Math Help PLEASE?

Question

Your teacher allows the class to have a pizza party. The pizza parlor is running a special: The first pizza in an order costs $15.99, the second (in the same order) costs half the price of the first, the third pizza costs half the price of the second, the fourth cost half the price of the third, and so on. If your class can only buy whole pizzas and has exactly $31.74 to spend (excluding tax and tip), what is the greatest number of pizzas that the class can purchase? Explain your reasoning.

This is a CONTEMPORARY MATH CLASS. Which is a professor is teaching future teachers how to teach math. And this is a weekly question that we get and we have to show how we solve it in steps such as
1. Understand
2. Plan
3. Solve
4. Evaluate
And answer #1 was no help.

Answer 1

The first pizza costs 15.99 or 15.99/(2^0), then the next is (15.99)/(2^1), then 15.99/(2^2), etc. So the total cost of n pizzas is the geometric series
∑[from k=0 to n][15.99 / (2^k)], which is
15.99 ∑[from k=0 to n][ (1/2)^k ]

The sum of the series part is [ 1 - (1/2)^(n+1) ] / (1 - 1/2), or
2[ 1 - (1/2)^(n+1) ] = 2 - (1/2)^n.

So the total amount spent on n pizzas is
15.99 (2 - (1/2)^n)

If this is to be less than or equal to 31.74, then:
15.99 (2 - (1/2)^n) < 31.74
1599 (2 - (1/2)^n) < 3174
(2 - (1/2)^n) < 3174/1599
-(1/2)^n < 3174/1599 - 2
-(1/2)^n < 3174/1599 - 3198/1599
-(1/2)^n < -24/1599
(1/2)^n > 24/1599
n*log(1/2) > log(24/1599)
n < log(24/1599) / log(1/2)
n < 6.057991722...

The highest integer less than this value is 6. So the class can by a maximum of 6 pizzas.

On a side note, if they had just a little more (like $32) then in theory they could have an infinite amount of pizzas, because
15.99 ∑[from k=0 to inf][ (1/2)^k ] =
15.99 * (1 / (1 - 1/2)) =
$31.98

Answer 2

I don't know what grade you are in but this is division. Start by writing down the price of the first pizza - $15.99. Then, you take the first price of $15.99 for the first pizza and divide it by half (or 2), you will get the second price. (see the note below about money that can't be evenly divided by 2) Write that one down below the first price. Then you take that price and divide it by half (or 2) and so on. Off to the right of the prices, run a continuous addition - by this I mean, when you add $15.99 and the price of the next pizza, place on the right the total of those two numbers. Then when you do the division for the third one, over on the right place the total of that amount and the number you wrote down before (the total of the first and second pizza price). Keep doing that until you see the amount in the right column reaches the $31.74.

NOTE: Now, you will find that some of the prices aren't exact. For instance they may not be totally divisible as a round number (it might be like $7.475.....) Any amount in that third decimal point that is 5 or higher, round up the number in the second decimal point. If it is lower than 5, leave it as it is. So $7.475 would be $7.48). Keep doing that with each pizza and you will land exactly on the $31.74.

Hope that helps. I don't want to do the work for you or you won't learn. I just wanted to show you how to do it yourself. Let me know if that makes sense to you.