Ratio and Proportion:

The concept of ratio and proportion is an important topic in mathematics. In this topic, the relationship between two or more quantities is expressed in the form of a ratio. A ratio is a comparison of two numbers by division. It is often represented as a fraction or using the symbol ":".

Compounded Ratio:

A compounded ratio is formed by multiplying two or more ratios together. For example, if we have two ratios A:B and C:D, then the compounded ratio would be (A/B) x (C/D).

Given ratios:

The given ratios are 2:7 and 7:8. To find the duplicate ratio of 7:8, we need to first find the duplicate ratio of the first ratio 2:7.

Duplicate Ratio:

The duplicate ratio of a ratio A:B is (A^2):(B^2). Using this formula, we can find the duplicate ratio of 2:7 as (2^2):(7^2) which simplifies to 4:49.

Compounded Ratio of Duplicate Ratios:

Now, we need to find the compounded ratio of the duplicate ratios of the given ratios i.e. (4:49) x (49:64).

Simplifying this, we get (4x49):(49x64) which further simplifies to 1:16.

Therefore, the compounded ratio of 2:7 and the duplicate ratio of 7:8 is 1:16.

Conclusion:

In conclusion, to find the compounded ratio of two given ratios, we need to first find the duplicate ratio of each of them and then multiply the duplicate ratios together to get the compounded ratio.

7:32