Ratios and Proportion

In their daily lives, people must count & compare the number of various goods as well as grasp the relationship between them. For example, we must measure both people’s heights and compare the findings to establish how tall 1 person is in contrast to another. While measuring someone’s height gives us an indication of their height, ratios allow us to see how much 1 person is taller than the other person.

Understanding ratio & proportion is essential for a range of tasks, including interpreting food recipes, estimating relative distances on road trips, creating metal alloys, & combine different chemicals to manufacture products, to name a few.

Ratios and Proportion

When comparing 2 quantities of the same type, the ratio is used. For 2 numbers, x and y, the ratio formula is written as x : y or x/y. 2 or more ratios all seem to be in proportion when they are equal. Fractions underpin the concepts of ratio and proportion. Ratio & proportion are the foundations for many other mathematical notions. Ratio and proportion can be used to solve a variety of everyday problems, such as comparing heights, weights, distances, and times, or adding ingredients to a recipe.

Ratios and Proportion Definitions

Ratio

A ratio is the comparison of 2 or more numbers which shows how they compare in terms of quantity. A dividend symbol (:) between 2 numbers represents a ratio of two separate quantities. For example, if one were to indicate the ratio of boys to females in a class, one could use 30:45 as a ratio of boys to girls, suggesting that the ratio of the boys to the girls in our hypothetical classroom is 2:3. The antecedent (number on the left) is known as an antecedent, while the divisor (number on right side) is regarded as a consequent.

Proportion

A mathematical formula which states that 2 ratios are equivalent is called proportion. While ratios aid in the comprehension of the relationship between 2 separate numbers, proportion aids in the comprehension of the relationship between 2 ratios. As a result, two ratios are said to be in proportion when their values are equal (or proportionate). When we need to represent proportions between 2 ratios, we can use the equals (=) or double dividend (::) symbol.

Formula For Ratio and Proportion

For two numbers a and b, ratio is given as

a:ba/b

Here, 

a = Antecedent

b = Consequent

The Proportion of two ratios a:b and c:d is given as

a:b ∷c:d→ab=cd

Here,

b & c = mean terms

a & d = extreme terms

In a: b = c: d, the values a and b must be of the be of the same same kind and have the same units, though c & d can be of the same kind & have the same units on their own.

Here, 

a×d=b×c

That is,

Product of Extremes = Product of Means

Properties of Ratio and Proportion

Properties of Ratio

a/b=pa/pb=qa/qb 

here, p & q≠0

When 2 ratios a: b and c: d are equal then

Invertendo

a/b=cd/ba=d/c 

Alternendo

a/b=cd/ac=b/d 

Componendo

a/b=cd/a+b/b=(c+d)d 

Dividendo

a/b=c/da-b/b=(c-d)d 

Properties of Proportion

1. Product of the Extremes = Product of the Means 
2. a, b, c, …. are in contin proportion then, a:b = b:c = c:d
3. a : b = b:c then b is the mean proportional and b2 = ac
4. The 3rd proportional of 2 numbers, a & b, is c, so that, a:b = b:c
5. d is 4th proportional to the numbers a, b, c when a:b = c:d

Variation

When two quantities a & b are linked in such a way that when one changes, the other changes as well, the 2 quantities are said to be in variation.

Variation is represented by .

There are two types of variation which are direct and inverse variations.

Conclusion

A proportion is an equation which defines the 2 given ratios are equal to each other, whereas a ratio is a comparison of 2 or more numbers which show their quantities in relation to each other. It facilitates the comprehension of the relationship between 2 ratios. In mathematics and other related areas, ratio & proportion are often used concepts to compare quantities and construct a link between them.

For two numbers a and b, ratio is given as

a:b→a/b

The Proportion of two ratios a:b and c:d is given as

p:q ∷r:spq=rs

The concept of variation refers to how one quantity changes in relation to one or more quantities.