The present ages of $A, B$ and $C$ are in the ratio of $8 : 14 : 22$ respectively. The present ages of $B, C$ and $D$ are in the ratio of $21 : 33 : 44$ respectively. Which of the following represents the ratio of the present ages of $A, B, C$ and $D$ respectively?

### Concept & Ratio Merging To combine two overlapping ratios, identify the common terms and scale the ratios so that the common terms have the exact same numerical value in both expressions. ### Step-by-Step Solution 1. We are given the first ratio for $A : B : C = 8 : 14 : 22$. 2. Simplify this ratio by dividing all terms by their greatest common divisor (which is $2$): $A : B : C = 4 : 7 : 11$ 3. We are given the second ratio for $B : C : D = 21 : 33 : 44$. 4. Identify the overlapping terms: $B$ and $C$. In the first ratio, the values for $B$ and $C$ are $7$ and $11$. In the second ratio, the values for $B$ and $C$ are $21$ and $33$. 5. Notice that the values in the second ratio are exactly $3$ times the values in the simplified first ratio ($7 \times 3 = 21$ and $11 \times 3 = 33$). 6. Multiply the entire first simplified ratio by $3$ to align it with the second ratio: $A : B : C = (4 \times 3) : (7 \times 3) : (11 \times 3) = 12 : 21 : 33$ 7. Now, both ratios share the exact same values for $B$ ($21$) and $C$ ($33$). We can directly combine them: $A : B : C : D = 12 : 21 : 33 : 44$ ### Exam Strategy & Shortcut Always simplify ratios first before trying to find the Least Common Multiple (LCM). Simplifying $8 : 14 : 22$ to $4 : 7 : 11$ immediately makes it obvious that scaling it by $3$ matches the $21$ and $33$ in the second ratio. ### Common Pitfall A frequent mistake is finding the LCM for just one variable (like $B$) without checking if the second overlapping variable ($C$) also aligns properly. If $C$ didn't align, the problem would be invalid or "Cannot be determined." ### Final Answer Therefore, the correct answer is **$12 : 21 : 33 : 44$**.