Learning Outcomes
In this post the reader will be able to understand the following topics.
1- What is the Ratio?
2- Equivalent Ratios.
3- Common friction, numerator, denominator, antecedent and Consequent.
4- About proportion and it's two types
-Direct proportion and
-Indirect proportion.
5- Compound proportion
Ratio:
The numerical comparison between the two(2) quantities of the same kind is called ratio.
Now we make it clear with an example:
John and Noni bought a book for 60 dollars. John paid 40 dollars and Noni paid 20 dollars. Now we can compare the amount paid by John and Noni to find the relation between the two amount. This comparison can be done by the following two (2) ways or methods.
A) By finding their difference.
( 40 dollars - 20 dollars) = 20 dollars
The difference of 20 dollars does not state the comparison with the other amount. Therefore it is not proper way to show the comparison between two quantities.
B) By writing them into a fraction.
Amount paid by Noni / Amount paid by John = 20/40 = 1/2
This fraction is showing that for every 1 dollar of Noni ,John paid 2 dollars. This fraction shows the clear comparison between these two amounts. Therefore it is a better way to show the comparison between the two amounts. It can be written by putting a colon (:) between two quantities.
Noni John
1 : 2
And read as 1 is to 2 but it can be read as 2 is to 1 , because by changing the order of elements of a ratio the value is also changed. Therefore 1:2 and 2:1 are two different ratios.
In general we can write it as x:y is not equal to y:x . A ratio can also be written for more than two(2) quantities, i.e
3:4:5 or x:y:z etc
Antecedent and Consequent:
If we have a ratio 2:4 , than the first element of this ratio is called antecedent and the second element is called consequent.
2 is called Antecedent and 4 is known as consequent.
Example:
Write the quantities in the form of x:y:Z
500 grams, 800 grams and 1 kilogramme
500 : 800 : 1000 (1kg= 1000gms)
Each is divided by 100 than
5 : 8 : 10
Equivalent Ratios:
As we know that a ratio is another form of a common fraction. So there are same rule for finding the equivalent fractions can be used to achieve equivalent Ratios.
Rule : When both the elements of a ratio are multiplied or divide by same number the value of a ratio is not changed.
Example
Relationship between ratio and fraction:
In fact ratio is the simplest form of common fraction(کسر عام), in which numerator denotes Antecedent and the denominator denotes consequent.
A ratio has no unit. It is just a number that indicates how many times one quantity is greater than the other.
Proportion:
The relation (تعلق) of equality of two ratios is called proportion.
For example, 1:2 = 2:4 and 2:4 = 3:6 or so on..
Further we mak it clear with the following example:
John bought two cups at the rate of 25 dollar per cup. How much did John pay for two cups?
Cost of 1 cup = 25 dollars
Cost of 2 cups = 25×2 = 50
Now we can write it as
Cups cost
1:2 = 25:50
From the above two equivalent Ratios are indicating the relation between two different quantities of cups and their prices. We can say that the two ratios are proportional to each other which are denoted by a symbol :: and can be written as 1:2 :: 25:50 and read as
1 is to 2 is proportional(تناسب) to 25 is to 50.
In above proportion the first and fourth elements are called extreme of a proportion and the second and third elements are called mean of a proportion, It can be shown as below.
Now we solve some examples of proportion.
So if there are four quantities a,b,c and D are in proportion then these are written as a:b :: c:d
Actually it is a relationship between two Ratios a:b and c:d
Proportion is of two (2) kinds:
1- Direct proportion
2- Indirect proportion
Direct proportion:
Direct proportion is a relationship in which one quantity increases or decreases in a same proportion by increasing or decreasing the other quantity.
Example:
A washerman irons 2 shirts in 10 minutes. How many shirts can the washerman iron in one hour ?
Inverse proportion:
Inverse proportion is a relationship in which one quantity increases by decreasing the other quantity and same quality decreases by increasing the other quantity.
Example:
A project can be completed by 150 working in 40 days. But project manager call 30 more workers after 16 days. Now in how many days will the remaining work be completed?
Compound proportion:
The relationship between two (2) or more proportion is known as compound proportion.
We try to understand compound proportion with the help of examples.
Example:
If 35 workers dig 805 cm³ of earth in 5 hours, how much of the earth will 30 workers dig in 6 hours ?
Example:
8000 rupees are sufficient for s family which have 4 members for 40 days. For how many days 15000 rupees will be sufficient for a family of 5 members?
Example:
If 4200 men have sufficient(خوراک) food for 32 (thirty two) days at a rate of 12 hectogram per person ,how many men may leave(چھوڑ جانا) so that the same food be sufficient for 42 ( forty two) at a rate of 16 hectogram per man ( person).
Solution:
As the number of days increases the number of persons decreases. So it is an inverse proportion.
As the quantity of food increases the number of persons decreases. So it is also inverse proportion.
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