BTGmoderatorDC wrote: ↑Sat Feb 29, 2020 8:16 pm
A baker makes chocolate cookies and peanut cookies. His recipes allow him to make chocolate cookie in batches of 7 and peanut cookies in batches of 6. If he makes exactly 95 cookies, what is the minimum number of chocolate chip cookies he makes?
A. 7
B. 14
C. 21
D. 28
E. 35
OA
E
Source: Manhattan Prep
Solution:
We can let c = the number of batches of chocolate chip cookies made and p = the number of batches of peanut cookies made and create the equation:
7c + 6p = 95
7c = 95 - 6p
c = (95 - 6p)/7
In order for c to be an integer, we need (95 - 6p) to be a multiple of 7.
Let’s start with the largest value of p and work our way down.
When p is 15, we have:
c = 5/7... this does not work
When p is 14, we have:
c = 11/7… this does not work
When p is 13, we have:
c = 17/7... this does not work
When p is 12, we have:
c = 23/7… this does not work
When p is 11, we have:
c = 29/7... this does not work
When p is 10, we have:
c = 35/7 = 5... this works!
So, the minimum number of batches of chocolate chip cookies is 5, and, thus, the minimum number of chocolate chip cookies is 5 x 7 = 35.
Alternate Solution:
Let’s test each answer choice, starting with the smallest value:
Answer Choice A: c = 7
If c = 7, then the number of peanut butter cookies is 95 - 7 = 88; however, since 88 is not divisible by 6, c = 7 is not possible.
Answer Choice B: c = 14
If c = 14, then the number of peanut butter cookies is 95 - 14 = 81; however, since 81 is not divisible by 6, c = 14 is not possible.
Answer Choice C: c = 21
If c = 21, then the number of peanut butter cookies is 95 - 21 = 74; however, since 74 is not divisible by 6, c = 21 is not possible.
Answer Choice D: c = 28
If c = 28, then the number of peanut butter cookies is 95 - 28 = 67; however, since 67 is not divisible by 6, c = 28 is not possible.
Since we eliminated every other answer choice, we know by this point that the correct answer is E; however, let’s verify this as an exercise:
Answer Choice E: c = 35
If c = 35, then the number of peanut butter cookies is 95 - 35 = 60, which is a possible value since 60 is divisible by 6.
Answer: E
