Two mixtures of A and B have A and B in the ratio 3:2 and 3:4. In what proportion should these two mixtures be mixed to get a new mixture in which the ration of A to B is 5:4?  Select.
A) 6:1  B) 5:4  C) 20:7  D) 10:9  E) 14:11
I got C but with a different, yet slightly longer approach, which I constantly use for proportion problems such as this one. Usually, this approach is very efficient when used with slightly difficult questions, but this problem is quite hard so I had to slightly tweak it. I will show you how I usually use the approach and later show you how to use it with this specific problem.   Example.   1 Liter of Solution A contains 45% alcohol, while 1 Liter of Solution B contains 20% alcohol. In what ratio must the two solutions be used to get a solution with 30% alcohol   Solution:   1.	45/100*[A/(A+B)]+20/100*[B/(A+B)] = 30/100   2.	Multiply 100 to both sides to arrive at 45A/(A+B) + 20B/(A+B) = 30   3.	Multiply (A+B) to both sides to arrive at 45A + 20B = 30A + 30B   4.	Distribute to arrive at 15A = 10B   5.	Thus the ratio is A/B = 10/15 = 2/3   Now using this same approach, we tackle Gopu106’s question. It is important to first think of X in the mixture as the alcohol in the problem above; hence, a mixture of X and Y in the ratio of 3:2 translates to X is 3/5 of the solution. Applying this concept to all three equations, we write:   1.	3/5*[A/(A+B)]+3/7*[B/(A+B)] = 5/9   2.	Now here is the tweak that must be made to continue with this approach. You must find the common denominator for all three numbers and organize the fractions accordingly. By finding the common denominator of 5,7,9 (or 315) we re-write the equations as follows   3.	189/315*[A/(A+B)]+135/315*[B/(A+B)] = 175/315   4.	Multiply 315 to both sides to arrive at 189A/(A+B) + 135B/(A+B) = 175   5.	Multiply (A+B) to both sides to arrive at 189A + 135B = 175A + 175B   6.	Distribute to arrive at 14A = 40B   7.	Thus the ratio is A/B = 40/14 = 20/7 or answer 20:7
<result>C</result>
Two mixtures of A and B have A and B in the ratio 3:2 and 3:4. In what proportion should these two mixtures be mixed to get a new mixture in which the ration of A to B is 5:4?