The Ratio Compound

The given problem involves different types of ratios - duplicate ratio, triplicate ratio, and sub-duplicate ratio. We need to find the ratio compound of these ratios.

Duplicate Ratio:
The duplicate ratio of a given ratio is obtained by multiplying both terms of the ratio by the same number. For example, the duplicate ratio of the ratio 4:5 would be 8:10.

Triplicate Ratio:
The triplicate ratio of a given ratio is obtained by multiplying both terms of the ratio by the square of a number. For example, the triplicate ratio of the ratio 1:3 would be 1^2:3^2, which simplifies to 1:9.

Sub-Duplicate Ratio:
The sub-duplicate ratio of a given ratio is obtained by taking the square root of both terms of the ratio. For example, the sub-duplicate ratio of the ratio 81:256 would be √81:√256, which simplifies to 9:16.

Finding the Ratio Compound:
To find the ratio compound, we need to combine the duplicate ratio, triplicate ratio, and sub-duplicate ratio of the given ratios.

Let's take the given ratios one by one and find their respective ratio compounds:

1. Duplicate Ratio of 4:5:
Duplicate ratio = 4 x 2:5 x 2 = 8:10

2. Triplicate Ratio of 1:3:
Triplicate ratio = 1 x 1^2:3 x 1^2 = 1:9

3. Sub-Duplicate Ratio of 81:256:
Sub-duplicate ratio = √81:√256 = 9:16

4. Sub-Duplicate Ratio of 125:512:
Sub-duplicate ratio = √125:√512 = 5:16

Now, let's combine these ratio compounds:

Ratio compound = (8:10) x (1:9) x (9:16) x (5:16)

To simplify this compound ratio, we multiply the numerators together and the denominators together:

Ratio compound = (8 x 1 x 9 x 5):(10 x 9 x 16 x 16)

Calculating the product, we get:

Ratio compound = 360:2,880

This ratio compound can be further simplified by dividing both terms by their greatest common divisor:

Ratio compound = 5:40

Therefore, the ratio compound of the given ratios is 5:40.

Summary:
To find the ratio compound, we combined the duplicate ratio, triplicate ratio, and sub-duplicate ratio of the given ratios. The final ratio compound obtained was 5:40.