A hall-room 39 m 10 cm long and 35 m 70 cm broad is to be paved with equal square tiles. Find the largest tile so that the tiles exactly fit and also find the number of tiles required.

Answer

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

### Step 1: Convert the dimensions of the hall-room into meters.
- Length (L) = 39 m 10 cm
- Breadth (B) = 35 m 70 cm

**Conversion:**
- 10 cm = 10/100 m = 0.1 m
- 70 cm = 70/100 m = 0.7 m

**Calculating the dimensions:**
- Length (L) = 39 m + 0.1 m = 39.1 m
- Breadth (B) = 35 m + 0.7 m = 35.7 m

### Step 2: Calculate the area of the hall-room.
- Area = Length × Breadth
- Area = 39.1 m × 35.7 m

**Calculating the area:**
- Area = 1395.87 m²

### Step 3: Find the largest square tile that can fit perfectly in the hall-room.
To find the largest square tile, we need to calculate the highest common factor (HCF) of the length and breadth.

**Convert the dimensions to centimeters for easier calculation:**
- Length = 39.1 m = 3910 cm
- Breadth = 35.7 m = 3570 cm

**Finding the HCF:**
- Factorizing 3910 and 3570:
- 3910 = 2 × 5 × 7 × 11
- 3570 = 2 × 3 × 5 × 7 × 17

**Finding the common factors:**
- Common factors = 2 × 5 × 7 = 70

Thus, the largest tile that can fit perfectly is:
- Side length of the tile = 70 cm = 0.7 m

### Step 4: Calculate the area of one square tile.
- Area of one tile = Side × Side
- Area of one tile = 0.7 m × 0.7 m = 0.49 m²

### Step 5: Calculate the number of tiles required.
- Number of tiles = Total area of the room / Area of one tile
- Number of tiles = 1395.87 m² / 0.49 m²

**Calculating the number of tiles:**
- Number of tiles ≈ 2846.5

Since we cannot have a fraction of a tile, we round up to the nearest whole number:
- Number of tiles = 2847

### Final Answers:
- The largest tile that can be used is 0.7 m × 0.7 m.
- The number of tiles required is 2847.

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