Ratio and Proportion

Ratio and proportion are two important concepts in mathematics. Ratio is a comparison between two quantities of the same kind. Proportion is the equality of two ratios.

Indices

Indices are a shorthand way of writing repeated multiplication. An index is also known as an exponent or power.

Logarithms

Logarithms are the inverse of exponential functions. They are used to solve equations involving exponents.

How are they related?

1. Using logarithms to solve equations involving ratios and proportions
2. Expressing ratios and proportions in terms of indices and vice versa
3. Using logarithms to simplify expressions involving indices
4. Using the laws of indices and logarithms to simplify expressions involving ratios and proportions

Example:

If the ratio of the length to the breadth of a rectangle is 3:2 and the area of the rectangle is 240 sq. units, find its length and breadth.

Solution:

Let the length of the rectangle be 3x and the breadth be 2x.

Then, the area of the rectangle = 3x x 2x = 6x^2

Given, the area of the rectangle is 240 sq. units.

Therefore, 6x^2 = 240

x^2 = 40

x = √40

x = 2√10

Therefore, the length of the rectangle = 3x = 6√10 units

And the breadth of the rectangle = 2x = 4√10 units.

Conclusion

Ratio and proportion, indices, and logarithms are closely related concepts in mathematics. They are used to solve a variety of problems involving equations, ratios, and proportions. Understanding these concepts is essential for success in mathematics.