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What are ratios and proportions?
A ratio is a relationship between two or more quantities expressed as a fraction (3/5) or in this form 1:3 - which is read as one to three (the colon ':' is read as 'to').
A proportion is similar to a ratio, but usually has a one for the denominator. So, the ratio, 3:4 could also be written as the proportion, .75 to 1. Proportion also has a general meaning - a relationship where one quantity varies directly with another is called proportional.
How much food a family needs is proportional to the number of people in the family. As the number of people go up, the amount of food also increases. However, the number of carpets in a house is usually not proportional to the number of people. A larger family might or might not have more carpets, you can't tell.
An interesting fact about ratios and proportions.
You can generate the golden ratio on a hand held calculator by starting with any number. Use the 1/x key to get the inverse. Now add one to that.
Take your new answer and repeat the process, 1/x first, then add 1.
The intermediate values will get closer and closer to Φ.
How are ratios used in the real world?
Ratios are used whenever we want to calculate one quantity when we know some other (when they vary proportionally). For instance, concrete is mixed at the ratio of 1 part water (by weight) to 4 parts concrete mix. The 1:4 ratio holds true for a 5 gallon pail of concrete as much as a 20 gallon wheelbarrow.
This is common in cooking, when you have a recipe for one quantity and you want to make some other amount. When you cut all the amounts of ingredients in half, you are reducing the proportion of them all by half. This is true for all simple ratios, each part has to be multiplied (or divided) by the same number for the ratio to remain true.
So, if you have a ratio of 1:4 (as in the concrete example) you could convert it to the equivalent ratios- 2:8, 5:20 and so on, as long as you multiplied each part by the same number.
A basic problem with ratios.
The average grades in math class for a particular school district are given by the following proportion in this order - A:B,C,D - as 5:15. This means that for every 5 students that gets an A, 15 students will get a lower grade. If this holds for a class of 40 students, how many will get A's?
Since 5 + 15 equals 20, the ratio given is for every 20 students. That is, out of 20 students, 5 will get A's and 15 will get something else. Since 40 is twice 20, we only need to multiply our ratio by two.
This gives, 5:15 X 2 = 10:30. (10 + 30 = 40) So, out of 40 students, we would expect 10 to get A's.
Who invented ratios and proportions?
Ratios and proportions have been known since before recorded history. One particular one was explored by the ancient Greeks - the Golden Ratio.
This can be seen in the following line drawing, where the length of the line segment BC is in the same ratio to AB as AB is to the whole line.
Algebraically, the relationship is: (ab+bc)/ab = ab/bc and there is important enough in mathematics to get it's own symbol, the Greek letter phi, Φ. Phi is equal to about 1.618033... Phi is irrational, and the digits go on infinitely, never settling down into a predictable pattern.