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Difference between revisions of "2021 WSMO Speed Round Problems/Problem 6"

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==Problem==
 
==Problem==
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A bag weighs 1 pound and can hold 16 pounds of food at maximum. Danny buys 100 packages of tomatoes and 300 packages of potatoes. Tomatoes come in packages that are <math>12</math> ounces each and potatoes come in packages that are <math>24</math> ounces each. If all of Danny's food must go in bags, how many pounds does Danny's total luggage weigh, including the bags? (Danny will use only as many bags as he needs).
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A bag weighs 1 pound and can hold 16 pounds of food at maximum. Danny buys 100 packages of tomatoes and 300 packages of potatoes. Tomatoes come in packages that are <math>12</math> ounces each and potatoes come in packages that are <math>24</math> ounces each. If all of Danny's food must go in bags, how many pounds does Danny's total luggage weigh, including the bags? (Note that Danny will use only as many bags as he needs and that packages have to stay together).
  
 
==Solution 1==
 
==Solution 1==
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Because bags weigh multiples of <math>12</math> ounces, a bag can hold at most <math>252</math> ounces, meaning that Danny uses at least <math>\frac{8400}{252}=\frac{100}{3}</math> bags, or <math>34</math> bags. A construction is to let the first <math>33</math> bags have <math>9</math> packages of potatoes and <math>3</math> packages of tomatoes, while the last bag has <math>3</math> packages of potatoes and <math>1</math> package of tomatoes.
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Since bags weigh multiples of <math>12</math> ounces, a bag can hold at most <math>252</math> ounces, meaning that Danny uses at least <math>\frac{8400}{252}=\frac{100}{3}</math> bags, or <math>34</math> bags. A construction is to let the first <math>33</math> bags have <math>9</math> packages of potatoes and <math>3</math> packages of tomatoes, while the last bag has <math>3</math> packages of potatoes and <math>1</math> package of tomatoes.
  
 
(We cannot use <math>33</math> bags because this construction is not valid!)
 
(We cannot use <math>33</math> bags because this construction is not valid!)
  
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==Solution 2==
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~pinkpig and asdf334
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Danny's packages weigh a total of <math>\frac{100\cdot12+300\cdot24}{16}=\frac{1200+7200}{16}=\frac{8400}{16}=525</math> pounds. This means that we need <math>\left\lceil\frac{525}{16}\right\rceil=33</math> bags to hold Danny's packages. Thus, the answer is <math>33+525=\boxed{558}</math>
 

Latest revision as of 08:38, 11 July 2022

Problem

A bag weighs 1 pound and can hold 16 pounds of food at maximum. Danny buys 100 packages of tomatoes and 300 packages of potatoes. Tomatoes come in packages that are $12$ ounces each and potatoes come in packages that are $24$ ounces each. If all of Danny's food must go in bags, how many pounds does Danny's total luggage weigh, including the bags? (Note that Danny will use only as many bags as he needs and that packages have to stay together).

Solution 1

Since bags weigh multiples of $12$ ounces, a bag can hold at most $252$ ounces, meaning that Danny uses at least $\frac{8400}{252}=\frac{100}{3}$ bags, or $34$ bags. A construction is to let the first $33$ bags have $9$ packages of potatoes and $3$ packages of tomatoes, while the last bag has $3$ packages of potatoes and $1$ package of tomatoes.

(We cannot use $33$ bags because this construction is not valid!)

~pinkpig and asdf334

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