Ratio, proportion and rates of change Study Guide

Dividing a quantity in a given ratio means splitting that quantity into parts, where the size of each part is proportional to the numbers in the ratio. For example, if you divide £100 in the ratio 2:3, it means for every £2 one person gets, the other person gets £3. The first step in this process is to find the total number of 'parts' in the ratio. You do this by adding together all the numbers in the ratio. In the 2:3 example, the total parts would be 2 + 3 = 5 parts. Next, you need to find the value of one part. This is achieved by dividing the total quantity by the total number of parts. So, for £100 divided into 5 parts, one part would be £100 / 5 = £20. Finally, to find the amount corresponding to each number in the ratio, you multiply the value of one part by each number in the ratio. For the 2:3 example, the first share would be 2 * £20 = £40, and the second share would be 3 * £20 = £60. You can check your answer by adding the individual shares together: £40 + £60 = £100, which is the original total quantity. This method works for ratios with any number of parts, such as 1:2:3 or 5:1. Understanding this method is crucial for solving many real-world problems involving fair distribution or scaling ingredients in recipes.

Ratios and fractions are fundamental mathematical concepts used to compare quantities and represent parts of a whole. A ratio compares two or more quantities of the same unit. For example, if there are 3 red apples and 2 green apples, the ratio of red apples to green apples is 3:2. This means for every 3 red apples, there are 2 green apples. Ratios can be simplified by dividing all parts by their greatest common factor, much like fractions. For example, a ratio of 6:4 simplifies to 3:2. Ratios can also be expressed in the form of 1:n or n:1, which can be useful for comparing. For example, a ratio of 3:2 can be written as 1:2/3 or 1.5:1.

A fraction represents a part of a whole. It consists of a numerator (the top number) and a denominator (the bottom number). The denominator indicates the total number of equal parts the whole is divided into, and the numerator indicates how many of those parts are being considered. For instance, in the fraction 3/5, the whole is divided into 5 equal parts, and we are considering 3 of them. Fractions can be converted to decimals or percentages, and vice versa. It is crucial to understand equivalent fractions, where different fractions represent the same value (e.g., 1/2 = 2/4 = 5/10).

There's a close relationship between ratios and fractions. If a ratio of A to B is A:B, then the fraction of the total that A represents is A/(A+B), and the fraction of the total that B represents is B/(A+B). For example, if the ratio of boys to girls in a class is 2:3, then the fraction of boys is 2/(2+3) = 2/5, and the fraction of girls is 3/(2+3) = 3/5. Mastering these initial concepts lays the groundwork for more complex problems involving proportions, percentages, and algebraic expressions of ratios.