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Hint: The ratios $A:B$ and $B:C$ are known to us. Make the value of $B$ in both the ratios same and then compare both the ratios.

Complete step-by-step answer:

According to the question, \[A:B = 2:3\] and \[B:C = 4:7\].

If we multiply the first ratio by 4 and second ratio by 3, we’ll get:

$

\Rightarrow A:B = \left( {2:3} \right) \times 4, \\

\Rightarrow A:B = 8:12 .....(i) \\

\Rightarrow B:C = \left( {4:7} \right) \times 3, \\

\Rightarrow B:C = 12:21 .....(ii) \\

$

Now, if we compare ratios $(i)$ and $(ii)$, the value of B in both the ratios is 12. So, we can easily combine both the ratios as:

$

\Rightarrow A:B = 8:12,B:C = 12:21 \\

\Rightarrow A:B:C = 8:12:21 \\

$

Therefore the required ratio $A:B:C$ is $8:12:21$.

Note: We can also solve the question by different method:

As we know that $A:B = 2:3$.

Let the value of A is $2x$ and that of B is $3x$.

Now it is given that \[B:C = 4:7\] and we have already taken the value of B as $3x$. So, we have:

$

\Rightarrow \dfrac{B}{C} = \dfrac{4}{7}, \\

\Rightarrow \dfrac{{3x}}{C} = \dfrac{4}{7}, \\

\Rightarrow C = \dfrac{{21}}{4}x \\

$

Thus we have:

$

\Rightarrow A = 2x,B = 3x{\text{ and }}C = \dfrac{{21}}{4}x, \\

\Rightarrow A:B:C = 2:3:\dfrac{{21}}{4}, \\

\Rightarrow A:B:C = 8:12:21 \\

$

Complete step-by-step answer:

According to the question, \[A:B = 2:3\] and \[B:C = 4:7\].

If we multiply the first ratio by 4 and second ratio by 3, we’ll get:

$

\Rightarrow A:B = \left( {2:3} \right) \times 4, \\

\Rightarrow A:B = 8:12 .....(i) \\

\Rightarrow B:C = \left( {4:7} \right) \times 3, \\

\Rightarrow B:C = 12:21 .....(ii) \\

$

Now, if we compare ratios $(i)$ and $(ii)$, the value of B in both the ratios is 12. So, we can easily combine both the ratios as:

$

\Rightarrow A:B = 8:12,B:C = 12:21 \\

\Rightarrow A:B:C = 8:12:21 \\

$

Therefore the required ratio $A:B:C$ is $8:12:21$.

Note: We can also solve the question by different method:

As we know that $A:B = 2:3$.

Let the value of A is $2x$ and that of B is $3x$.

Now it is given that \[B:C = 4:7\] and we have already taken the value of B as $3x$. So, we have:

$

\Rightarrow \dfrac{B}{C} = \dfrac{4}{7}, \\

\Rightarrow \dfrac{{3x}}{C} = \dfrac{4}{7}, \\

\Rightarrow C = \dfrac{{21}}{4}x \\

$

Thus we have:

$

\Rightarrow A = 2x,B = 3x{\text{ and }}C = \dfrac{{21}}{4}x, \\

\Rightarrow A:B:C = 2:3:\dfrac{{21}}{4}, \\

\Rightarrow A:B:C = 8:12:21 \\

$