A square room is covered by a number of whole rectangular slabs of sides 60cm and 42cm. Calculate the least possible area of the room in square meters.
Answer
To calculate the least possible area of the room in square meters, we need to find the greatest common divisor (GCD) of the sides of the rectangular slabs.
The GCD of 60 cm and 42 cm can be found by using the prime factorization method.
Prime factorization of 60: 2^2 * 3 * 5
Prime factorization of 42: 2 * 3 * 7
To find the GCD, we take the product of the common prime factors raised to the lowest power:
GCD = 2 * 3 = 6
So, the GCD of the sides of the rectangular slabs is 6 cm.
To convert the area from cm^2 to m^2, we divide by 10000 (1 m = 100 cm, so 1 m^2 = 10000 cm^2).
Now, we can calculate the least possible area of the room by dividing the area of each rectangular slab by the GCD and summing them up:
Area of the room = (60 cm / 6) * (42 cm / 6) * (number of slabs) / 10000
To find the least possible area, we need to find the minimum number of slabs that can cover the entire room. Since the room is a square, the side length of the room should be a multiple of the GCD.
Let's assume the side length of the room is x meters.
x meters = 6 cm * n (where n is an integer)
Since the length and width of the room are the same, the area of the room = x^2 = (6 cm * n)^2
To find the least possible value of n, we need to find the smallest integer multiple of n that can accommodate the entire room.
60 cm = 6 cm * 10 (n = 10)
42 cm = 6 cm * 7 (n = 7)
So, the least possible number of slabs would be 10 * 10 = 100 slabs.
Area of the room = (60 cm / 6) * (42 cm / 6) * 100 slabs / 10000
= 10 * 7 * 100 / 10000
= 70 / 100
= 0.7 square meters
Therefore, the least possible area of the room is 0.7 square meters.
To calculate the least possible area of the room in square meters, we need to determine the dimensions of the room that would allow for the most efficient use of the rectangular slabs.
Since the room is covered by whole rectangular slabs of sides 60cm and 42cm, we can assume that the room dimensions are multiples of these slab dimensions.
To find the least possible area, we need to find the least common multiple (LCM) of 60 and 42. The LCM is the smallest number that is divisible by both 60 and 42 without leaving a remainder.
To find the LCM, we can use the prime factorization method:
Prime factorization of 60: 2^2 * 3 * 5
Prime factorization of 42: 2 * 3 * 7
To find the LCM, we take the highest power of each prime factor:
LCM = 2^2 * 3 * 5 * 7 = 420
Therefore, the LCM of 60 and 42 is 420.
Now we can calculate the area of the room. Since the room dimensions are multiples of the rectangular slabs, we can divide 420 by the common factors of 60 and 42, which are 2 and 3:
Area of the room = (420 / 2) * (420 / 3) = 210 * 140 = 29400 square centimeters
To convert this to square meters, we divide by 10000 (since there are 10000 square centimeters in a square meter):
Area of the room = 29400 / 10000 = 2.94 square meters
Therefore, the least possible area of the room is 2.94 square meters.
0.6 m * 0.4 m = 0.24 ^2 /slab
how any slabs?
n*.6 in length
m*.4 in width
6*n = 4*m min common multiple
12
6*2 and 4*3
so
2*.6 and 3*.4
1.2 * 1.2 =1.44