For another picnic, you want to make hamburgers with pickles, again without having any left over. You need to balance the number of packages of buns (which usually contain 8 buns) with the number of packages of hamburger patties (which usually contain 12 patties) and the number of jars of pickles (which contain 18 slices). Assume that each hamburger needs three pickle slices.
3,2,4
What is the smallest number of packages of buns, packages of patties, and jars of pickles, respectively?
In Part C, 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 300 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?
39,26,52
For the first part of the question, to balance the number of packages, we need to find the least common multiple (LCM) of 8 (buns), 12 (patties), and 18 (pickles).
LCM(8, 12, 18) = 72
So, the smallest number of packages of buns, patties, and pickles needed is 72 each.
For the second part of the question, if we want to feed at least 300 people while maintaining the same ratio of 3:2:4, we need to multiply the number of packages by the ratio factor.
Ratio factor = 300 / (3 + 2 + 4) = 300 / 9 = 33.33 (approximately)
Now, we multiply the ratio factor with the original number of packages:
Buns = 3 * 33.33 = 99.99 (approximately 100 packages)
Patties = 2 * 33.33 = 66.66 (approximately 67 packages)
Pickles = 4 * 33.33 = 133.33 (approximately 133 jars)
Therefore, the minimum number of packages of buns, patties, and jars of pickles needed to feed at least 300 people with the same ratio is 100 packages of buns, 67 packages of patties, and 133 jars of pickles.
To balance the number of packages of buns, packages of patties, and jars of pickles in Part A, we need to find the smallest common multiple of 8, 12, and 18.
The prime factorization of 8 is 2^3.
The prime factorization of 12 is 2^2 * 3.
The prime factorization of 18 is 2 * 3^2.
To find the smallest common multiple, we take the highest power of each prime factor from the three numbers:
2^3 * 3^2 = 8 * 9 = 72.
Therefore, we need at least 72 packages of buns, 72 packages of patties, and 72 jars of pickles to make hamburgers with pickles without having any left over.
To determine the minimum number of packages of buns, packages of patties, and jars of pickles needed to feed at least 300 people while maintaining the proper ratio, we need to scale up the previous minimum quantities.
The ratio of packages of buns, packages of patties, and jars of pickles is 3:2:4, or 3x:2x:4x where x is the scaling factor.
We need to find the smallest x that satisfies the inequality 4x ≥ 300 since we want to feed at least 300 people.
Dividing both sides of the inequality by 4, we get:
x ≥ 300/4
x ≥ 75.
Therefore, the minimum number of packages of buns, packages of patties, and jars of pickles needed to feed at least 300 people while maintaining the proper ratio is 3x:2x:4x where x is equal to or greater than 75.
Substituting x = 75, we get:
3 * 75 = 225 packages of buns
2 * 75 = 150 packages of patties
4 * 75 = 300 jars of pickles.
So, the minimum number of packages of buns, packages of patties, and jars of pickles needed to feed at least 300 people while maintaining the proper ratio is 225 packages of buns, 150 packages of patties, and 300 jars of pickles.
To determine the smallest number of packages of buns, packages of patties, and jars of pickles, respectively, we need to find the least common multiple (LCM) of the given ratios.
The ratio we are given is 3:2:4, which means that for every 3 packages of buns, we need 2 packages of patties and 4 jars of pickles.
To find the LCM, we can use the product of the highest powers of each prime factor present in the given ratios.
First, let's find the prime factors of each number:
- 3: prime factor is 3.
- 2: prime factor is 2.
- 4: prime factors are 2 and 2.
Now, we take the highest powers of each prime factor:
- 3: power of 3 is 1.
- 2: power of 2 is 1.
- 4: power of 2 is 2.
Next, we calculate the LCM by multiplying the prime factors raised to their highest powers:
LCM = (3^1) * (2^1) * (2^2) = 3 * 2 * 4 = 24.
Therefore, the smallest number of packages of buns, packages of patties, and jars of pickles needed is 24.
Moving on to the second part of the question:
If you want to feed at least 300 people and maintain the proper ratio of 3:2:4, we should scale up the previous solution.
The ratio 3:2:4 means that for every 3 packages of buns, we need 2 packages of patties and 4 jars of pickles. We need to find the LCM of 3, 2, and 4, and then multiply it by a factor to reach a number that can feed at least 300 people.
To find the factor, divide 300 by the LCM:
Factor = 300 / 24 ≈ 12.5
Since we can't have half of a package or jar, we need to round this up to the nearest whole number, which is 13.
Finally, we multiply the LCM by the factor to get the minimum number of packages of buns, packages of patties, and jars of pickles required:
Number of packages of buns = LCM * Factor = 24 * 13 = 312
Number of packages of patties = 2 * 13 = 26
Number of jars of pickles = 4 * 13 = 52
Therefore, you would need a minimum of 312 packages of buns, 26 packages of patties, and 52 jars of pickles to feed at least 300 people while maintaining the proper ratio.