# How long does it take to consume the same amount of food

For a group of 32 students food lasts for 45 days. For how many days will the same food last for 72 students?

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Are you considering a proportional case? – Sigur Aug 27 '12 at 14:45
yes, I think so – Joe Aug 27 '12 at 14:46

32 students ~ 1/45 days
72 students  ~ 1/x days
x=32*45/72 = 20


Inverse proportionality

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you rock, thanks – Joe Aug 27 '12 at 15:01
@Joe:you're welcome ;) A cartoon explanation can be found here :emathematics.net/porcentajes1.php?tp=3 – Zeta.Investigator Aug 27 '12 at 15:07
The answer is correct, but the reasoning seems peculiar: Why is it $x$ students, when the question is asking for the number of days? – huon Aug 27 '12 at 17:16
@dbaupp fixed it.thanks ;) – Zeta.Investigator Aug 27 '12 at 17:22

total students are $32$ and food last for $45$ days. Total quantity of food is $32\cdot 45$

Same food to be distributed to $72$ studentss (assuming rate is equal)

Therefore total food will last for 72 students for $\frac{32\cdot 45}{72}= 20$ days.

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Think of a student as needing (precisely) one kg of noodles a day. Since the noodles will last $45$ students for $32$ days, we must have $(32)(45)$ kg of noodles. This will last $72$ students a total of $\frac{(32)(45)}{72}$ days. Calculate.

There are many other approaches, some more algebraic than others. Whatever approach you take, note that it is obvious that (under the assumption that food need for each student is fixed) the food will last $72$ students fewer than $32$ days. So after you do a calculation, do an informal reality check: if your "answer" is greater than $32$, it must be wrong.

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It is not obvious that the food will last 72 students fewer than 32 days. Assume the first group of students were big college student-athletes and the second group of students were meek little kindergartners. Without making assumptions this question can not be answered. – emory Aug 27 '12 at 18:44
@emory: Thanks. the assumption had already mentioned in the first paragraph ($1$ kg of noodles). But it is a good idea to keep stressing underlying assumptions, so now it is also mentioned in the second paragraph. – André Nicolas Aug 27 '12 at 18:58