+25 Rosencrantz and Guildenstern mix red, blue, and white paint in the following ratio:$$\text{red}:\text{blue}:\text{white}=3:2:1.$$They have $48$ cans of red paint, $40$ cans of blue paint, and $15$ cans of white paint. The cans are all the same size. What is the greatest number of cans of mixed paint they can make with the same ratio of colors, and why? How much paint of each color will be left over?
We can find the greatest number of cans of mixed paint they can make by finding the greatest common factor of the number of cans of each color. The greatest common factor of 48, 40, and 15 is 12. Therefore, they can make 12 cans of mixed paint. They will have 48−12=36 cans of red paint left over, 40−12=28 cans of blue paint left over, and 15−12=3 cans of white paint left over.
3 + 2 + 1 ==6 sum of all ratios
48 + 40 + 15 ==103 cans of all 3 colors.
103 / 6 ==17+1/6 cans of any one color.
But we have maximum of 15 cans of white paint, therefore:
15 x 6 ==90 cans be mixed up.
15 cans of red paint ==15 x 3 ==45 cans of red paint
15 cans of blue paint==15 x 2==30 cans of blue paint
15 cans of white paint ==15 x 1==15 cans of white paint.
So, we mix: 45 red + 30 blue + 15 white ==90 cans in total
48 - 45 ==3 cans of red paint left over
40 - 30 ==10 cans of blue paint left over
15 - 15 ==0 cans of white paint left over.
Rosencrantz and Guildenstern can make the greatest number of cans of mixed paint if they use all of the red paint, because it is the most. They can use 48 cans of red paint, which is 12 groups of 4 cans.
To make the same ratio of colors, they need to use 8 groups of 4 cans of blue paint and 3 groups of 4 cans of white paint. This will leave them with 20 cans of blue paint and 12 cans of white paint.
Therefore, the greatest number of cans of mixed paint they can make is 12.
Since the ratio of red paint to blue paint to white paint is 3:2:1, the greatest number of cans of mixed paint Rosencrantz and Guildenstern can make is the least common multiple of 3, 2, and 15, or 30 cans. To make the greatest number of cans, Rosencrantz and Guildenstern will use 30 cans of red paint, 20 cans of blue paint, and 10 cans of white paint. The amount of red, blue, and white paint left over is 48−30=18 cans, 40−20=20 cans, and 15−10=5 cans, respectively. Therefore, Rosencrantz and Guildenstern will have 18 cans of red paint, 20 cans of blue paint, and 5 cans of white paint left over.
To see why 30 is the greatest number of cans of mixed paint Rosencrantz and Guildenstern can make, notice that if they make 29 cans of mixed paint, they will have 1 can of red paint left over, and if they make 31 cans of mixed paint, they will have 2 cans of white paint left over.
