Ratio, proportion and rates of change Study Guide

A ratio compares the sizes of two or more quantities. For example, a ratio of 1:2 for apples to oranges means there is one apple for every two oranges. Equivalent ratios are ratios that represent the same proportional relationship, even though the numbers themselves might be different. Think of it like equivalent fractions: 1/2 is the same as 2/4, which is the same as 3/6. Similarly, the ratio 1:2 is equivalent to 2:4, which is equivalent to 3:6.

To find equivalent ratios, you multiply or divide all parts of the ratio by the same non-zero number. For a ratio a:b, if you multiply both 'a' and 'b' by a number 'k', you get the equivalent ratio (a*k):(b*k). If you divide both 'a' and 'b' by a common factor 'c', you get the equivalent ratio (a/c):(b/c). It's crucial that you apply the operation to *every* part of the ratio, not just one part. Simplifying a ratio means finding an equivalent ratio where the numbers are as small as possible, which involves dividing all parts by their greatest common factor (GCF). For instance, the ratio 6:9 can be simplified by dividing both numbers by their GCF, which is 3, to get 2:3. This simplified form is also known as the ratio in its simplest form or lowest terms. This concept is fundamental for comparing different ratios and solving problems involving proportional sharing.

Dividing a quantity in a given ratio means sharing it out proportionally according to the numbers in the ratio. For example, if you divide 20 apples between two people in the ratio 3:2, it means for every 3 apples one person gets, the other person gets 2 apples. The first step in this process is to find the total number of 'parts' in the ratio. This is done by adding up all the numbers in the ratio. In the example 3:2, the total number of parts would be 3 + 2 = 5 parts. Next, you need to find the value of one 'part'. This is achieved by dividing the total quantity (e.g., 20 apples) by the total number of parts (e.g., 5). So, 20 ÷ 5 = 4 apples per part. Finally, to find each individual share, you multiply the value of one part by each number in the original ratio. For the 3:2 ratio, the first person gets 3 × 4 = 12 apples, and the second person gets 2 × 4 = 8 apples. You can always check your answer by adding up the individual shares; they should sum to the original total quantity (12 + 8 = 20 apples). This method is applicable to any quantity and any ratio, whether it involves two or more parts.