Let's check to see if it is possible for their ratios to get towards 50% in the future.
Age of Bob can be twice that of his brother's age only in the future, or in the past.
Step 1. Let's look in the future:
i.e. For current year, ratio of ages is 12/20 = 0.6 or 60%,
Two years later, ratio of ages is 14/22 = 0.6363 or 63%,
Eight years later, ratio of ages is 20/28 = 0.714 or 71%
It cannot be twice(50%) in the future, because the ratio of their ages increases as time progresses in the future. And will never get back towards 50%. So it must be in the past that the ratio of their ages was 50%.
Step 2. Let's look in the past.
i.e. lets choose a random past year, say 6 years ago, Bob was 14, and his brother was 6.
So ratio of their ages was, 6/14 = 0.428 = 42%. So it must be in the past, but more recent than 6 years ago.
Now that we know it must be in the past, let us choose Bobs age as our unknown x, and let's setup an equation based on what we know. Bob is 20, brother is 12. Now 20 - 12 = 8 years difference now.
If Bobs' age is x years old , then his brother will have age x - 8 years old.
Step 3. We want to find the age when Bob was twice as old as his brother.
So we create the equation x = 2 * ( x -8)
x = 2x-16 ,from distributive property
16=x ,from combining like terms
So when Bob is 16, his brother is 16-8 = 8 years old.
At Bob's age of 16, his brother was 8 years old. So Bob was twice as old as his brother back then.
Now how far from Bob's current age of 20 years is 16 years?
So we are looking for 16 - 20 = -4 years. This is 4 years ago.
Solution: 4 years ago.