Diana is making packages of mixed desserts with all the leftover desserts from her bake sale. She has the following deserts: 45 brownies 18 donuts 90 cookies
Diana must put the same number of items in each box. What is the greatest number of boxes Diana can make without having any desserts left over?
How many of each dessert will be in each box?
9 is the largest common factor for the greatest number of boxes. Work it from there.
Well, let's do some dessert math! To find the greatest number of boxes Diana can make without any leftovers, we need to find the greatest common factor of 45, 18, and 90.
Now, the GCF is like the superhero of numbers, swooping in to save the day! It helps us split things up equally.
So, the GCF of 45, 18, and 90 is... drumroll, please... 9!
That means Diana can make 9 boxes. But how many of each dessert will be in each box? To find out, we divide the total number of each dessert by the number of boxes. So in each box, she can put:
45/9 brownies = 5 brownies
18/9 donuts = 2 donuts
90/9 cookies = 10 cookies
So, Diana can make 9 boxes, and each box can have 5 brownies, 2 donuts, and 10 cookies. Enjoy the dessert mix and try not to eat them all at once!
To find the greatest number of boxes Diana can make without having any desserts left over, we need to find the greatest common divisor (GCD) of the given quantities of desserts.
45, 18, and 90 can all be divided evenly by 9, so the GCD is 9.
Therefore, Diana can make the greatest number of boxes, with 9 desserts in each box. To find out how many of each dessert will be in each box, we divide the quantity of each dessert by the GCD:
45 brownies ÷ 9 = 5 brownies per box
18 donuts ÷ 9 = 2 donuts per box
90 cookies ÷ 9 = 10 cookies per box
So, Diana can make the greatest number of boxes filling each box with 5 brownies, 2 donuts, and 10 cookies.
To figure out the greatest number of boxes Diana can make without having any desserts left over, we need to find the greatest common divisor (GCD) of the numbers 45, 18, and 90.
One quick way to find the GCD is to list the prime factors of each number and find the highest power of each prime factor that appears in all three numbers.
Let's list the prime factors of each number:
45 = 3 x 3 x 5
18 = 2 x 3 x 3
90 = 2 x 3 x 3 x 5
Now, find the highest power of each prime factor that appears in all three numbers:
- The highest power of 2 is 1 (appears in 18 and 90)
- The highest power of 3 is 2 (appears in 45, 18, and 90)
- The highest power of 5 is 1 (appears in 45 and 90)
Therefore, the GCD of 45, 18, and 90 is 2^1 x 3^2 x 5^1 = 2 x 9 x 5 = 90.
This means that Diana can make the greatest number of boxes without having any desserts left over is 90. Each box will contain the same number of items, which is the GCD, so each box will have:
- 90/45 = 2 brownies
- 90/18 = 5 donuts
- 90/90 = 1 cookie
Therefore, each box will have 2 brownies, 5 donuts, and 1 cookie. Diana can make a total of 90 boxes with no desserts left over.