The problem provided states:

"The ratio of ages of the father and mother was 11:10 when their son was born. The ratio of ages of the father and mother will be 19:18 when the son will be twice his present age. What is the ratio of the present ages of the father and mother?"

To solve this, let's define the following:

• Let the present age of the father be $F$, the present age of the mother be $M$, and the present age of the son be $S$.

Step 1: Information from the problem

• When the son was born, the ratio of the ages of the father and mother was 11:10. This means, at that time: $\frac{F-S}{M-S} = \frac{11}{10}$ which can be written as: $10(F-S) = 11(M-S)$ This gives the equation: $10F - 10S = 11M - 11S \quad \text{(Equation 1)}$

• The second condition is that when the son will be twice his present age, the ratio of the ages of the father and mother will be 19:18. At that time, the father's and mother's ages will be $F + S$ and $M + S$, respectively. So, we have: $\frac{F+S}{M+S} = \frac{19}{18}$ This can be written as: $18(F+S) = 19(M+S)$ Expanding this gives: $18F + 18S = 19M + 19S \quad \text{(Equation 2)}$

Step 2: Solve the system of equations

We now have two equations:

1. $10F - 10S = 11M - 11S$
2. $18F + 18S = 19M + 19S$

From these, we can solve for $F$ and $M$, which will give us the ratio of their present ages. Let me solve this system.The solution for the present ages of the father and mother in terms of the son's age $S$ is:

• Father's age $F = \frac{15S}{4}$
• Mother's age $M = \frac{7S}{2}$

Step 3: Find the ratio of their present ages

The ratio of the father’s age to the mother’s age is: $\frac{F}{M} = \frac{\frac{15S}{4}}{\frac{7S}{2}} = \frac{15}{14}$

Thus, the ratio of the present ages of the father and mother is $15:14$.

The correct option is (1) $15:14$.

Would you like further details on the solution, or do you have any other questions? Here are five related questions you might want to explore:

1. How would the solution change if the son were a different age?
2. Can we solve this problem using an alternative method, like graphing the equations?
3. What if the ratio of the father and mother's ages when the son was born was different?
4. How would we handle this problem if there were more than two age conditions?
5. Can you explain the general process for solving ratio-based age problems?

Tip: When working with ratios in algebraic problems, converting them into linear equations allows for more direct manipulation and solution.