Reviewing Representations of Ratios

Segway Sam is visiting his favorite supermarket. The free samples at the Steal-a-Deal market are incredible! Sam never seems to buy anything and has a tendency to go a bit overboard with the free samples. He decides to keep track of his calories while snacking using his new, fancy fitness watch. Reviewing representations of ratios will help Sam keep his free-sample eating-habits in check. Sam takes a good look at the cheese cubes. His fancy fitness watch tells him the caloric content of the cheese in the form of a ratio. 3 cheese cubes are 15 calories. We can write the ratio of cheese cubes to calories as 3 to 15 or in fraction form, 3 over 15. We could also think about calories to cheese cubes instead, giving us an associated ratio 15 to 3 or in fraction form, 15 over 3. We could also reduce these ratios to their unit rate: the ratios equivalent to these ones in which one of the terms is equal to one. Here we have that the unit rate of 3 over 15 is 1 over 5, or 1 cube per 5 calories. The unit rate of this associated ratio is then 5 over 1, or 5 calories per cube. He excitedly grabs 2 cheese cubes, giving him 2 times 5 calories, or 10 calories. Good deal! Segway Sam rolls on to his next target. Yo! Jackpot! Tater tots! Let’s take a look at the table given by Sam's fancy watch. 3 tots have 66 calories. Heavy tots! But wait a minute, are we sure this is a table of equivalent ratios? A table is a ratio table if every ratio in the table reduces to the same fraction in simplest form. We see that 3 over 66 reduces to 1 over 22. 6 over 132 also reduces to 1 over 22, the same with 9 over 198, and 30 over 660. This means that all of these ratios are equivalent ratios, and that this table is a ratio table. Sam grabs 3 tots, adding 66 to his calorie count, and Segway's off! Hot dog, what have we here? Tasty mini-dogs, right off the grill. The watch says that each dog is 12 calories. Sam's got his eye on four dogs though, how many calories is that? We can use a tape diagram to figure this out. We draw one square to represent one mini-dog, and we draw 12 squares to represent 12 calories. So each square represents 1, showing us visually that for every one mini-dog, there are 12 calories. How do we use this tape diagram to figure out how many calories four are? Since 1 dog to 12 calories is an equivalent ratio to 4 dogs to however many calories, we know that for every 4 mini-dogs there will be 12 times 4 calories. Which adds up to 48 calories. Hrm. That's a bit too many calories for Sam. If he takes 3 dogs, then by writing a 3 in every calorie square and summing them up gives us 36 calories. Sam can live with that. Sam still has room for dessert! And dessert is served! Piping hot chocolate chip cookies, ready to melt in your mouth! Sam's thinking of taking 20 cookies! We’re gonna need a graph to see how many cookies and calories he is willing to scarf down! The given table shows us that 3 cookies gives him 90 calories, and so on. We can plot the points on a graph. y', the dependent variable, is calories. That makes sense because Sam's calories depend on the number of cookies he eats. The number of cookies is 'x', the independent variable. A straight line through the origin means we’re graphing a set of equivalent ratios! The unit rate shows up on the graph as the 'y'-coordinate of the point (1,30). The unit rate is also known as the constant of proportionality, as it is the constant which every ratio is equal to. So in our case, the number of calories, 'y', over the number of cookies, 'x', is always equal to 30. Notice that this is also the slope of the line. The unit rate, or constant of proportionality, and the slope are always equal to each other. We multiply by 'x' on both sides to get the equation 'y' equals '30x'. Now we use our equation to calculate the calorie count for 20 cookies. Substituting 20 for 'x' gives us 'y' equals 30 times 20. y' equals 600. This means that 20 cookies gives Sam 600 calories of sweet excess! He stuffs 20 cookies in his pack, and he’s off and rolling! Man, knowing how to represent ratios with fractions, tables, tape diagrams, graphs and equations sure makes calorie counting easy. Wait a minute, Steal-A-Deal security doesn't seem to like Sam's sample-eating-habits! He's on Sam's trail and riding a turbo Segway! Well, I guess this is as good of a time as any for Sam to segue into...watch out!