Equivalent ratios

Daniel and Rachel train by running laps of a track. Every morning, Daniel runs 3 laps. And every morning, Rachel runs 2. You already know how to write that. Daniel's laps to Rachel's. The ratio is 3 to 2. That's the pattern. For every 3 laps Daniel runs, Rachel runs 2. Keep your eye on that. The mornings will add up, but the pattern won't move. Morning 2. Daniel runs 3 more. Rachel runs 2 more. Now count them. Daniel has run 6 laps, and Rachel has run 4. So the ratio of Daniel's laps to Rachel's is 6 to 4. But look at the mornings. Each one is still 3 for every 2. The pattern never changed. So 6 to 4 and 3 to 2 are the same comparison. Same ratio, written with different numbers. Here's what happened to the numbers. 6 divided by 2 is 3. And 4 divided by 2 is 2. Both numbers, divided by the same number. That's the move that keeps a ratio the same. And it works the other way. Multiply both by 2, and 3 to 2 turns back into 6 to 4. Now look at the numbers 3 and 2. No number bigger than 1 divides them both. There's no more common factor between 3 and 2. So the numbers can't get any smaller. 3 to 2 is the simplest form. The smallest whole numbers this ratio can be written with. When a question asks for simplest form, that's the job. Divide both numbers by the same number, again and again, until you can't. Now they keep training. Morning after morning, same pattern. One day, Rachel counts up. She has run 8 laps. We haven't drawn Daniel's laps yet. But we don't need to see them. The ratio is still 3 to 2. Your turn. 3 to 2, is the same as something, to 8. What's the missing number? Try it first. Pause if you like. Start with the number you can see. Rachel's side. 2 became 8. That's times 4. So Daniel's number is times 4 as well. Both numbers, multiplied by the same number. 3 times 4 is 12. The missing number is 12. Daniel has run 12 laps. And there they are. 4 mornings, 3 laps each. 3 to 2. 6 to 4. 12 to 8. Three ways to write the same comparison. Multiply both numbers by the same number, or divide both by the same number. The ratio doesn't change.