Question

The least number of square tiles required to pave the floor of a room 15 m 91 cm and 9 m 46 cm broad is 

• A. 814
• B. 820
• C. 840
• D. 844

Answer 1

The Correct Option is A

To solve the problem of finding the least number of square tiles required to pave the floor of the room, we need to find the greatest square tile size that perfectly covers the given dimensions of the room with minimal wastage.

The dimensions of the room are given as 15 m 91 cm length and 9 m 46 cm breadth. Let's first convert these dimensions into centimeters:

1 meter = 100 centimeters, so:

• Length = \(15 \times 100 + 91 = 1591\) cm
• Breadth = \(9 \times 100 + 46 = 946\) cm

To find the largest square tile that can be used, we need to calculate the Greatest Common Divisor (GCD) of 1591 and 946.

Using the Euclidean algorithm for GCD:

• 1591 divided by 946 gives a quotient of 1 and a remainder of 645. \(\quad [1591 \mod 946 = 645]\)
• 946 divided by 645 gives a quotient of 1 and a remainder of 301. \(\quad [946 \mod 645 = 301]\)
• 645 divided by 301 gives a quotient of 2 and a remainder of 43. \(\quad [645 \mod 301 = 43]\)
• 301 divided by 43 gives a quotient of 7 and a remainder of 0. \(\quad [301 \mod 43 = 0]\)

Therefore, the GCD of 1591 and 946 is 43 cm. This means the largest possible square tile that will fit perfectly is a 43 cm × 43 cm tile.

Next, we calculate the number of such tiles needed:

• Number of tiles along the length = \( \frac{1591}{43} = 37 \) tiles
• Number of tiles along the breadth = \( \frac{946}{43} = 22 \) tiles

Total number of tiles = \(37 \times 22 = 814\)

Thus, the least number of square tiles required to cover the floor is 814.