Question

for this picnic, you need to feed at least 700 people. if you want to feed everyone, but still have equal numbers of buns and hot dogs, what is the minimum packages of buns and hot dogs you need, respectively?
express your answer as two integers separated by a comma.
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part c

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 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?
express your answer as three integers separated by commas.
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Explanation:

Step1: Define missing picnic 1 details

Assume standard packaging: buns come 8 per package, hot dogs 10 per package (common for such problems, as the original question omitted this critical info). Let $b$ = bun packages, $h$ = hot dog packages. We need $8b = 10h$ (equal counts) and $8b \geq 700$.

Step2: Simplify the equality

Simplify $8b=10h$ to $4b=5h$. This means $b$ must be a multiple of 5, $h$ a multiple of 4. Let $b=5k$, $h=4k$.

Step3: Solve for minimum $k$

Substitute into the inequality: $8(5k) \geq 700$
$40k \geq 700$
$k \geq \frac{700}{40} = 17.5$
Since $k$ is integer, $k=18$.

Step4: Calculate bun and hot dog packages

$b=5(18)=90$, $h=4(18)=72$

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Step5: Define picnic 2 variables

Let $B$ = bun packages (8 buns each), $P$ = patty packages (12 patties each), $K$ = pickle jars (18 slices each). Each hamburger uses 1 bun, 1 patty, 3 slices. So total buns = total patties, total slices = 3×total buns.
$8B = 12P$ and $18K = 3(8B)$

Step6: Simplify bun-patty equality

$8B=12P$ simplifies to $2B=3P$. So $B$ is multiple of 3, $P$ multiple of 2. Let $B=3m$, $P=2m$.

Step7: Simplify pickle-bun equality

$18K=24B$ simplifies to $3K=4B$. Substitute $B=3m$:
$3K=4(3m) \implies K=4m$

Step8: Find smallest $m$

The smallest positive integer $m=1$. Calculate values:
$B=3(1)=3$, $P=2(1)=2$, $K=4(1)=4$

Answer:

First picnic (buns, hot dogs): 90, 72
Second picnic (buns, patties, pickles): 3, 2, 4