I believe that the following reflects the intent of the problem:

If a = 3bc and abc≠0, what is the value of c?

(1) a = 10-b
(2) 3a = 4b

a = 3bc
(1/3)a = bc
(1/3)(a/b) = c
c = (1/3)(a/b).

To determine the value of c, we need to know the value of a/b.
Question stem, rephrased:
What is the value of a/b?

Statement 1: a = 10-b
If b=1, then a=9.
In this case, a/b = 9/1 = 9.
If b=2, then a=8.
In this case, a/b = 8/2 = 4.
Since a/b can be different values, INSUFFICIENT.

Statement 2: 3a = 4b
3(a/b) = 4
a/b = 4/3.
SUFFICIENT.

The correct answer is B.

vinhaha wrote:If a = 3bc, what is the value of c?

(1) a = 10 - b

(2) 3a = 4b

Here is the solution to the question as posted.

Statement 1:

You can plug in various numbers to see that c could have various values.

a = 9 b = 1: 9 = 3c c = 3

a = 5 b = 5: 5 = 15c c = 1/3

Insufficient.

Statement 2:

From this statement we know that a = 4/3b.

So if a = 3bc, then, if an and b are not 0, then 3c = 4/3, and we could find the value of c.

However, if a = b = 0, then 3a = 4b = 0, and a = 3bc = 0 for any value of c.

Insufficient.

Statements Combined:

If a = 10 - b, and 3a = 4b, then 3a = 30 - 3b and 4b = 30 - 3b.

So we could determine the values of a and b, neither of which is 0, and from there determine the value of c.

Sufficient.

The correct answer is C.

vinhaha wrote:Hi all, can anyone help me with this question? I can figure out that (1) is insufficient, but i cannot proof (2). many thanks in advance!!

If a = 3bc, what is the value of c?

(1) a = 10-6
(2) 3a = 4b

Yeah, this is tough one. Condition (2) looks like it could be sufficient, but as the others pointed out, if a and b are both zero, both the original equation and the equation in condition (2) would be satisfied for any value of c. Therefore, (2) is not sufficient alone. But combined with (1), together the two statements are sufficient.

If we were given any additional prompt to the effect that a is a nonzero number, or the product bc is nonzero, or the product abc is nonzero, then (2) alone would be sufficient, because it would only be satisfied in the case where c equals 4.

Note that you can tell the combination of (1) & (2) will be sufficient because (1) is basically another way of saying a is nonzero. If a = 4, then a cannot equal zero. So really (1) could have given us any number of new information that when combined with (2) would be sufficient. It could have said a < 0, or a is a positive integer, or a is a prime number. All of those would also be consistent with a being a nonzero number.

800_or_bust wrote:If we were given any additional prompt to the effect that a is a nonzero number, or the product bc is nonzero, or the product abc is nonzero, then (2) alone would be sufficient, because it would only be satisfied in the case where c equals 4.

It's probably easiest to see this with equations.

a = 3bc
3a = 4b

gives

a = 3bc
a = (4/3)b

gives

3bc = (4/3)b

gives

0 = (4/3)b - (3c)b

gives

0 = b * (4/3 - 3c)

So we have two solutions, b = 0 or (4/3 - 3c) = 0, that is, b = 0 or c = 4/9.