RATIO AND PROPORTION

Ratio: A ratio is a way of comparing two or more quantities. It expresses the relationship between two numbers or quantities and is often written in the form of "a:b" or as a fraction "a/b". The key aspect of a ratio is that it represents the relative size or magnitude of the quantities being compared. For example, if there are 3 red balls and 5 blue balls, the ratio of red to blue balls is 3:5.

Compound ratio 

A compound ratio is a ratio obtained by combining two or more simple ratios through multiplication. It essentially compares the product of the antecedents (first terms) of the individual ratios with the product of their consequents (second terms).

Here's an illustration:

Suppose we have two simple ratios:

• Ratio 1: A : B = 2 : 3 (read as "A is to B as 2 is to 3")
• Ratio 2: C : D = 4 : 5

The compound ratio of these two ratios would be:

• Compound Ratio: (A x C) : (B x D) = (2 x 4) : (3 x 5) = 8 : 15

Therefore, the compound ratio tells us that the relationship between A and C is similar to the relationship between 8 and 15, while the relationship between B and D is similar to the relationship between 3 and 5

Proportion: A proportion is an equation that states two ratios are equal. In a proportion, the cross products are equal. The general form of a proportion is written as (a/b)$=(c/d)$, where $a$, $b$, $c$, and $d$ are numbers. This implies that the ratio of $a$ to $b$ is equal to the ratio of $c$ to $d$. Proportions are used to solve problems involving ratios.

For example, consider the following proportion: (2/3)=(4/6). This means that the ratio of 2 to 3 is equal to the ratio of 4 to 6.

Proportions express the equality of two ratios, meaning they share the same relationship between their terms

We can show a proportion with double colons (::) or by setting the ratios equal (=). Imagine mixing paint, a 2:3 ratio of red to yellow is equivalent to a 4:6 ratio of both colors. 

In proportions, the product of the middle terms (b and c) equals the product of the outer terms (a and d). This can be written as b × c = a × d

Key Properties:

1. Cross-Products Property: In a proportion (a/b)​=(c/d), the product of the means (ad) is equal to the product of the extremes (bc). Mathematically, this is expressed as $ad=bc$.

2. Scaling: Ratios and proportions are often used to scale quantities. For example, if the ratio of the side lengths of two similar triangles is 2:3, it means one triangle is an enlargement of the other, with each side length multiplied by the same factor.

Understanding ratios and proportions is crucial in various mathematical applications, from basic arithmetic and problem-solving to more advanced mathematical concepts like algebra and statistics. They find extensive use in everyday situations and are foundational in many areas of mathematics

Ratio and Proportion Formula

Ratio:

1. Basic ratio: A ratio can be written in two ways:

   • a:b, where a and b are the two quantities being compared. This reads as "a to b".
   • a/b, the fraction of a to b.

2. Equivalence of ratios: Two ratios are equal if they represent the same proportion. Mathematically, a:b = c:d if and only if a/b = c/d.

Proportion:

1. Proportion formula: A proportion can be expressed as an equation comparing two ratios:

   • a:b::c:d, where a/b is equal to c/d.
   • You can also use an equal sign: a/b = c/d.

2. Properties of proportions:

   • Cross multiplication: If a:b::c:d, then ad = bc.
   • Means and extremes: If a:b::c:d, then ab = cd.
   • Sum and difference of means and extremes:
     • (a + b) : (a - b) :: (c + d) : (c - d)
     • (a + b) : (b - a) :: (c + d) : (d - c)