Ratio, proportion and rates of change Study Guide

Ratio and proportion are fundamental mathematical concepts used to describe relationships between quantities. A ratio compares two or more quantities of the same type. For example, if you have 3 red balls and 2 blue balls, the ratio of red to blue balls is 3:2. Ratios can be simplified, much like fractions. If you have 6 red balls and 4 blue balls, the ratio is 6:4, which simplifies to 3:2 by dividing both sides by 2.

Proportion, on the other hand, deals with the equality of two ratios. If two quantities are in proportion, it means their ratios are equivalent. For instance, if the ratio of red to blue balls in one bag is 3:2, and in another bag, it's 6:4 (which simplifies to 3:2), then the quantities of red and blue balls in both bags are in proportion. This concept is crucial for scaling recipes, mixing materials, and solving problems involving direct and inverse relationships.

When solving ratio problems, common strategies include finding the 'total parts' (adding the numbers in the ratio), finding the value of 'one part' (dividing the total amount by the total parts), and then multiplying to find the specific quantity for each part. For example, if a sum of money is divided in the ratio 3:2, there are 3+2=5 parts. If the total sum is £100, then one part is £100/5 = £20. The first person gets 3 * £20 = £60 and the second gets 2 * £20 = £40.

Proportion problems often involve direct proportion (as one quantity increases, the other increases at the same rate) or inverse proportion (as one quantity increases, the other decreases). A key technique for direct proportion is to find the value for a single unit and then scale it up. For example, if 3 pencils cost £1.50, then 1 pencil costs £0.50, and 7 pencils would cost 7 * £0.50 = £3.50. Understanding these core methods allows you to effectively tackle a wide range of ratio and proportion problems.

Direct proportion describes a relationship between two quantities where if one quantity increases, the other quantity increases by the same factor, and if one quantity decreases, the other quantity decreases by the same factor. This means their ratio remains constant. We often express this relationship using the notation $y \propto x$, which reads as 'y is directly proportional to x'. To turn this proportionality into an equation, we introduce a constant of proportionality, usually denoted by 'k'. So, $y \propto x$ becomes $y = kx$. This constant 'k' tells us how much 'y' changes for every single unit change in 'x'. For example, if you buy more apples, the total cost increases directly with the number of apples. If 1 apple costs 50p, then 2 apples cost £1, 3 apples cost £1.50, and so on. Here, the cost is directly proportional to the number of apples, and the constant of proportionality 'k' would be 50p per apple. To find 'k', you just divide y by x ($k = y/x$). Once you have determined the constant of proportionality, you can use the equation $y = kx$ to find unknown values of either 'y' or 'x' given the other. This concept is fundamental in many real-world applications, from calculating costs based on quantity to understanding physical laws.