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Word Problems/Calculation Solution

Question
Hello:

The dorm charges \$6300 for 35 students for 24 days. In how many days will the dorm charges be \$3375 for 25 students?

I think it is 18 days.

Here is an answer and solution:
6300 dollars/(35 students × 24 days) = 6300 dollars/(840 student days) = 7.50 dollars/(student day)

3375 dollars/(25 students) × 1 student day/(7.50 dollars) = 18 days

25 students will be charged \$3375 for 18 days.

Can you explain the development of the above calculation? For example, why divide 6300 dollars by 35 students X 24 days and so forth.
What is the reasoning used for determining the answer as is shown in the calculation?

The dorm charged a total of \$6,300 for 35 students over 24 days.
To find the answer, keep track of the units involved.

If we're looking for dollars per student per day, that is dollars / (students * days).
That is why they put 6,300 / (35 * 24).  Now it can be seen that 35 * 24 is 700 + 140 = 840.
Looking at 6,300 / 840, that reduces to 630 / 84.  Those are both divisible by 21, which gives
30 / 4 = 7.5 dollars / (student day).

On the next one, "3375 dollars/(25 students) × 1 student day/(7.50 dollars) = 18 days",
note that dollars are in the numerator and the denominator, as wells as students,
so the result will be in days for each student.

The next line, "25 students will be charged \$3375 for 18 days", is a restatement of what was just calculated.

As I learned in physics, its all in what units are desired.

The number in the problem is 7.5 dollars / (student day).

If the cost over 2 days, was wanted, we would need to multiply to cancel days,
giving 2 * 7.5 = 15 dollars / student.

If the cost for 4 students was wanted, again we would multiply to cancel students,
giving 7.5 * 4 dollars / day = 30 dollars / day.

If we wanted to know how long it would take to generate 150 dollars, we would take
150 / (7.5 / (student day)).  That would give 20 student days.

On that last question, if we had 2 students, we would get 20 days/2 = 10 days.
If we had to have it in 4 days, we would have 20 students / 4 = 5 students.

Questioner's Rating
 Rating(1-10) Knowledgeability = 10 Clarity of Response = 10 Politeness = 10 Comment Thanks for the reply and explanation!

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