Proportion: 

When two ratios are equal, then the four quantities involved in the two ratios are said to be proportional.

When a, b, c, and d are in proportion, then a and d are called EXTREMES, and b and c are called MEANS.

If a: b = c: d, we write, a: b :: c:d and say that a, b, c, d are in proportion. Here a and b are called extremes, and b and c are called means terms.

Thus a:b :: c:d ⇒ (a × d) = (b × c)
Ex. 2:3 :: 5:7 ⇒ (2 × 7)= (3 × 5)

How to Use Proportions in Math

Proportions have several uses in math, including finding percentages and finding sizes of similar triangles (or other shapes). 

Proportions are denoted by the symbol  ‘::’ or ‘=’.

The proportion can be classified into the following categories, such as:

• Direct Proportion
• Inverse Proportion
• Continued Proportion

Now, let us discuss all these methods in brief:

Direct Proportion

The direct proportion describes the relationship between two quantities, in which the increases in one quantity, there is an increase in the other quantity also. Similarly, if one quantity decreases, the other quantity also decreases. Hence, if “a” and “b” are two quantities, then the direction proportion is written as a∝b.

Inverse Proportion

The inverse proportion describes the relationship between two quantities in which an increase in one quantity leads to a decrease in the other quantity. Similarly, if there is a decrease in one quantity, there is an increase in the other quantity. Therefore, the inverse proportion of two quantities, say “a” and “b” is represented by a∝(1/b).

Continued Proportion

Consider two ratios to be a: b and c: d.

Then in order to find the continued proportion for the two given ratio terms, we convert the means to a single term/number. This would, in general, be the LCM of means.

For the given ratio, the LCM of b & c will be bc.

Thus, by multiplying the first ratio by c and the second ratio by b, we have

First ratio – ca : bc

Second ratio – bc : bd

Thus, the continued proportion can be written in the form of ca: bc: bd

Examples

1. Which of the following numbers should be added to 13, 43, 23, and 73 So that they are in proportion?

2. Find the value of ‘a’ in the following proportion 36 : 108 : : x : 12.

Fourth proportional: 

If 2:3 :: 5:7, then 7 is called the fourth proportional to 2,3,5

Example: If a : b: c is 2 : 5 : 3 and c :  d : e is 2 : 3 : 5  then find a : b : c : d : e ?

Third proportional: 

If 2:3 :: 5:7, then 5 is called the third proportional to 2 and 3

Example: If A : B is 2 : 5, B: C is 3: 4, then find A : B : C.

A Practical Exercise:

The following practical exercise is designed for students to apply their knowledge of Proportion in Math and understand its utility in a real-life context.

Case:

You and your family have a doll shop in Illinois. For years, you have made a single doll, the Dardie. The Dardie has always been an iconic doll that has been played with by hundreds of children in your neighborhood. Every Dardie is hand-made by you and your family members and is crafted exceptionally well. However, last year, your family decided that it was time for Dardie to get a dog as a companion: Barky was thus created – A dog doll. Your family is curious to know how well the Barky doll has been selling relative to the total doll sales. The sales data is below.

Required:

Compute the following proportions based on the sales data provided:

1. The proportion of Barky Sales relative to Total Sales this year (Provide your answer as the smallest possible fraction).
2. The proportion of Barky Sales relative to Total Sales last year (Provide your answer as the smallest possible fraction).
3. Has the proportion of Barky Sales increased, decreased, or remained unchanged year over year?

Solution:

1.

= Barky Sales this year / Total sales this year

= 95/285

= 1/3

2.

= Barky Sales last year / Total sales last year

= 60/200

= 3/10

3.

Let’s bring the answer in #1 above in the same denominator #2.

1/3 vs 3/10

(1*3.3)/(3*3.3) vs 3/10

3.3/10 vs 3/10

The answer is that the proportion has increased since 3.3/10 is greater than 3 /10.