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

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

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New post 15 Jun 2017, 23:41
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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

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

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New post 16 Jun 2017, 00:01
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Bunuel wrote:
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


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).
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Re: At a party, John collected $96 to buy p equally-priced pizzas. When  [#permalink]

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New post 16 Jun 2017, 00:14
Bunuel wrote:
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


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

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New post 16 Jun 2017, 02:22
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


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Re: At a party, John collected $96 to buy p equally-priced pizzas. When  [#permalink]

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New post 16 Jun 2017, 03:23
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Bunuel wrote:
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


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)
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Re: At a party, John collected $96 to buy p equally-priced pizzas. When  [#permalink]

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New post 16 Jun 2017, 04:12
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
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Re: At a party, John collected $96 to buy p equally-priced pizzas. When  [#permalink]

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New post 16 Jun 2017, 04:37
VeritasPrepKarishma wrote:
Bunuel wrote:
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


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)



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?
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Re: At a party, John collected $96 to buy p equally-priced pizzas. When  [#permalink]

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New post 16 Jun 2017, 12:53
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
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Re: At a party, John collected $96 to buy p equally-priced pizzas. When  [#permalink]

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New post 16 Jun 2017, 20:45
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 :)
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At a party, John collected $96 to buy p equally-priced pizzas. When  [#permalink]

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New post 17 Sep 2017, 02:04
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
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Re: At a party, John collected $96 to buy p equally-priced pizzas. When  [#permalink]

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New post 02 Oct 2018, 14:46
Hello! Is there any easy way to realize when a problem is going to become a quadratic when attempted algebraically? Thank you!
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Re: At a party, John collected $96 to buy p equally-priced pizzas. When  [#permalink]

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New post 03 Oct 2018, 17:51
Bunuel wrote:
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


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
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Re: At a party, John collected $96 to buy p equally-priced pizzas. When   [#permalink] 03 Oct 2018, 17:51
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