1. A ratio is a statement of the number or quantity of one sort of thing to the number or quantity of another sort of thing. The things being talked about are generally described either as part of the ratio itself, or as a description of what the ratio is.
The given image has a number of possible descriptors, and we can write ratios between many of them. It may be easiest to talk about the numbers of squares of different colors (or shades of gray), but we can also talk about lengths, the number of lines, the size of the largest or smallest square or rectangle with a particular charcteristic, and so on.
Let's start with something simple. In the first row of the image are 1 white square, 1 black square, and 2 gray squares. We can write that as a ratio saying, "The ratio of square colors in the first row is white : black : gray = 1 : 1 : 2."
Then in the second row, the ratio of white : black : gray squares is 1 : 0 : 3.
In the whole image, the ratio of white to black to gray squares is 4 to 2 to 10.
Note that all the numbers in this last ratio are divisible by 2. This means we can remove a factor of 2 from each of them to reduce the ratio to simplest terms: white : black : gray squares = 2 : 1 : 5.
So far, we've just been counting squares. We can get a little more exotic and talk about areas, for example. Consider the largest all-gray square. It is the 2×2 square in the center of the figure, so has an area of 4 square units. The area of the entire figure is 4×4 = 16 square units. Then the ratio of the area of the largest gray square to the area of the whole figure is 4 : 16. After we reduce this to lowest terms, it is largest square gray area : whole area = 1 : 4.
We have shown above a couple of different kinds of ratios. One is the ratio of number of parts to number of parts. The other (the last one) is the ratio of a part to the whole. These can be converted back and forth. Consider the first ratio we talked about: white : black : gray = 1 : 1 : 2. The number of squares in the whole row can be found by adding the numbers in the ratio: 1+1+2 = 4. Then the ratio of the number of gray squares to the total number of squares in the first row is gray : total = 2 : 4 = 1 : 2, the latter form being in lowest terms.
2. Billy's ratio should be 3 apples : 1 pepper, or apples : peppers = 3 : 1. The order of numbers in the ratio must correspond to the order of objects being talked about.
