

**Factorization of 525:**
- 525 = 3 × 5 × 5 × 7 = 3 × 5^2 × 7

**Finding the common factors:**
The common factors are:
- 3 (from both)
- 7 (from both)

Thus, the HCF = 3 × 7 = 21 cm.

### Step 3: Calculate the area of the rectangular courtyard
Now, we calculate the area of the rectangular courtyard using the formula:
Area = Length × Breadth

Area = 378 cm × 525 cm

Calculating this gives:
Area = 198450 cm².

### Step 4: Calculate the area of one square tile
The area of one square tile with a side of 21 cm is:
Area of one tile = side × side = 21 cm × 21 cm = 441 cm².

### Step 5: Calculate the number of tiles
To find the number of tiles that can fit in the courtyard, we divide the area of the courtyard by the area of one tile:

Number of tiles = Area of courtyard / Area of one tile
Number of tiles = 198450 cm² / 441 cm².

Calculating this gives:
Number of tiles = 450.

### Final Answer:
The largest size of the tile that can be used is **21 cm**, and the number of such tiles is **450**.

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