Ratio and Proportion

A ratio is a way to compare two quantities that have the same unit. We represent a ratio using the colon symbol (:) as:

a : b

Here,

a is the Antecedent

b is the Consequent

The ratio a : b can also be written as a/b. When we multiply both terms by the same number k, we get equivalent ratios. So, a : b means ak : bk, and its simplest form is still a/b.
We read a : b as “a ratio b” or “a to b.”

• Multiplying both terms by the same number does not change the ratio.
  Example: a : a = na : na
   
• Dividing both terms by the same number also keeps the ratio unchanged.
  Example: a : b = a/n : b/n
   
• If two ratios are equal, their reciprocals are also equal.
  Example: If a : b = c : d, then b : a = d : c
   
• If two ratios are equal, their cross-products are equal.
  Example: If a : b = c : d, then a × d = b × c
   
• Different values can have the same simplified ratio.
  Example:
  50 : 60 = 5 : 6
  
  100 : 120 = 5 : 6
  
  Even though the values differ, the simplified ratio 5 : 6 remains the same.

Ratios come in different forms. Learning the types helps students understand which type to use in different situations. We will now learn about the various types:

• Equivalent ratios: Two ratios that are the same when simplified.
  
  For example, 5:10 and 1:2

• Part-to-part ratio: When one part is compared to another in a whole, it is said to be in a part-to-part ratio.
  
  For example, the ratio of blue ribbons to white ribbons in a shop is 3:5.

• Compound ratio: When we compare two or more ratios by taking their products.
  
  For example: \((5:6) × (3:8) = 15: 48\).

• Part-to-whole ratio: When a part is compared to the whole or the total amount, the ratio is part to whole.
  
  For example, if there are 5 blue ribbons and 3 red ribbons, the total number of ribbons is 8. Now, the part-to-whole ratio of blue ribbons to total ribbons is 5:8.

A proportion shows that two ratios are equal. When this happens, the ratios are said to be proportionate. Proportions are written using a double colon (::).
If the ratios a : b and c : d are equal, we write:

a : b :: c : d

In this form:

a and d are the extreme terms,

b and c are the mean terms.

• If two ratios are proportional, then comparing the first terms and the second terms yields equal ratios.
  When a/b = c/d, then a/c = b/d.
   
• Reciprocals of proportional ratios also form a proportion.
  If a/b = c/d, then b/a = d/c.
   
• In any proportion, the product of the extremes equals the product of the means.
  If a : b :: c : d, then a × d = b × c.
   
• Adding the numerator and denominator of each ratio maintains the proportion.
  If a/b = c/d, then (a + b)/b = (c + d)/d.
   
• Subtracting the denominator from the numerator also preserves the proportion.
  If a/b = c/d, then (a − b)/b = (c − d)/d.

If two ratios are identical, then they are proportional to each other. Proportions are of two types, based on which they compare two ratios:

Direct proportion

When two ratios have a direct relationship, they are directly proportional to each other. If a change in a quantity occurs, the other quantity also changes proportionally. We often use the symbol ‘∝’ to denote the proportionality. For example: The number of chocolates you buy increases, and the amount you have to pay also increases.

Inverse proportion

If two quantities have an inverse relationship that is, one quantity increases the other decreases, and vice versa, they are inversely proportional to each other. 
It can be written as a ∝ \(\frac 1b\).

Let’s explore the key formulas for ratios and proportions in detail.

1. Compound Ratios

When two ratios are multiplied, the resulting ratio is called a compound ratio.
Example: If a : b and c : d are two ratios, the compound ratio is:
ac : bd

2. Special Ratios

Duplicate Ratio: For a : b, the duplicate ratio is a² : b²

Sub-duplicate Ratio: For a : b, the sub-duplicate ratio is √a : √b

Triplicate Ratio: For a : b, the triplicate ratio is a³ : b³

3. Proportion Formulas

If a : b = c : d, the following formulas help solve proportion problems:

Addendo: (a + c) : (b + d)

Subtrahendo: (a – c) : (b – d)

Dividendo: (a – b)/b = (c – d)/d

Componendo: (a + b)/b = (c + d)/d

Alternendo: a : c = b : d

Invertendo: b : a = d : c

Componendo and Dividendo: (a + b) : (a – b) = (c + d) : (c – d)

4. Proportionality

Direct Proportion: If a is proportional to b, then a = k × b, where k is a constant.

Inverse Proportion: If a is inversely proportional to b, then a = k / b, where k is a constant.

5. Equivalent Ratios

Multiplying or dividing both terms of a ratio by the same number gives an equivalent ratio.
Example: a : b = n × a : n × b or a/n : b/n

To understand comparisons of quantities in mathematics, it is essential to know the difference between ratios and proportions. While a ratio compares two quantities, a proportion checks whether two ratios are equal. The chart below highlights the key differences in a simple, easy-to-read format.
 

Ratio and Proportion enable children to solve many real-life problems. To grasp it easily, let’s look at a few tips and tricks:

• If a ratio is expressed as x: y: z, the total can be calculated by simply adding x +y +z.
   
• Do not forget to express the ratio in its fractional form. For example: \(4:6 = \frac {4}{6}\).
   
• You can multiply the denominator and numerator of a fraction with the same number to get the same ratios. For example, 2: 5 is equivalent to 4:10 (when multiplied by 2).
   
• Cross-multiplication can be used to identify the missing values. For example: If \(y:3 = 6: 9\) then:
  \(y × 9 = 3 ×6 = 9y =18\)
  \(y = \frac {18}{9} = 2\)
   
• Ratios can be expressed in any of these three forms: 5 to 6, 5:6, or \(\frac 56\).
   
• Show ratios using fruits or toys, then let the child record comparisons on a ratio and proportion worksheet for simple understanding and practice.
   
• Use snack-sharing or classroom activities to explain equal relationships, and allow students to check their answers later with a ratio and proportion calculator.
   
• Give students the colored counters or blocks to visually compare quantities, helping them understand ratios and proportions through concrete hands-on experience.
   
• Show students methods such as cross-multiplication, the unitary method, and scaling techniques so they can choose the approach that best suits each problem.

Ratios and proportions are two fundamental concepts that have multiple real-life applications. Let’s look at a few:

• Understanding real-world situations helps children adjust the ingredient proportions when cooking their favorite dish.
   
• They will be able to comprehend the conversion of currency, which are ratios of various currencies.
   
• To determine the discounts, we apply the ratios to understand the amount that can be saved.
   
• We use proportion to evaluate the population growth. For example, demographers conclude that population growth is directly proportional to the consumption of resources.
   
• Using the ratios, we can calculate the investment returns or the growth rate.