Question

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

Answer 1

Solutions  

If we remove the extra chip from the special cookie, all cookies have the same number of chocolate chips for a total of 999 chips. We look at factorization's of 999.

The question states that the number of cookies in the jar is between 12 and 36 so this implies that the only factorization of 999 that works is
( 3 x  3 x 3) (37)  .
 
Thus the only divisor of 999 between 12 and 36 is 27. From this, we see that there are 27 cookies. An ordinary cookie has 999 / 27 = 37 chocolate chips, and the special cookie has 38 chocolate chips. The required sum is 27 38 65 + = .
 
ANSWER: (A)

Answer 2

Answer : (A) 65

Solution :

Let the total number of cookies be 'x'

Total Choco chips = 1000

So, number of choco chips if we remove the extra choco chip = 1000 - 1 = 999

Number of choco chips on each cookies except the one with one extra choco chip = 999/x

Now, to get a whole number, x must be a factor of 999

Given, 36 > x > 12

Factor of 999 between 36 and 12 :
⇒ 27 

So, the value of x must be 27.

Now, number of choco chips on the exceptional cookie = 999/x + 1 
                                                                                        = 999/27 + 1
                                                                                        = 37 + 1 
                                                                                        = 38

So, 27 + 38 = 65 

So, the answer is (A) 65.