The ratio of the present ages of A and B is 3 : 7, whereas after n years it will be 1 : 2. Find the difference between their ages after 2n years, if the ratio of their ages 15 years ago was 1 : 4.

Let's solve this step by step:

Step 1: Let's assume the present ages of A and B are 3x and 7x respectively.

Step 2: According to the problem, after n years, the ratio of their ages will be 1:2. So, we can write it as (3x+n)/(7x+n) = 1/2.

Step 3: Solving this equation, we get 2(3x+n) = 7x+n, which simplifies to 6x+2n = 7x+n. This further simplifies to x = n.

Step 4: Now, we know that the ratio of their ages 15 years ago was 1:4. So, we can write it as (3x-15)/(7x-15) = 1/4.

Step 5: Substituting x = n in the above equation, we get (3n-15)/(7n-15) = 1/4.

Step 6: Solving this equation, we get 4(3n-15) = 7n-15, which simplifies to 12n-60 = 7n-15. This further simplifies to 5n = 45, so n = 9.

Step 7: Now, we need to find the difference between their ages after 2n years. So, the age of A after 2n years will be 3x+2n and the age of B after 2n years will be 7x+2n.

Step 8: Substituting x = n = 9 in the above, we get the age of A after 2n years as 39+29 = 45 and the age of B after 2n years as 79+29 = 81.

Step 9: Therefore, the difference between their ages after 2n years will be 81 - 45 = 36 years.

Let's solve this step by step:

Step 1: Let's assume the present ages of A and B are 3x and 7x respectively.

Step 2: According to the problem, after n years, the ratio of their ages will be 1:2. So, we can write it as (3x+n)/(7x+n) = 1/2.

Step 3: Solving the above equation, we get 2(3x+n) = 7x+n, which simplifies to 6x+2n = 7x+n. This further simplifies to x = n.

Step 4: Now, we know that the ratio of their ages 15 years ago was 1:4. So, we can write it as (3x-15)/(7x-15) = 1/4.

Step 5: Substituting x = n in the above equation, we get (3n-15)/(7n-15) = 1/4. Solving this equation, we get n = 10.

Step 6: Now, we need to find the difference between their ages after 2n years. So, the ages of A and B after 2n years will be 3x+2n and 7x+2n respectively.

Step 7: Substituting x = n = 10 in the above, we get the ages of A and B after 2n years as 60 and 140 respectively.

Step 8: Therefore, the difference between their ages after 2n years will be 140 - 60 = 80 years.