Question 166505
What you are essentially doing here is finding the LCM (lowest common multiple) of 24, 20, and 16. Remember, the LCM is the SMALLEST number that two other whole numbers can go into. For instance, 15 is the LCM of 3 and 5 since both 3 and 5 can go into 15 evenly (and 15 is the smallest that 3 and 5 can go into)




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So let's find the LCM of 24, 20, and 16



First, list the prime factorization of each number. If you need help with prime factorization, check out this <a href="http://www.algebra.com/algebra/homework/divisibility/factor-any-number.solver">solver</a>.



24: 2*2*2*3



20: 2*2*5



16: 2*2*2*2



Now highlight the unique factors. Remember, highlight the factors that occur most frequently



24: 2*2*2*<font color="red">3</font>



20: 2*2*<font color="red">5</font>



16: <font color="red">2*2*2*2</font>



Now multiply all of the highlighted factors


2*2*2*2*3*5=240


So the LCM of 24, 20, and 16 is 240




Note: there are many ways to find the LCM. Let me know if this method does not make sense to you. 


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Now take the LCM and divide it by 24 to get the number of plate packages:


{{{240/24=10}}}



So she'll need to buy 10 plate packages to get 240 plates


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Now take the LCM and divide it by 20 to get the number of napkin packages:


{{{240/20=12}}}



So she'll need to buy 12 napkin packages to get 240 napkins


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Finally, take the LCM and divide it by 16 to get the number of paper cup packages:


{{{240/16=12}}}



So she'll need to buy 15 paper cup packages to get 240 paper cups



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Answer:



So she needs to buy 10 plate packages, 12 napkin packages, and 15 paper cup packages to get 240 plates, 240 napkins, and 240 cups.







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# 2



Since there are 360 students and 360 is greater than 240 (the number of plates, napkins, and cups), this means that there won't be enough to go around.



So she'll have to buy another batch of 10 plate packages, 12 napkin packages, and 15 paper cup packages to get another 240 plates, 240 napkins, and 240 cups.



Now add the two batches to get

240+240=480


Since 480 is greater than 360, this means that she'll have enough (after she buys another batch of items)