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Ratio and proportion concepts
IACE
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1. Definition of Ratio
When we need to express one number as a fraction of another, we utilise ratios. If we have two numbers, say x and y, the ratio of x to y is calculated as xy and expressed as x:y. The first term in the ratio is known as the antecedent, while the second term is known as the consequent.
2. Definition of proportion
Proportion is an equation that defines that two specified ratios are equivalent. The proportion expresses the equivalence of two fractions or ratios, i.e. Equivalent Ratios. The sign (::) or equal to (=) represents proportions. There are different types of proportions such as

Direct proportion
The direct proportion defines the relationship between two quantities in which when one number rises, the other quantity increases as well. Similarly, if one quantity falls, the other falls as well. As a result, if “a” and “b” are two numbers, the direction percentage is expressed as ab.

Inverse proportion
The inverse proportion explains the relationship between two quantities in which an increase in one causes a reduction in the other. Similarly, a reduction in one item results in a rise in the other amount. As a result, a(1/b) represents the inverse proportion of two quantities, say “a” and “b”.

Continued proportion
Consider the following two ratios: a b and c:d.
These means are then converted to a single term/number to get the continuous proportion for the two specified ratio terms. In general, this is the LCM of means.
The LCM of b and c for the given ratio is bc.
As a result of multiplying the first ratio by c and the second ratio by b, we get
The first ratio is ca: bc.
The second ratio is bc: bd.
As a result, the continuous percentage can be expressed as ca:bc: bd
3. Formula of Ratio and Proportion.
3.1 Ratio: Assume we have two quantities (or two numbers or two entities) and we need to find the ratio of these two; the ratio formula is as follows;
a: b ⇒ a/b
For Example: In Ratio 5:7 is represented by 5/7 where 5 is antecedent and 7 is consequent.
3.2 Proportion: Let us now assume that the two proportions are a:b ::c:d. The terms ‘b’ and ‘c’ are known as means or mean terms,’ whereas ‘a’ and ‘d’ are known as ‘extremes or extreme terms.’
a:b::c:d or \( \frac{a}{b} \times \frac{c}{d} \)
For example: In a city x and y the ratio of males and females is 7:5 and 9:4 or 7/5 and 9/4
where 7 and 4 are extremes and 5 and 9 are means.
4. Difference between ratio and proportion
Let us understand the difference between the ratio and proportion
Ratio  Proportion 
It is used to compare two different quantities that belong to one similar thing i.e., the same unit or same parameter.  It is used to compare two different ratios which belong to two different units. 
The ratio is expressed as 2:3  Proportion is expressed as 2:3::4:7 
Ratio is an expression  Proportion is an equation 
5. Examples
 Are the ratio of 1:5 and 4:5 is said to be proportionate?
Sol) 1/5 = 0.2 and 4/5 =0.8 since both the ratios are not equal they are not proportionate.
 If their total income of Arjun is 25,000 his total expenses are 15,000 and Rajesh’s yearly income is 3,60,000 and his total expenses are 15,000 then find out the ratio of savings of Arjun and Rajesh.
Sol) Arjun’s monthly savings is 25,000 – 15,000 = 10,000
Rajesh monthly savings is 3,60,000/12= 30,000 – 15,000 = 15,000
Then the ratio of Savings will be 10: 15 or 2:3.
 X and y together they have Rs 1500. If 2/4 x’s amount is equal to 1/4 of y’s amount, how much does y have?
\(\text{Sol) } x \left( \frac{2}{4} \right) = y \left( \frac{1}{4} \right)\)
\(x = y \left( \frac{1}{4} \times \frac{4}{2} \right)\)
\(x = y \left( \frac{1}{2} \right)\)
\(x:y = 1:2\)
\(\text{Y’s share is } \frac{2}{3} \times 1500 = 1000\)
 Two numbers are respectively 30% and 40% more than a third number. Then the ratio of the two numbers will be?
Sol) Let us assume the third number is x
Then, first number = 130% of \(x= \frac{130x}{100}=\frac{13x}{10}\)
Second number = 140% of \(x = \frac{140x}{100}=\frac{14x}{10}\)
The ratio of the first two numbers will be \(\frac{13x}{10}: \frac{14x}{10}\)= \(13x: 14x=13:14\)
6. FAQ's
1. what is the formula of the ratio?
Ans: The formula of ratio is a: b ⇒ a/b
2. what is the formula of proportion?
Ans The formula of proportion is a:b::c:d or \( \left(\frac{a}{b} \times \frac{c}{d} \right) \)
3. What are the types of proportions?
Ans There are different types of proportions such as
 Direct proportion
 Inverse proportion
 Continued proportion
4. what are the means and extremes in proportion?
Ans Suppose a:b:: c:d. The terms ‘b’ and ‘c’ are known as means or mean terms,’ whereas ‘a’ and ‘d’ are known as ‘extremes or extreme terms.’
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