Two ratios that are equal are said to be in proportion.
For example the ratio 6:8 is 3: 4, and the ratio 12: 16 is also 3: 4.
So, the two ratios, 6: 8 and 12: 16 are said to be in proportion.
We say “the ratio of 6 to 8 is same as that of 12 to 16”, or in short as:
“6 is to 8 as 12 is to 16”
In general, four different numbers a, b, c and d are in proportion if
“the ratio a to b is same as the ratio of c to d”
and it is denoted as
a: b :: c : d
(and read as “a is to be is same as c is to d”)
For example, 2 is to 3 as 4 is to 6 is denoted as 2: 3 :: 4: 6
In the proportion a: b :: c : d,
a and d are called extremes and b and c are called means.
And a Very Important Rule:
For example in the proportion 2: 3 :: 4: 6,
Product of Extremes 2 and 6 is 12 and product of means 3 and 4 is also 12.
There is another kind of proportion.
Three numbers a, b and c are said to be in continued proportion if
a : b :: b : c
In this proportion,
a is called the first proportional,
b is called the Mean proportional,
c is called the third proportional.
Again, applying the rule of “product of means is equal to product of extremes” in the proportion a: b :: b : c, we have
b2 = a × c,
so b = √ (a × c)
The mean proportional of two numbers is the square root of their product.
There are two types of Proportions:
Two quantities are said to be in Direct Proportion, if a change in one brings a similar change in the other.
Change includes increase and decrease.
An increase or decrease in one causes a corresponding increase or decrease in the other.
A box of pencils costs $10, then 2 same boxes will cost $20.
Here, the two quantities are number of boxes of pencils and their cost.
As the number (of boxes of pencils) “N”increases, then price “P”also increases.
So, there is a direct proportion between Number and Price. It is denoted as
N α P
Two quantities are in Indirect Proportion, if an increase or decrease in one brings an opposite change in the other quantity.
An amount of $120 is to be equally divided between 3 men. Each one will get $40
If one man leaves the group, then the remaining 2 men will receive $60 each.
Fewer men leaves greater share for each.
So, there is an indirect proportion between Number and Share (of each)
N α 1/A
N is number of persons and A is the amount of each person’s share
A few more examples:
To reach your school earlier, you will have to ride faster.
So, speed and time are indirectly related.
Greater the speed, lesser is the time taken to cover a same distance.
3 men, all of same efficiencies, can build a wall in 1 hour.
Then, 1 man alone will take 3 hours
Again, between Number of men and time taken to finish a same work, there is an indirect proportion.
To score high, you need to put greater efforts,
So, Result and Effort are directly related (Agree/Disagree?)