Ms. winter makes homemade bath soaps and bottles of lotion. in her inventory, she had 48 bath soaps and 64 bottles of lotion. she wants to use all of her inventory to make as many gift baskets as she can use using the same number of bath soaps and the same number of lotion bottles in each gift basket. she will be able to make at most ? gift baskets. each basket will contain ? bath soaps and ? bottles of lotion.

Keep in mind that each basket will have S bottles of bath soap and L bottles of lotion, and that S and L are the same for every basket.

The trick here is to determine which common factors 48 and 64 have.
Factors of 48 are 2, 3, 4, 6, 8, 12, 16, 24 and 48.
Factors of 64 are 2, 4, 8, 16.
Note that the common factors are 2, 4, 8 and 16.
Also note that there are a total of 48+64 bottles, all of which must be used.

Since there are more bottles of Lotion than there are bottles of Soap,
the number of Lotion bottles in each basket will be greater than the number of Soap bottles in that basket.  S and L together equal 48+64 = 112.

We want to maximize the number of baskets that can be created.  Call this number N.

N(S+L) = 112.  Because there are more Lotion bottles than there are Soap bottles, S = (48/64)L, or       S = 3L/4

Imagine that N baskets are created.  Each basket will contain L bottles of Lotion and 3L/4 bottles of Soap.

L and S must both be integers, so that we're dealing only in whole, full bottles of lotion and soap.

Let's make a table here:

N         L          S=3L/4          Total # of bottles of S and L
16       4              3                 16(4+3) =112

This was my first guess!        We could make 16 baskets, each containing
                                               4 bottles of lotion and 3 bottles of soap.

Try again:

N         L          S=3L/4          Total # of bottles of S and L
16       4              3                 16(4+3) =112
12       8              6                  12(8+6) = 168 (this doesn't work since 
                                                there are only 112 bottles, total.

The constraints here are as follows:  

1) S is always (3/4) of L:  S = 3L/4, and both S and L must be integers.
2) N(S+L) must equal 112.

There's probably an algebraic solution to this problem.  You might want to think about this and see whether you can create and solve a system of linear equations that produces the same optimal results:

"16 baskets containing 8 bottles of lotion and 6 bottles of soap"
satisfies the requirement that S = 3L/4, and also the requirement that N(S+L)=112.