1. **Problem Statement:** We are asked to draw sets of counters representing given ratios and find how many different ways these can be done. 2. **Understanding Ratios:** A ratio compares quantities of different items. For example, red:blue = 5:6 means for every 5 red counters, there are 6 blue counters. 3. **Part a) i) red:blue = 5:6** - We can represent this ratio by drawing 5 red counters and 6 blue counters. - Different ways to do this involve scaling the ratio by multiplying both parts by the same number. - For example, 10 red and 12 blue (scale by 2), 15 red and 18 blue (scale by 3), etc. 4. **Part a) ii) blue:green = 3:4** - Draw 3 blue counters and 4 green counters. - Similarly, scale by any positive integer to get other ways: 6 blue and 8 green, 9 blue and 12 green, etc. 5. **Part a) iii) red:blue:green = 10:12:16** - Draw 10 red, 12 blue, and 16 green counters. - Scaling by integers gives other sets: 20:24:32, 30:36:48, etc. 6. **How many different ways?** - Infinite ways exist by scaling ratios by any positive integer. 7. **Part b) Draw one set of counters that satisfies all 3 ratios:** - We want red:blue = 5:6, blue:green = 3:4, and red:blue:green = 10:12:16. - Check if these ratios are consistent. - From red:blue = 5:6 and blue:green = 3:4, find combined red:blue:green. - Scale red:blue = 5:6 to match blue in blue:green = 3:4. - Blue in first ratio is 6, in second is 3, so multiply second ratio by 2: blue:green = 6:8. - Now red:blue:green = 5:6:8. - Given red:blue:green = 10:12:16, which is exactly double 5:6:8. - So the consistent combined ratio is 10:12:16. - Draw 10 red, 12 blue, and 16 green counters. - Different ways to do this are scaling by integers: 20:24:32, 30:36:48, etc. **Final answer:** - There are infinite ways to represent each ratio by scaling. - The combined ratio satisfying all three is 10:12:16 and its multiples.