Ratio and proportion concepts

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 

5. Examples

1. 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. 

2. 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.

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?

\(\text{Sol) } x \left( \frac{2}{4} \right) = y \left( \frac{1}{4} \right)\)

\(x = y \left( \frac{1}{4} \times \frac{4}{2} \right)\)

\(x = y \left( \frac{1}{2} \right)\)

\(x:y = 1:2\)

\(\text{Y’s share is } \frac{2}{3} \times 1500 = 1000\)

4. 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 \(\frac{13x}{10}: \frac{14x}{10}\)= \(13x: 14x=13:14\)