Short Answer

A duplicate ratio is a ratio formed by squaring both terms of a given ratio. If the original ratio is a:b, then the duplicate ratio is a²:b². For example, if the ratio is 2:3, its duplicate ratio is 4:9.

Duplicate ratio helps in comparing squared values of quantities. It is useful in mathematical problems where quantities are related in squared form.

Detailed Explanation:

Duplicate ratio

Duplicate ratio is a special type of ratio in mathematics. It is obtained by squaring both terms of a given ratio. This means we multiply each term of the ratio by itself. If the original ratio is a:b, then the duplicate ratio becomes a²:b².

Meaning of duplicate ratio

The word “duplicate” means double in power, not in value. In duplicate ratio, we do not simply multiply the terms by 2, but we square them. For example, if the ratio is 3:4, then the duplicate ratio is 3²:4², which becomes 9:16.

This shows that the duplicate ratio represents the square relationship of the original quantities.

Method of finding duplicate ratio

To find the duplicate ratio, we follow a simple method. First, take the given ratio. Second, square both numbers. Third, write the new values as a ratio.

For example, if the ratio is 5:6, then the duplicate ratio is 5²:6² = 25:36. This method is easy and direct.

Simplification of duplicate ratio

After finding the duplicate ratio, it can be simplified if both numbers have a common factor. For example, if the duplicate ratio is 16:36, it can be simplified to 4:9 by dividing both numbers by 4.

Simplifying helps in expressing the ratio in its simplest form.

Importance of duplicate ratio

Duplicate ratio is important in problems where squared relationships are involved. It helps in comparing areas, squares, and other quantities that depend on the square of values. It is also useful in advanced mathematics.

Use in mathematics

In mathematics, duplicate ratio is used in geometry and algebra. For example, when comparing areas of similar shapes, the ratio of their areas is the duplicate ratio of the ratio of their sides.

Real life applications

In real life, duplicate ratio can be used in situations involving area and scaling. For example, if the length of an object is doubled, its area increases according to the square of the ratio. This concept is useful in design, construction, and measurements.

Common mistakes

A common mistake is multiplying the terms by 2 instead of squaring them. Another mistake is forgetting to square both terms. It is important to apply the correct rule to get the right result.

Conclusion

Duplicate ratio is formed by squaring both terms of a given ratio. It represents the square relationship between quantities and is useful in mathematical and real-life problems involving areas and scaling.