Using linear equations and inequalities to determine exact values help.

[Kaizen]

Greeting all,

I am having trouble with a certain question in what has up until now been a remarkably easy chapter.
The question in question, states the following:

22) A cricket social club committee booked four tables at a local restaurant for a casual lunch for their members. The organisers were unsure in advance how many people were attending but on the day three of four people sat at each table.
A) Construct an inequality to describe the number of people from the cricket club who actually attended the lunch.
B) The lunch consisted of two courses for a fixed amount per person. If the total bill the cricket social club had to pay was exactly $418.50, how many people from the cricket club attended the lunch, and what was the fixed charge per person?

So far, i have completed A), by doing the following:
A) 4x3(less than or equal to)x(less than or equal to)4x4
I have consulted my book's answer sheet and found it to be correct.

However, i simply cannot figure out how to complete B) as it seems to demand an exact answer from an inequality.
I have consulted the answer sheet for B), and it seems the answer is 15x, i am simply curious as to how to get there.
*I have divided 418.50 by 16, 15, 13 and 12, and 15 gives the 'cleanest' answer, with only a single decimal value, however, the question has not specified the need for such a thing*
Thanks for your time.

 

[ksdhart]

Part (b) relies on your knowledge that currency is made up of discrete units. What I mean by that is that, when paying for something, the cost will always be some number of cents. It is impossible to pay a fraction of a cent for anything. You know the total was $418.50, and you know that each person paid exactly the same amount. You also have an inequality that limits the potential values of x to 12, 13, 14, 15, or 16. So, let's try a few cases and see what happens:

Suppose there were 16 people. Each person then must pay $418.5/16 = 837/32 = 26.15625. Well, we know that can't be, because, as previously established, one cannot spent a fraction of a cent. Some people would need to pay $26.16 and some $26.15, which violates the "rules" of the problem.

Suppose there were 13 people. Each person then must pay $418.5/13 = 837/26. And that's not even a rational number, so it's definitely not the answer.

And so on, until you arrive at 15 being the only possible value for x where everyone can pay exactly the same amount.

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EDIT: Not sure what I was thinking, but 837/26 is indeed a rational number, as it can be represented as a fraction. Thanks Ishuda for alerting me to my error.

 

[Kaizen]

Thanks ksdhart,
It seems like my original reasoning, that is to divide as you did, was correct, and it was simply a refusal to assume that led me to being incapable of finding the answer.
~Kaizen