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.
What is the smallest number of packages of buns, packages of patties, and jars of pickles, respectively?
The answer above is correct if each burger only needed 1 pickle, but in the question it says it needs 3 each so you have to triple the pickle number.
To determine the smallest number of packages of buns, packages of patties, and jars of pickles needed, we need to find the least common multiple (LCM) of the quantities of each item per package.
The number of buns per package is 8.
The number of patties per package is 12.
The number of pickle slices per jar is 18, and each hamburger needs 3 pickle slices.
To find the LCM, we can follow these steps:
Step 1: Find the LCM of 8 and 12.
- The prime factors of 8 are 2 * 2 * 2.
- The prime factors of 12 are 2 * 2 * 3.
- Taking the maximum count of each prime factor gives us 2 * 2 * 2 * 3 = 24.
- So, the LCM of 8 and 12 is 24.
Step 2: Find the LCM of 24 (from step 1) and 18.
- The prime factors of 24 are 2 * 2 * 2 * 3.
- The prime factors of 18 are 2 * 3 * 3.
- Taking the maximum count of each prime factor gives us 2 * 2 * 2 * 3 * 3 = 72.
- So, the LCM of 24 and 18 is 72.
Therefore, the smallest number of packages of buns, packages of patties, and jars of pickles needed to avoid having any leftovers is 72.
To determine the smallest number of packages of buns, packages of patties, and jars of pickles needed, we need to find the least common multiple (LCM) of the number of buns, patties, and pickle slices required.
Given:
- Each package of buns contains 8 buns.
- Each package of patties contains 12 patties.
- Each jar of pickles contains 18 slices.
We need to find the LCM of 8 (buns), 12 (patties), and 3 (pickles per hamburger).
Step 1: Find the LCM of 8 and 12.
The multiples of 8 are: 8, 16, 24, 32, 40, 48, 56, 64, ...
The multiples of 12 are: 12, 24, 36, 48, 60, 72, ...
We can see that 24 is the smallest common multiple of both numbers.
Step 2: Find the LCM of 24 and 3.
The multiples of 24 are: 24, 48, 72, 96, ...
The multiples of 3 are: 3, 6, 9, 12, 15, 18, 21, 24, ...
We can see that 24 is the smallest common multiple of both numbers.
Therefore, the smallest number of packages of buns, packages of patties, and jars of pickles required would be:
- 24 packages of buns (24 x 8 = 192 buns)
- 24 packages of patties (24 x 12 = 288 patties)
- 24 jars of pickles (24 x 18 = 432 slices)
Note: With this combination, you would have exactly the right amount of each ingredient without any leftovers.
Buns......meat.......pickle
..8/pk....12/pk......18/pk
..4 hamb..12 hamb....6 hamb
The least common denominator for these three numbers is 24. Try it.
pk buns = 24/4 = 6 pk
pk patties = 24/12 = 2 pk
pk pickles = 24/6 = 4 pk
See if that works.
buns: 6 pk x 8 buns/pk x 1 hamb/2 buns = 24 hamburgers.
patties: 2 pk x 12 patt/pk x 1 hamb/patt = 24 hamburgers.
pickles: 4 pk x 18 pick/pk x 1 hamb/3 pick = 24 hamburgers.
Check my work.