Visualizing the Percent of a Quantity

Every night, three brother raccoons go out grocery shopping. Butch, Bailey, and Bonsai are very picky eaters. They have strict rules about what they will and will not eat. To correctly sort their, uh, groceries, the brothers use models and diagrams to visualize the percent of a quantity. After their shopping spree, the brothers dump out their bounty and start sorting! Butch has a sweet tooth and only eats sugary foods. He collected 60 items total, and 18 of these items are sweet. What percent of his items will he eat? This problem contains three components: the whole, the part, and the percent. If you know two out of three of these components, you can always find the third. The 60 items Butch collected are "the whole" in this problem. That's the total number of items. 18 of these items are sweet. Those 18 items are "the part". So what is a percent? A percent is what the part would be if the whole were 100. It's an equivalent ratio to part over whole, where the numerator is the percent and the denominator is 100. In this problem, the part and whole are known, and the percent is unknown. Let's use a double number line to visualize and find the percent. A double number line shows a series of pairs that all have the same ratio. The bottom number line will represent the "whole" and the top number line will represent the "part". We want to know the part when the whole is 100. That will give us our percent. First we count on the bottom number line, from 0 to 100, by an increment that will have 60, as well as 100, as a multiple. Let's count the bottom number line by increments of 20, since 60, as well as 100, are multiples of 20. We could have also chosen to count by increments of 10, or 5, as well. We know that when the whole is equal to 60 that the part is equal to 18, and that when the whole is equal to 0 that the part is also equal to 0. We must count on the top number line by an increment which has 18 as a multiple. Since there are three intervals before 18, we divide 18 by 3 to get that each interval should increase by 6 on the top number line. So we count on the top number line by increments of 6. Looks like there are 30 parts per 100, or 30 percent. That means 30 percent of Butch's items are sweet. Sugar overload! The second brother, Bailey, only eats red foods. Bailey collected 40 items, and 30 percent of them are red. Again, we have two components, and we can find the third. We know the whole: 40 items. And we know the percent: 30 percent. What's missing? The part. The part in this problem is how many of Bailey's items are red. We can visualize this problem using a 10 by 10 grid that represents the whole. Let's divide up our 40 items evenly among the 100 squares in this grid. We can determine how many items one square gets by dividing 40 by 100. Doing this, we see that one square gets .4 of an item. As there are 100 squares in this grid, 30 squares of the grid represent 30 parts per 100, or 30 percent. Now we can find the part. How many items are there in total in these 30 squares? We see that adding the items in these 30 squares together, or calculating 30 times .4, gives us 12 items total. So now we know that 12 out of 40 of Bailey's items are red. Bonsai is a vegan. He won't eat any animal products, not even leather shoes! 70% of what Bonsai collected is vegan, and he has 14 vegan items. In this problem, we know the percent, 70%, and the part, 14 vegan items. Looks like Bonsai forgot how many items he collected in the first place! Don't worry, Bonsai! Because we know the values of two of the quantities, we can find the third, the whole. Let's use a tape diagram to visualize this problem. We draw a piece of tape, and we would like to split it up into evenly sized pieces. In general we can split up our piece of tape anyway we choose, as long as it is split into evenly sized pieces. In our case, let's choose to split our piece of tape into 10 pieces, each piece representing 10 percent. Then we shade in 7 pieces of our tape to represent 70 percent. This shaded section of the tape represents Bonsai's 14 vegan items. So how many items does each piece of tape represent? 14 items divided evenly into 7 pieces means 2 items per piece. Counting up all the items in each piece, we see that all 10 pieces together represent a total of 20 items. That's the whole. So Bonsai collected a total of 20 items! We have seen that percent problems deal with three quantities: the part, the whole, and the percent. And if we know two of the three quantities, then we can solve for the last. We can use visual tools, like double number lines, 10 by 10 grids, and tape diagrams to help visualize and solve percent problems. Use the one that makes the most sense to you! Oh, Looks like Bonsai collected a red, vegan, sweet caramel apple! Oh no...