equations

jsasipriya wrote:Sixty cookies were to be equally distributed to x campers. When 8 campers did not want the cookies, the other campers each received 2 more cookies. Which of the following equations could be used to find the number of campers x?

x2 – 8x – 240 = 0
x2 – 8x + 240 = 0
x2 + 8x – 240 = 0
x2 + 8x + 240 = 0
x2 – 4x – 120 = 0

Each camper would have got 60/x cookies.
Remaining campers = x - 8
(x - 8) campers will get (60/x + 2) cookies
Then (60/x + 2)(x - 8) = 60 or 60 + 2x - 480/x - 16 = 60 or 2x - 480/x - 16 = 0
2x^2 - 480 - 16x = 0
x^2 - 8x - 240 = 0

[spoiler]The correct answer is (A).[/spoiler]

jsasipriya wrote: ↑

Thu Jul 15, 2010 3:42 am

Sixty cookies were to be equally distributed to x campers. When 8 campers did not want the cookies, the other campers each received 2 more cookies. Which of the following equations could be used to find the number of campers x?

x2 – 8x – 240 = 0
x2 – 8x + 240 = 0
x2 + 8x – 240 = 0
x2 + 8x + 240 = 0
x2 – 4x – 120 = 0

Solution:

First, let’s list the factors of 60:

1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.

We see that 2, 4, and 12 are 8 less than 10, 12, and 20, respectively. Let’s try 12 and 20 first.

Let’s say there are 20 campers. So each would have gotten 60/20 = 3 cookies. However, since 8 campers didn’t want cookies, the 60 cookies would then be distributed to 12 campers and each would get 60/12 = 5 cookies. Since this is exactly two cookies more than they would have gotten if everyone had wanted cookies, we see that there must be 20 campers.

Alternate Solution:

When 60 cookies are distributed to x campers, each camper gets 60/x cookies. Since 8 campers didn’t want any cookies, there were (60/x)*8 = 480/x cookies to be redistributed. Since these cookies were distributed to x - 8 campers, each camper got extra (480/x)/(x - 8) = 480/[x(x - 8)] cookies. We are told that this is equal to 2; hence:

480/[x(x - 8)] = 2

240/[x(x - 8)] = 1

x(x - 8) = 240

x^2 - 8x - 240 = 0

(x - 20)(x + 12) = 0

x = 20 or x = -12

Since the number of campers cannot be negative, we see that x must be 20. Thus, there are 20 campers.

Answer: E