If A:B = 1 : 3, B:C = 4 : 3 and C : D = 6 : 7, then A:B:C:D is

Correct option is A

Given:
A:B = 1:3
B:C = 4:3
C:D = 6:7
Formula Used:
To find the combined ratio A : B : C : D, express all ratios in terms of a common value.
We make the common terms B and C equal in all ratios by taking their LCM.
Solution:
Make B equal in A:B and B:C:
A:B = 1:3
B:C = 4:3
LCM of B = 3 and B = 4 is 12
Convert both ratios:
A:B = (1 \times 4) : (3 \times 4) = 4:12
B:C = (4 \times 3) : (3 \times 3) = 12:9
Now, A:B:C = 4:12:9.
Make C equal in B:C and C:D:
B:C = 12:9
C:D = 6:7
LCM of C = 9 and C = 6 is 18
Convert both ratios:
B:C = (12 \times 2) : (9 \times 2) = 24:18
C:D = (6 \times 3) : (7 \times 3) = 18:21
Now, B:C:D = 24:18:21.
From A:B:C = 4:12:9 and B:C:D = 24:18:21, ensure all values are in common terms:
Since B = 12 in the first ratio and  B = 24in the second, multiply the first ratio by 2:
​$$A:B:C = (4 \times 2) : (12 \times 2) : (9 \times 2) = 8:24:18$$​​
Now, combining A:B:C:D = 8:24:18:21.
Thus, A:B:C:D = 8:24:18:21.

Alternate Solution:

$$\begin{array}{cccc}\mathbf{A} & \mathbf{:} & \mathbf{B} & \mathbf{:} & \mathbf{C} & \mathbf{:} & \mathbf{D} \\\hline\mathbf{1} & \mathbf{:} & \mathbf{3} & \mathbf{:} &{3} & \mathbf{:} & {3} \\{4} & \mathbf{:} & \mathbf{4} & \mathbf{:} & \mathbf{3} & \mathbf{:} & {3} \\{6} & \mathbf{:} & {6} & \mathbf{:} & \mathbf{6} & \mathbf{:} & \mathbf{7} \\\hline{24} & \mathbf{:} & {72} & \mathbf{:} &{54} & \mathbf{:} &{63} \\\end{array}$$​
Simplify the final ratio, we get

A: B: C: D = 8 : 24: 18 : 21