Ratio and proportion

Understanding Ratio and Proportion

• A ratio is a method of expressing a relationship between two or more quantities. It can be written in a number of ways, including ‘as a to b’, ‘as a:b’, or ‘as a/b’.
• A proportion is an equation stating that two ratios are equivalent - for example, if a:b = c:d, then a is to b as c is to d.
• The terms of a ratio can be simplified in the same way as a fraction, by dividing both sides by their highest common factor.

Working with Ratios

• Sharing in a given ratio: Split the total quantity into parts as per the given ratio, then distribute the total quantity according to these parts.
• Comparing ratios: Before comparing, it’s crucial to make sure that the ratios are in simplest form. It might be necessary to convert the ratios so that they have a common term.
• Combining ratios: When two ratios are equal to the same thing, they can be set equal to each other. This allows us to combine the ratios into a single ratio.

Direct and Inverse Proportions

• Direct proportion means as one quantity increases, another quantity also increases at the same rate, or if one quantity decreases, the other decreases at the same rate. The general form of direct proportion is y = kx, where k is the constant of proportionality.
• Inverse proportion means as one quantity increases, the other decreases at the same rate. The general form of inverse proportion is y = k/x or xy=k, where k is the constant of proportionality.

Key Concepts in Ratio and Proportion

• Know the difference between a ratio and a fraction. A fraction is a part of a whole, while a ratio is a comparison or relationship between two whole quantities.
• Understand the concept of ‘part-to-part ratio’ (for example, boys to girls) and ‘part-to-whole ratio’ (for example, boys to total students), which are used in different ways in ratio problems.
• Learn the ‘unitary method’ to solve ratio and proportion problems, which involves converting quantities to a rate per one unit.

Common Pitfalls in Ratio and Proportion

• It’s easy to confuse terms in a proportion. Always remember that terms that are diagonally opposite in a proportion are equal.
• When solving problems using proportional relationships it is important to keep units consistent. Misapplication of units can lead to wrong answers.
• Practice is key when it comes to ratio and proportion, particularly when working with more complex problems involving direct and inverse proportion. Always remember to check your work.