The primary relationship between measurement and ratio/proportion is in converting units.
For example converting from American standard units to metric requires proportions. To change from inches to centimeters we use the fact that there are 2.54cm per inch.
We also use conversions within a system. For example, converting feet to yards or miles or converting centiliters to liters.
Another connection is using similarity to figure out lengths and angles. This is especially important in scale drawings such as maps and blueprints.
Measurement is related to ratio and proportion in real life in many ways. Here are three types of examples:
First consider conversions. In one example if you compare inches to feet you are looking at a 12:1 ratio and you could solve for the length in feet of an 18 inch model using a proportion 12/1 = 18/x
In my favorite example changes in temperature are proportional at the rate of 5°C increase for every 9°F.
Secondly consider rates, a special type of ratio, often have to do with measurement Miles/hr, Dollars/ft², Miles/gallon
Thirdly there are special cases like scale models, maps, and similar figures where a scale factor exists between proportional objects.
1) A ratio is expressed as the relation between two real numbers. example would be 4:2 Take not that 2^2 : 4 is also a ratio, because 2^2 = 4.
2) Ratio can also be a proportion, because it is desrcibing the relation of two numbers. Example, what is the proportion of the width and height, if the ratio of the W to H is 3:2 the proportion would be the same thing.
3) Now when you talk about proportions, how do you know how long each side is? You have to MEASURE it, and thats their connection.