We can think of the liquids in the red bucket as liquids A, B, C and E, where E represents the totality of every other kind of liquid that is not A, B, or C. In order to determine the percentage of E contained in the red bucket, we will need to determine the total amount of A + B + C and the total amount of E.

It is TEMPTING (but incorrect) to use the following logic with the information given in Statement (1).

Statement (1) tells us that the total amount of liquids A, B, and C now in the red bucket is 1.25 times the total amount of liquids A and B initially contained in the green bucket.

Let's begin by assuming that, initially, there are 10 ml of liquid A in the green bucket. Using the percentages given in the problem we can now determine that the composition of the green bucket was as follows:

10% A = 10 ml

10% B = 10 ml

80% E = 80 ml

Since there were 20 total ml of A and B in the green bucket, we know from statement (1) that there must be 25 ml of A + B + C now in the red bucket (since 25 is 1.25 times 20).

From this we can deduce that, there must have been 5 ml of C in the blue bucket. We can use the percentages given in the problem to determine the exact initial composition of the blue bucket:

10% C = 5 ml

90% E = 45 ml

Since the liquid in the red bucket is simply the totality of all the liquids in the green bucket plus all the liquids in the blue bucket, we can use this information to determine the total amount of A + B + C (25 ml) and the total amount of E (80 + 45 = 125 ml) in the red bucket. Thus, the percentage of liquid now in the red bucket that is NOT A, B, or C is equal to 125/150 = 83 1/3 percent.

This ratio (or percentage) will always remain the same no matter what initial amount we choose for liquid A in the green bucket. This is because the relative percentages are fixed.

We can generalize that given an initial amount x for liquid A in the green bucket, we know that the amount of liquid B in the green bucket must also be x and that the amount of E in the green bucket must be 8x. We also know that the amount of liquid C in the blue bucket must be .5x, which means that the amount of E in the blue bucket must be 4.5x.

Thus the total amount of A + B + C in the red bucket is x + x + .5x = 2.5x and the total amount of liquid E in the red bucket is 8x + 4.5x = 12.5x. Thus the percentage of liquid now in the red bucket that is NOT A, B, or C is equal to 12.5x/15x or 83 1/3 percent.

However, the above logic is FLAWED because it assumes that the green bucket does not contain liquid C and that the blue bucket does not contain liquids A or B.

In other words, the above logic assumes that knowing that there are x ml of A in the green bucket implies that there are 8x ml of E in the green bucket. Remember, however, that E is defined as the totality of every liquid that is NOT A, B, or C! While the problem gives us information about the percentages of A and B contained in the green bucket, it does not tell us anything about the percentage of C contained in the green bucket and we cannot just assume that this is 0. If the percentage of C in the green bucket is not 0, then this will change the percentage of E in the green bucket as well as changing the relative amount of liquid C in the blue bucket.

For example, let's say that the green bucket contains 10 ml of liquids A and B but also contains 3 ml of liquid C. Take a look at how this changes the logic:

Green bucket:

10% A = 10 ml

10% B = 10 ml

3% C = 3 ml

77% E = 77 ml

Since there are 20 total ml of A and B in the green bucket, we know from statement (1) that there must be 25 ml of A + B + C in the red bucket (since 25 is 1.25 times 20).

Since the green bucket already contributes 23 ml of this total, we know that there must be 2 total ml of liquids A, B and C in the blue bucket. If the blue bucket does not contain liquids A or B (which we cannot necessarily assume), then the composition of the blue bucket would be the following:

10% C = 2 ml

90% E = 18 ml

Note, however, that if the blue bucket does contain some of liquids A or B, then the composition of the blue bucket might also be the following:

10% C = 1 ml

10% A = 1 ml

80% E = 8 ml

Notice that it is impossible to ascertain the exact amount of E in the red bucket - since this amount will change depending on whether the green bucket contains liquid C and/or the blue bucket contains liquids A or B.

Thus statement (1) by itself is NOT sufficient to answer this question.

Statement (2) tells us that the green and blue buckets did not contain any of the same liquids. As such, we know that the green bucket did not contain liquid C and that the blue bucket did not contain liquids A or B. On its own, this does not help us to answer the question. However, taking Statement (2) together with Statement (1), we can definitively answer the question.

The correct answer is C: BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

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Thanks & Regards,

Anaira Mitch