To solve this question quickly, let’s analyze the information given and find a shortcut approach:

1. Present Ratio of Ages:

   • The present ratio of ages of A, B, and C is $4 : 5 : 6$.

2. Past Ratio of Ages:

   • 18 years ago, the ratio of their ages was $11 : 16 : 21$.

3. Set up the Equation:

   • Let the present ages of A, B, and C be $4x$, $5x$, and $6x$ respectively.
   • 18 years ago, their ages would have been $4x - 18$, $5x - 18$, and $6x - 18$.

4. Form Ratios for Past Ages:

   • According to the problem, 18 years ago, these ages were in the ratio $11 : 16 : 21$.
   • So, we can set up the following equations: $\frac{4x - 18}{11} = \frac{5x - 18}{16} = \frac{6x - 18}{21}$

5. Solve Using Two Ratios:

   • Let's just use the first two ratios to find $x$: $\frac{4x - 18}{11} = \frac{5x - 18}{16}$
   • Cross-multiplying, we get: $16(4x - 18) = 11(5x - 18)$
   • Simplifying this will give us the value of $x$.

6. Calculate Total Present Age:

   • Once $x$ is determined, plug it back to find the present ages as $4x$, $5x$, and $6x$.
   • Add these values to get the total present age of A, B, and C.

With these steps, you can quickly reach the answer by focusing on simplifying the ratios and finding $x$ through one equation, rather than solving all three. If you work out the calculation, the answer should be 150 years.

Would you like further details on any step?

Related Questions:

1. How to solve similar age ratio problems with different time intervals?
2. What if the ratio difference isn't given directly in the problem?
3. How to set up equations quickly in age-related problems?
4. How does the cross-multiplication method work in ratio problems?
5. Are there alternative shortcuts for solving proportion-based questions?

Tip:

In age problems, setting up ratios with fewer variables can save time by reducing the number of equations needed to solve for unknowns.