If $A : B = 2 : 3$, $B : C = 4 : 5$ and $C : D = 6 : 7$, then $A : B : C : D$ is

### Concept & Chaining Multiple Ratios When combining more than two ratios, apply the equalization technique sequentially. First combine $A:B$ and $B:C$ to find $A:B:C$, and then combine that result with $C:D$. ### Step-by-Step Solution 1. We have the ratios: $A : B = 2 : 3$ $B : C = 4 : 5$ $C : D = 6 : 7$ 2. First, combine $A : B$ and $B : C$. The common term is $B$ (values $3$ and $4$). The LCM is $12$. Multiply $A : B$ by $4 \Rightarrow 8 : 12$ Multiply $B : C$ by $3 \Rightarrow 12 : 15$ Combined ratio: $A : B : C = 8 : 12 : 15$ 3. Now, combine the new $A : B : C$ ratio with $C : D$. The common term is $C$ (values $15$ and $6$). The LCM of $15$ and $6$ is $30$. 4. Scale the $A : B : C$ ratio to make $C = 30$. Multiply all terms by $2$: $A : B : C = (8 \times 2) : (12 \times 2) : (15 \times 2) = 16 : 24 : 30$ 5. Scale the $C : D$ ratio to make $C = 30$. Multiply both terms by $5$: $C : D = (6 \times 5) : (7 \times 5) = 30 : 35$ 6. Since $C$ is now $30$ in both sequences, we can link them completely: $A : B : C : D = 16 : 24 : 30 : 35$ ### Exam Strategy & Shortcut Use the block-filling method. Write out a grid: $A : B : C : D$ $2 : 3 : 3 : 3$ (Fill right with $3$) $4 : 4 : 5 : 5$ (Fill left with $4$, right with $5$) $6 : 6 : 6 : 7$ (Fill left with $6$) Multiply down the columns: $A = 2 \times 4 \times 6 = 48$ $B = 3 \times 4 \times 6 = 72$ $C = 3 \times 5 \times 6 = 90$ $D = 3 \times 5 \times 7 = 105$ Divide everything by the common factor $3$: $16 : 24 : 30 : 35$. ### Common Pitfall Trying to find a single multiplier for all terms at once often leads to confusion. Always break the process down into manageable pairs, or strictly adhere to a proven visual shortcut. ### Final Answer Therefore, the correct answer is **$16 : 24 : 30 : 35$**.