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

Question

At a party, John collected $96 to buy p equally-priced pizzas. When he called the store, however, they were running a promotion for $2 off of each pizza, so he was able to buy 4 more pizzas than he expected for the same $96. How much was the cost of each pizza after the discount?

A. $6
B. $8
C. $9
D. $10
E. $12

A. $6
B. $8
C. $9
D. $10
E. $12

sashiim20 · Answer

Let price per pizza be  x.

Total number of pizza  = p

Price of  x  pizza's before discount  = $96

px = 96  ------------ (i)

Total number of pizza after discount  = p + 4

Price decrease on each pizza after discount  = x - 2

Given Price of pizza's after discount is  $96 . ie;

(p+4)(x-2) = 96 
 px + 4x -2p - 8 = 96 
 96 + 4x - 2p -8 = 96  ---------- (Substituting value of  px  from (i))
 4x - 2p - 8 = 0 
 4x - 2p = 8 
 2x - p = 4 
 p = 2x + 4  ---------- (ii)

Substituting value of  p  from (ii) in equation (i), we get;

(2x+ 4)x = 96 
 2x^2 + 4x = 96 
 2x^2 - 4x - 96 = 0 
 x^2 - 2x - 48 = 0 
 x^2 - 8x + 6x - 48 = 0 
 x(x - 8) + 6 (x - 8) = 0 
 (x+6)(x+8) = 0 
 x = 8, -6

Price cannot be negative. Therefore price of each pizza before discount was  $8 .

Price of pizza after discount is  8-2 = $6.  
 Answer (A).

Shruti0805 · Answer

Let the price for each pizza be x.
Thus,  p*x = 96  {no. of pizza*price of each pizza = Total sum of money John had} 
=> p = \frac{96}{x}  -- eqn. (1)

After the discount, reduced price of each pizza =  (x-2) 
and thus, he could buy 4 extra pizza. So, no. of pizzas becomes  (p+4) 
thus the equation changes to : (p+4)*(x-2) = 96
=> p+4 = \frac{96}{(x-2)}  -- eqn (2)

Equating eqn. 1 and 2-

\frac{96}{(x-2)} = \frac{96}{x} + 4 
=> \frac{96}{(x-2)} - \frac{96}{x} = 4

Taking LCM and solving results in-
 4x(x-2) = 192 
=>  4x^2 - 8x = 192 
=>  x^2 -2x - 48 = 0 
=>  (x-8)(x+6) = 0 
=>  x = 8 or -6 .

As the price cant be a negative value,  x= 8 
As x was the original price for the pizzas, i.e., price before discount, price after discount =  (x-2)  =  6

Option A.

pushpitkc · Answer

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

KarishmaB · 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)

arshiyakhanna · Answer

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

laxpro2001 · Answer

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?

sonikavadhera · Answer

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

stonecold · Answer

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

ravjai81 · Answer

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

gmat800live · Answer

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

ScottTargetTestPrep · Answer

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

alexisrakh · Answer

$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.

Posted from my mobile device

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

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

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions

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

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards