Ratios

The given problem involves four types of ratios: duplicate, triplicate, sub duplicate, and sub triplicate ratios. Let's analyze each ratio individually and then determine the compound ratio.

Duplicate Ratio (4:5)

The duplicate ratio is formed by doubling both terms of a given ratio. In this case, the duplicate ratio of 4:5 would be 8:10, since 4 x 2 = 8 and 5 x 2 = 10.

Triplicate Ratio (1:3)

Similarly, the triplicate ratio is formed by tripling both terms of a given ratio. So, the triplicate ratio of 1:3 would be 3:9, as 1 x 3 = 3 and 3 x 3 = 9.

Sub Duplicate Ratio (81:256)

The sub duplicate ratio is formed by taking the square root of the first term and the square root of the second term of a given ratio. In this case, the sub duplicate ratio of 81:256 would be 9:16, since the square root of 81 is 9 and the square root of 256 is 16.

Sub Triplicate Ratio (125:512)

Lastly, the sub triplicate ratio is formed by taking the cube root of the first term and the cube root of the second term of a given ratio. So, the sub triplicate ratio of 125:512 would be 5:8, as the cube root of 125 is 5 and the cube root of 512 is 8.

Compound Ratio

To determine the compound ratio, we need to multiply all four ratios together. Let's multiply each corresponding term of the ratios calculated above:

Duplicate ratio: 8:10
Triplicate ratio: 3:9
Sub duplicate ratio: 9:16
Sub triplicate ratio: 5:8

Multiplying all the first terms together, we get: 8 x 3 x 9 x 5 = 1080
Multiplying all the second terms together, we get: 10 x 9 x 16 x 8 = 11520

Therefore, the compound ratio is 1080:11520, which can be simplified by dividing both terms by their greatest common divisor (GCD), which is 240.

Thus, the simplified compound ratio is 9:96.

In summary, the compound ratio of the given duplicate ratio, triplicate ratio, sub duplicate ratio, and sub triplicate ratio is 9:96.

Sub triplicate ratio of 125:512 is 5:8