A scale factor expresses the relationship between two lengths. It can be expressed as a ratio or a positive number. Let's look at some examples. If the original image is 4 inches, and the scale drawing is 8 inches, we can say that the scale factor is 4:8, but we must simplify that ratio, making it 1:2. We can also write the scale factor as a number by dividing 8 by 4 for a scale factor of 2.
That example is pretty straightforward, but let's look at something a bit more complicated. Let's say we have an original image that is 4 inches and a scale drawing that is 9.5 inches, and we want to find the scale factor. We cannot write the ratio 4:9.5, because ratios do not allow decimals or fractions. We must simplify the ratio. This involves getting rid of the decimal. To do this, we multiply 9.5 by 2 and get 19. What we do to one element of the ratio, we must do to the other, so we multiply 4 by 2 for 8. Our ratio looks like this: 8:19. Because 19 is a prime number, we cannot simplify further. This scale factor of 8:19, then, tells us that for every 8 inches in the original, the scale drawing will have 19 inches. We can also express the scale factor as 2.375 (19 divided by 8).
We'll try one more example. We have an original image that is 1.25 inches and a scale drawing that is 4 inches. Again, we cannot write a ratio with decimals, so we can't leave our scale factor as 1.25:4. We must simplify. This time, we must get rid of the decimal by multiplying each side by 4, so we have 5:16. We cannot simplify this ratio any further, but we can express it by the number 3.2 (16 divided by 5).