Hi,
Bag A ->
Red : White = 1:3. So, white is a multiple of 3
White : Blue = 2:3. So, white is a multiple of 2
So, white is a multiple of 2 and 3.
So, Red:White:Blue = 2:6:3 or 2a:6a:3a

Bag B ->
Red : White = 1:4 or b:4b
6a+4b = 30
So, only possible positive integral solutions are:
a=1,b=6
a=3,b=3
So, number of red balls in A = 2a is either 2 or 6
Only 6 is in the given options. As it is a could be question, Answer is D.
This is the general approach.

You can try plugging values from options. We know that 2a is even. So, A,B are out.
Use C: 2a= 4, So, 6a = 12. The number of remaining white balls is 30-12 = 18. But, this is not divisible by 4 because number of balls in bag B(4b) is divisible by 4.
Similarly, try D. It will satisfy.

There is one more way which may not be always applicable but you will able to see if you are very comfortable with numbers.

kaps786 wrote:Bag A contains red, white and blue marbles such that the red to white marble ratio is 1:3 and the white to blue marble ratio is 2:3. Bag B contains red and white marbles in the ratio of 1:4. Together, the two bags contain 30 white marbles. How many red marbles could be in bag A?

A) 1 B) 3 C) 4 D) 6 E) 8

OA follows...

We can plug in the answers, which represent the number of red marbles in Bag A.
Since R:W = 1:3, the number of white marbles in Bag A is 3 times the number of red marbles in Bag A.
Thus, the answers imply that the number of white marbles in Bag A could be:

3*1 = 3
3*3 = 9
3*4 = 12
3*6 = 18
3*8 = 24.

Since the total number of white marbles between the two bags is 30, the results above imply that the number of white marbles in Bag B could be:

30-3=27
30-9=21
30-12=18
30-18=12
30-24=6.

In Bag B, R:W = 1:4, implying that the number of white marbles in Bag B must be a multiple of 4.
In the list above, only 12 -- the result of answer choice D -- is a multiple of 4.

The correct answer is D.