The cookies in a cookie jar...

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bazooka-worm

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Oct 7, 2008
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Hello everyone, Im in grade 9 and as I was clearing my old binders and such I came across a question from last year which happens to be uncompleted, so I decided to read it over and sadly I still can't do it. Can anyone help? ps. I forgot my password & email to my older account (bazookaworm) is there anyway i can recover it?

The cookies in a jar contain a total of 1000 chocolate chips. All but one of these cookies contains the
same number of chips; it contains one more chip than the others. The number of cookies in the jar is
between one dozen and three dozen. What is the sum of the number of cookies in the jar and the
number of chips in the cookie with the extra chocolate chip?
(A) 65 (B) 64 (C) 63 (D) 66 (E) 67

So I know there is between 12-36 cookies in the jar. I need to know how many cookies are in the jar and the number of chips in the cookie with the extra chocolate chip. I don't know where to start, can someone point me in the right direction? I think I was born with horrid problem solving skills lol. It's not like we learn how to do this stuff in class!

Thanks In Advance
 
let x= number of cookies
but 12<x<36

let y = number of chips in majority of cookies

total chips= [x-1]y+[y+1]=1000
xy-y+y+1=1000
xy=999

let us look at the number 999 in terms of its prime products
basic algorithim of arithmetic," any number can be written as a finite product of primes"
999=3,333
999=3,3,111
999=3,3,3,37

if x=1 y=999 out side of given data
if x=3 y=333 out side of given data
if x=9 y =111 outside of given data
if x=27 y=37 outside of given data

thus there is no solution

let us look at suggested solutions
number of cookies = 63 or 64 or 65 or 66 or 67

no.of cookies = 63
then
62 y+y+1=1000
63y=999
y=15.8 partial chip

no. of cookies=64
y=15.6

no of cookies =65
y=15.4

no of cookies= 66
y=15.1

no of cookies = 67
y=14.9

none satisfy the given values
Arthur
 
I get A)65 as answer, Arthur.

27 cookies:
26 cookies with 37 chips each = 962
1 cookie with 38 chips = 38
962 + 38 = 1000

cookies + chips on cookie with extra = 27 + 38 = 65

999 (chips) / 27 (cookies) = 37 (chips) is the only integer solution for 12 to 36 cookies; add a chip to 27th
 
werewolf you are right!!!!
I made a stupid error.

let x= no.of cookies 12<x<36
let chips in a cookie be y
total chips =1000

then
[x-1]y+[y+1]=1000 here is where I made my error
[x-1]= no of cookies with y chips
[y+1= extra chips in remaining cookie
stupid error

[x-1]y+y+1=1000
xy=999

the prime divisors of 999=3,3,3,37
if y=1 then x=999 out of spec
if y=3 then x=333 out of spec
if y=9 then x= 111 out of spec
if y=27 then x= 37 out of spec
if y=37 then x=27 answer
check
26 cookies with 37 chips and one cookie with 38 chips=1000

no.of chips in a cookie 37
no of cookies 27
27x37+1=999+1=1000
no of cookies =36
 
Well Arthur, you made your 1st mistake this year :wink:
 
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