At a party, John collected $96 to buy p equally-priced pizzas. When

Since John bought p equally priced pizzas, with 96$,
and we assume each of the pizza cost 'c'
From the question stem pc = 96
96(when prime factorized) gives \(2^5*3\)
The cost of the pizza's cannot be \(9(3^2)\) and \(10(2*5)\). So Option C and D are out!

The second part of the question, gives us the price during promotion
(p + 4)(c - 2) = 96

Now evaluating the answer options,
Since price of pizza (after discount) is 8$, John would have bought 12 pizza's during the promotional period.
However, price before discount would be 10$ and he would have bought 8 pizza's which doesn't give us 96$ as total cost.
Hence, Option C is also not right

If price after discount is 6$, he would have bought 16 pizza's during the promotional period.
However, price before discount would be 8$ and he would have bought 12 pizza's which gives us 96$ as total revenue.
Hence, Option A is our answer

Instead of taking variables and solving the equations, this question can be best done by jumping to the options directly.

John collected $96 to buy p pizzas, each of same price. So obviously, 96 will be divisible by p, and by the un-discounted price of the pizza.
The discounted price is $2 less and bought 4 more pizzas i.e. p + 4 pizzas. The discounted price would be divisible by 96 too.

Looking at the options, 6 and 8 could be the discounted and un-discounted prices.
At $6, you would get 96/6 = 16 pizzas
At $8, you would get 96/8 = 12 pizzas (4 less)
(A match)

Hence discounted price is $6

Answer (A)

We can do 'plug-in answers' as well. If total no. of pizzas is p, then price per pizza is 96/p. Price after discount is 96/p-2. Plug in for 10: 96/p-2 = 10, => p = 8. Now, originally price per pizza was 96/8 = 12 (rs2 off) but 96/10 = 9.6 (not four extra pizzas). So we look at a lesser no. Plug-in for 6, 96/p-2 = 6, => p = 12. Now price per pizza = 96/12 = 8. (rs 2 off) and 96/6 = 16, which is 4 extra pizzas. Ans is A

That's how I went about it, however my concern was assuming that the $96 had to be divisible by both p and p+4 (especially the latter). They wouldn't have to say that there was any $ left over after in either case, right?

96/p = cost before discount
after discount: 96/p - 2

thus: (96/p -2)*(p+4) = 96

p = 12,
price after discount: $6
Answer OA

Excellent Question Bunuel.

Here is what I did on this one ->


Cost 1 = Cost 2

96 => (96/p - 2 ) (p+4)
Solving this we get => p=12 or p=-16

Hence p=12

So the cost after the discount would be 96/12 - 2 => 8-2 => 6

Hence A

Great Question

Price after discount as per options given are: 6, 8, 9, 10, 12
Price before discount: 8,10,11,12,14
Price after discount and before discount should divide 96. Combination of 6 and 8 does it, therefore answer is A

Hello! Is there any easy way to realize when a problem is going to become a quadratic when attempted algebraically? Thank you!

We can let the price of each pizza before the discount = n and create the equations:

pn = 96

p = 96/n

and

(p + 4)(n - 2) = 96

pn + 4n - 2p - 8 = 96

Substituting p = 96/n into the second equation, we have:

(96/n)n + 4n - 2(96/n) - 8 = 96

96 + 4n - 192/n - 8 = 96

4n - 8 - 192/n = 0

Multiplying by n, we have:

4n^2 - 8n - 192 = 0

n^2 - 2n - 48 = 0

(n - 8)(n + 6) = 0

n = 8 or n = -6

Since the price of a pizza can’t be negative, n = 8. However, this is the price before the discount. The price of each pizza after the discount is 8 - 2 = 6 dollars.

Answer: A

$96 should be divisible by the cost after (x) and before (x+2) the discount. From the answers the only one that satisfies this is 6.

Answer A.

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