Lesson Introduction to Proportions

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This Lesson (Introduction to Proportions) was created by by Shruti_Mishra(0) About Me : View Source, Show
About Shruti_Mishra: I am a maths graduate from India and am currently persuing masters in Operations Research.
When we talk of ratio, it represents the relationship between two numbers or quantities.
Proportion is an extension of ratios. A Proportion is a statement which shows that two ratios are equal. In other words, when the ratio of two terms is equal to the ratio of two other terms, then these four terms are said to be in proportion.

For example:
4%2F6+=+2%2F3, 3%2F9+=+2%2F6, 14%2F18=+7%2F9 are all in proportion.

Notation:
If a%2Fb+=+c%2Fd, then a, b, c, d are in proportion. a, b, c, d, are called the first, second, third and fourth proportionals respectively. The terms ‘a’ and ‘d’ are called the extremes, while ‘b’ and ’c’ are called the means.

Property
Product of Extremes = Product of Means

(This can be seen by cross multiplication also)

Solving Proportions
If one of the four numbers in a proportion is unknown, we can find the unknown quantity by equating the cross products, i.e. Product of Extremes = Product of Means

Example
Q. If x/90 =2/3, find x.
Solution:
Using 'Product of Extremes = Product of Means'
x * 3 = 2* 90 = 180
=> x= 60.
Therefore x = 60


Continued Proportion:
When a%2Fb+=+b%2Fc, then a, b, c are said to be in continued proportion and b is called the geometric mean or mean proportional between ‘a’ and ‘c’.
We get:
b%5E2+=+a+%2A+c
=>b+=+sqrt%28ac%29

Similarly, if a%2Fb+=+b%2Fc+=+c%2Fd+=+d%2Fe=... , then a, b, c, d, e are said to be in continued proportion. From here we can easily find out the value of a/d = (a/b)* ( b/c) * ( c/d).


Example:
Q. If a/b = 2/3; b/c = 4/5; c/d = 7/9, find a/d.
Solution:
a/d = (a/b)*(b/c)*(c/d)=(2/3)*(4/5)*(7/9) = 56/135

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