Marco is making mosaic garden stones using red, yellow and blue tiles. he has 45 red tiles, 90 blue tiles and 75 yellow tiles. each stone must have the same number of each color tile. what is the greatest number of stones marco can make? how many of each color tile will marco use in each stone? how can marco use the gcf to find out how many tiles he has in all?

What is the greatest number of stones marco can make?

You have to find the greatest number that divides the three kinds of tiles at the same time, because this will permit Marco to use the minimum number of each tile and so he will be able to build the maximum number of stones.

The greatest number that divides a set of numbers is the greatest common factor (GCF).

To find the greatest common factor you have to decompose every number in the prime factors and multiply only the common factors each raised to its lower exponent.

So,

45 = 3^2*5

90 = 2*3^2*5

75 = 3*5^2

The common factors with their minimum exponents are: 3 and 5. Whose multilplication is 15.

Then the maximum number of stones is 15.

How many of each color tile will Marco use in each stone?

Just divide the number of each stone by 15

Red: 45/15 = 3

Blue: 90/15 = 6

Yellow: 75/3 = 25

How can Marco use the gcf to find out how many tiles he has in all?

By multiplying the GCF times the number of tiles in each stone, he will find out how many tiles of each kind he has in all.

The Greatest Common Factor: G C F (45, 90, 75) = 15

The greatest number of stones Marco can make is 15.

45 : 15 = 3

90 : 15 = 6

75 : 15 = 5

He will use: 3 red, 6 blue and 5 yellow tiles in each stone.