Earlier you determined that the proper ratio of packages of buns, packages of patties, and jars of pickles is 3:2:4. If you want to feed at least 540 people, but also maintain the proper ratio, what minimum number of packages of buns, packages of patties, and jars of pickles do you need, respectively? (
(In each package there are 8 buns, 12 patties, and 18 slices of pickles in each jar (with 3 pickles put on each sandwich)
Please show work if you can!!!! Thank you
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To determine the minimum number of packages of buns, packages of patties, and jars of pickles needed, we need to find the least common multiple (LCM) of the ratios 3:2:4, and then divide that by the respective quantities in each package.
The LCM of 3, 2, and 4 is 12.
To find the minimum number of packages of buns, divide the LCM (12) by the number of buns per package (8):
12 buns ÷ 8 buns/package = 1.5 packages of buns
However, since we cannot have a fraction of a package, we will need to round up to the nearest whole number. So, we need at least 2 packages of buns.
To find the minimum number of packages of patties, divide the LCM (12) by the number of patties per package (12):
12 patties ÷ 12 patties/package = 1 package of patties
Therefore, we need at least 1 package of patties.
To find the minimum number of jars of pickles, divide the LCM (12) by the number of pickles per jar (18):
12 pickles ÷ 18 pickles/jar = 0.67 jars of pickles
Again, since we cannot have a fraction of a jar, we will need to round up to the nearest whole number. So, we need at least 1 jar of pickles.
Therefore, to feed at least 540 people while maintaining the proper ratio, we need a minimum of 2 packages of buns, 1 package of patties, and 1 jar of pickles.
To determine the minimum number of packages of buns, packages of patties, and jars of pickles needed to feed at least 540 people while maintaining the proper ratio, we need to follow these steps:
Step 1: Determine the number of packages needed for each ingredient by finding the least common multiple (LCM) of the ratios.
The given ratio is 3:2:4, which means for every 3 packages of buns, we need 2 packages of patties and 4 jars of pickles. To find the LCM of 3, 2, and 4, we can list the multiples of each number until we find a common multiple:
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27...
Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20...
Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36...
From the list, we can see that 12 is the smallest common multiple. So, for every 12 packages of buns, we need 8 packages of patties and 12 jars of pickles.
Step 2: Calculate the total number of people each set of ingredients can feed.
Each package of buns contains 8 buns, which can make 8 sandwiches.
Each package of patties contains 12 patties, which can make 12 sandwiches.
Each jar of pickles contains 18 slices, which can make 6 sandwiches (3 pickles per sandwich).
So, for every 12 packages of buns, we can make 8 x 12 = 96 sandwiches.
For every 8 packages of patties, we can make 12 x 8 = 96 sandwiches.
For every 12 jars of pickles, we can make 6 x 12 = 72 sandwiches.
Step 3: Determine the number of sets required to feed the desired number of people.
To feed at least 540 people, we need to divide the desired number of people by the number of people each set can feed:
Number of sets required = Desired number of people / Number of people each set can feed
For each set, we can make 96 sandwiches. Therefore:
Number of sets required for buns = 540 / 96 = 5.625 (approximately)
Number of sets required for patties = 540 / 96 = 5.625 (approximately)
Number of sets required for pickles = 540 / 72 = 7.5 (approximately)
Since we can't have fractional sets, we round up each value:
Number of sets required for buns = 6
Number of sets required for patties = 6
Number of sets required for pickles = 8
Step 4: Calculate the minimum number of packages needed by multiplying the number of sets by the number of packages per set.
Minimum number of packages of buns = 6 sets × 12 packages per set = 72 packages
Minimum number of packages of patties = 6 sets × 8 packages per set = 48 packages
Minimum number of jars of pickles = 8 sets × 12 jars per set = 96 jars
Therefore, the minimum number of packages of buns, packages of patties, and jars of pickles needed, respectively, are 72, 48, and 96.