A baker makes a combination of chocolate chip cookies and peanut butte

Say x is the number of chocolate cookies and y is the number of peanut cookies Bob makes. Notice that since chocolate cookies are in batches of 7 and peanut cookies are in batches of 6 then x must be a multiple of 7 and y must be a multiple of 6.

Given: x+y=95 --> y=95-x. We want to minimize x, so we need to find the minimum value of a multiple of 7 (x) that must be subtracted from 95 to get a multiple of 6 (y). The minimum value turns out to be 35=5*7: 95-35=60=6*10.

Answer: E.

The first thing I noticed is that the answer choice must be odd. Since 6 is even, the quantity of peanut butter cookies must be even, so in order to add to 95, which is odd, the quantity of chocolate cookies must be odd. That eliminates answer choices B and D.

To find the answer, I started with the lowest odd answer choice, subtracted that number from 95, and found if it was divisible by 6.

A) 95 - 7 = 88. 88 is not divisible by 6 because 8+8 is not divisible by 3. Eliminate A.
C) 95 - 21 = 74. 74 is not divisible by 6 because 7+4 is not divisible by 3. Eliminate C.

This leaves only answer choice E remaining.

The answer is E.

Hope this method helps!

We require to check divisibility of (95 - OA) by 6

95 is not divisible by 6, however 95+1 = 96 is divisibly by 6

So, adding 1 to the OA, only 35+1 = 36 is divisible by 6

Answer = E

\(x \geqslant 1\,\,\,{\text{choco}}\,\,{\text{batches}}{\text{,}}\,\,\,\,\,{\text{7}}\,\,{\text{choco/batch}}\)

\(y \geqslant 1\,\,\,{\text{pean}}\,\,{\text{batches}}{\text{,}}\,\,\,\,\,{\text{6}}\,\,{\text{pean/batch}}\)

\(7x + 6y = 95\,\,\,\,\,\left( * \right)\)

\({\text{? = }}{\left( {{\text{7x}}} \right)_{\,\min }}\)

\({\left( {{\text{7x}}} \right)_{\,\min }}\,\,\, \Leftrightarrow \,\,\,{x_{\min }}\,\,\,\)

\({\left( {multiple\,\,of\,\,6} \right)_{\max }} = {\left( {6y} \right)_{\max }}\mathop = \limits^{\left( * \right)} \,\,95 - 7x\)

\(\begin{gathered}\\
x = 1\,\,\, \Rightarrow \,\,\,95 - 7x = 88\,\,{\text{not}}\,\,{\text{divisible}}\,\,{\text{by}}\,\,3\, \hfill \\\\
x = 2\,\,\, \Rightarrow \,\,\,95 - 7x = {\text{odd}}\,\,\left( {{\text{not}}\,\,{\text{divisible}}\,\,{\text{by}}\,\,2} \right) \hfill \\\\
x = 3\,\,\, \Rightarrow \,\,\,95 - 7x = 74\,\,{\text{not}}\,\,{\text{divisible}}\,\,{\text{by}}\,\,3 \hfill \\\\
x = 4\,\,\, \Rightarrow \,\,\,95 - 7x = {\text{odd}}\,\,\left( {{\text{not}}\,\,{\text{divisible}}\,\,{\text{by}}\,\,2} \right) \hfill \\\\
x = 5\,\,\, \Rightarrow \,\,\,95 - \boxed{7x = 35} = {\text{60}}\,\,\underline {{\text{divisible}}\,\,{\text{by}}\,\,6!} \hfill \\ \\
\end{gathered}\)


The above follows the notations and rationale taught in the GMATH method.

Regards,
fskilnik.

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

LCM OF 95 is 5 and 19
so smaller number is 5 or multiple of 5 which is 35

LCM OF 95 is 5 and 19
so smaller number is 5 or multiple of 5 which is 35

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