A certain type of wooden board is sold only in lengths of multiples of 25 cm from 2 to 10 metres . A carpenter needs a large quantity of this type of board in `1.65` m length , For the minimum waste , the length to be purchased should be

Options

A. `3.30` m
B. `6.60` m
C. `8.25` m
D. `9.95` m

Answer

To solve the problem step by step, we need to determine the minimum length of wooden board that the carpenter should purchase, given that the board is sold in lengths of multiples of 25 cm and the carpenter needs a length of 1.65 m.

### Step 1: Convert the required length from meters to centimeters.
1. The required length is given as 1.65 m.
2. We know that 1 meter = 100 centimeters.
3. Therefore, to convert 1.65 m to centimeters:
\[ 1.65 \text{ m} \times 100 \text{ cm/m} = 165 \text{ cm} \]

### Step 2: Identify the lengths available for purchase.
1. The board is sold only in lengths that are multiples of 25 cm.
2. The possible lengths from 2 m to 10 m in centimeters are:
- 200 cm (2 m)
- 225 cm (2.25 m)
- 250 cm (2.5 m)
- 275 cm (2.75 m)
- 300 cm (3 m)
- 325 cm (3.25 m)
- 350 cm (3.5 m)
- 375 cm (3.75 m)
- 400 cm (4 m)
- 425 cm (4.25 m)
- 450 cm (4.5 m)
- 475 cm (4.75 m)
- 500 cm (5 m)
- 525 cm (5.25 m)
- 550 cm (5.5 m)
- 575 cm (5.75 m)
- 600 cm (6 m)
- 625 cm (6.25 m)
- 650 cm (6.5 m)
- 675 cm (6.75 m)
- 700 cm (7 m)
- 725 cm (7.25 m)
- 750 cm (7.5 m)
- 775 cm (7.75 m)
- 800 cm (8 m)
- 825 cm (8.25 m)
- 850 cm (8.5 m)
- 875 cm (8.75 m)
- 900 cm (9 m)
- 925 cm (9.25 m)
- 950 cm (9.5 m)
- 975 cm (9.75 m)
- 1000 cm (10 m)

### Step 3: Find the least common multiple (LCM) of the required length and the board length.
1. We need to find the LCM of 25 cm and 165 cm.
2. First, we factor both numbers:
- \( 25 = 5^2 \)
- \( 165 = 3 \times 5 \times 11 \)
3. The LCM is found by taking the highest power of each prime factor:
- For 5, the highest power is \( 5^2 \).
- For 3, the highest power is \( 3^1 \).
- For 11, the highest power is \( 11^1 \).
4. Therefore, the LCM is:
\[ LCM = 5^2 \times 3^1 \times 11^1 = 25 \times 3 \times 11 = 825 \text{ cm} \]

### Step 4: Convert the LCM back to meters.
1. Now, we convert 825 cm back to meters:
\[ 825 \text{ cm} \div 100 = 8.25 \text{ m} \]

### Conclusion
The minimum length of wooden board that the carpenter should purchase is **8.25 m**.

---