Question

what is the minimum number of square marbles required to tile a floor of length 5 m 78 cm and width 3 m 74 cm?

Answer 1

The marbles used to tile the floor are square marbles. Therefore, the length of the marble = width of the marble.
As we have to use whole number of marbles, the side of the square should a factor of both 5 m 78 cm and 3m 74. And it should be the highest factor of 5 m 78 cm and 3m 74.

5m78cm=578cm and 3m74cm=374cm.
The HCF of 578 and 374 = 34.

Hence, the side of the square is 34.

The number of such square marbles required =578×37434×34=17×11= 187 marbles

Answer 2

Answer:

The minimum number of square marbles required to tile the floor is 187.

Step-by-step explanation:

Given the length of a floor, $l = 5 m~ 78 cm = 578cm$

The width of a floor, $b=3 m~ 74 cm=374cm$

To find the minimum number of square marbles required to tile a floor,  we need to find first minimum size of the square.

This is given by the greatest common divisor of the two numbers.

Greatest common divisor(GCD) of two numbers is the greatest possible number that divides each number.

So, GCD of 578 and 374 is

$374)~578~(1\\ ~~~~~~~~\underline{374}\\~~~~~~~~204)~374~(1\\ ~~~~~~~~~~~~~~~\underline{204}\\~~~~~~~~~~~~~~~170)~204~(1\\ ~~~~~~~~~~~~~~~~~~~~~~\underline{170}\\~~~~~~~~~~~~~~~~~~~~~~~~{34})~170~(5\\ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\underline{170}\\~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~{0}$

So, GCD of 578 and 374 is 34.

So, side of the square is 34.

Area of the square marble is

$\mbox{side}\times \mbox{side}=34\times34 ~\mbox{sq.cm}$

Area of the given rectangular floor is

$\mbox{length}\times \mbox{breadth}=578\times374 ~\mbox{sq.cm}$

Then, the minimum number of square marbles required to tile a floor is

$\dfrac{\mbox{Area of floor}}{\mbox{Area of square}} =\dfrac{578\times374}{34\times34}$

                       $= {17\times11}=187$

Therefore, the minimum number of square marbles required to tile the floor is 187.

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