What is the best formula for solving ratio and proportion problems?
Ratio and proportion are a section that is studied in mathematics in school. Well before taking a plunge into the description of ratio and proportion along with its significance let’s first know why do we even need that and what is the role of ratio and proportion in our daily life. Well, it sometimes appears that it is unimaginable to determine how we can use the fundamental principles of mathematics in day-to-day life situations. The concept of ratio is very much acquainted to everyone, but one may find trouble to define it in a mathematical way of definition. An example of cooking where we need a ratio of various ingredients in a proper form to get the best taste out of the dish can be achieved. Here one can estimate the quantity and accordingly can scale the ingredients on a large or small level of food preparation. Similarly, one may also have gone through the notion of a ratio. For example, one may recognize a kid to be smaller than the normal full-grown adult in the same way that the same adult is smaller than a professional bodybuilder, even though the three sizes of individuals are different. In math and statistics, proportion, percentage, ratio, etc. questions thrive. Thus, in general ratios and proportions are ways to describe the relationships between and among several quantities. In a simple term, we may define ratio to be an ordered form of pair of numbers a and b, written a/b where b cannot be equal to zero whereas proportion can be defined as a mathematical equation in which two ratios are set equal to each other. We here are going to break the best way to solve a ratio and proportion. In many situations, the comparison between two different quantities is efficient via division. The ratio is nothing but a simplified form of two quantities and gives us the relation of how many times one quantity is equal to the other quantity. The ratio can be represented in three forms i.e., a to b, a: b, a/b.
It should always be noted that ratio and proportion do exist in quantities of the same attributes and properties and also their units have to be the same. Coming to the proportion as stated above it is equality of the two fractions or ratios. For example, the time taken by a car to cover 50km in one hour is equal to the time taken by car to travel 150 km in 3 hours. In such a way, we can say that 50km/hr=150km/3hrs. We can categorize the proportion into the following types:
- Direct proportion (increase in one quantity leads to increase in other quantity)
- Inverse proportion (increase in one quantity leads to decrease in other quantity)
- Continued proportion (in order to determine the continued proportion for given ratios, it is needed to convert to a single term by LCM,
Say for a: b and c: d, b, and c LCM would be bc. Thus multiplying the first ratio by c and second by b, we have continued proportion to be ca:bc bd.
Assume that, we have two quantities and we need to find the ratio of these two, Here the formula for ratio is defined as;
a: b ⇒ a/b
Here a and b could be any two quantities.
a: b ⇒ a/b
Here a and b could be any two quantities.
Example: In ratio 3:7, is represented by 3/7, where 3 is antecedent and 7 is consequent.
If we happen to multiply and divide each term of the ratio by the same number, it doesn’t affect the ratio.
Example: 3:7 = 6:14
Now, let us assume that, in proportion, the two ratios are a:b & c:d. Here it should be noted that the two terms ‘b’ and ‘c’ are called ‘means,’ whereas the terms ‘a’ and ‘d’ are called ‘extremes.’
a/b=c/d or a:b :: c:d
Now, we will have a look at one more example of a number of students in a classroom. The first ratio of the number of girls to boys is 3:5 while that of the other is 4:8. Here the proportion can be written as:
3 : 5 :: 4 : 8 or 3/5 = 4/8
It should be noted that here, 3 & 8 are the extremes, while 5 & 4 are the means.
Important properties of Proportion
Let us look at the important properties of proportion:
- If a : b = c : d, then we find that a + c : b + d
- If a : b = c : d, then we find that a – c : b – d
- If a : b = c : d, then we find that a – b : b = c – d : d
- If a : b = c : d, then we find that a + b : b = c+d : d
- If a : b = c : d, then we find that a : c = b: d
- If a : b = c : d, then we find that b : a = d : c
- If a : b = c : d, then we find that a + b : a – b = c + d : c – d
Fourth, third, and mean proportion
If a : b = c : d, then:
- d is known as the fourth proportional to a, b, c.
- c is known as the third proportion to a and b.
- The mean proportion between a and b is √(ab).
Now below are some tricks and shortcuts through which we can simplify our ratio and proportion situations.
- If u/v = x/y, then we can say uy = vx
- If u/v = x/y, then we can say v/u = y/x
- If u/v = x/y, then we can say u/x = v/y
- If u/v = x/y, then we can say (u+v)/v = (x+y)/y
- If u/v = x/y, then we can say (u-v)/v = (x-y)/y
- If u/v = x/y, then we can say (u+v)/ (u-v) = (x+y)/(x-y)
- If a/(b+c) = b/(c+a) = c/(a+b) and a+b+ c ≠0, then we can say a =b = c
Attempt this question: Write True ( T ) or False ( F ) against each of the following statements :(a) 16 : 24 :: 20 : 30 (b) 21: 6 :: 35 : 10 (c) 12 : 18 :: 28 : 12(d) 8 : 9 :: 24 : 27 (e) 5.2 : 3.9 :: 3 : 4 (f) 0.9 : 0.36 :: 10 : 4
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