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Business Mathematics

Ratios

A ratio is a simple mathematical term used for comparing two or more quantities that are measured in the same units. Whether it be a 4000000:1 contrast ratio formula for a new TV that you are buying, or a 16:9 aspect ratio of a movie you’re watching, we encounter ratios in our everyday life. Subsequently, we use ratios in proportion problems too you’ll encounter later on, which simply deals with the equality of two ratios. So let’s start understanding the basic meaning of ratios!

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Ratio

ratio formula

Ratios are the comparison of two quantities or more quantities (having the same units) that we express as a fraction. The concept of equivalent fractions allows the ratios of different physical quantities to be the same sometimes. Thus, a ratio is a general term independent of a unit and we use it across multiple platforms. Consider the following example –

⇒ A shopkeeper sells mangoes for 50 rupees per kg and buys them at a wholesale price of 30 rupees per kg. What is the ratio of his profit to the cost price per kg?

Solution: For 1 kg of mangoes,
The profit = Selling Price – Cost Price
= 50 – 30 = 20 rupees
Then we write the ratio of his profit to the cost price per kg as –

 $$\frac{\text{Profit per kg}}{\text{Cost Price per kg}}$$

$$ = \frac{20}{30}$$
$$= \frac{2}{3}$$

This is the fractional form of the ratio formula. We can equivalently express it as $$\frac{10}{15}$$
or any other equivalent fraction. But conventionally, we only use the reduced form of the fraction. Notation-wise, we then say that the ratio of the shopkeeper’s profit to the cost price per kg = 2 : 3

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From this example, we can build up to mention the following properties and conventions concerning ratios –

Properties of Ratios

  • We express a ratio only between two quantities of the same units. (rupees per kg in the above example)
  • We use the symbol ‘:’ to denote ratios.
  • We call two ratios as equivalent if their corresponding fractions are equivalent.
  • For ratios written as a : b, the first term i.e. a is known as the antecedent and the second term i.e. b is known as the consequent.
  • The order of the terms in ratios is very important i.e. the positions of antecedent and consequent are not interchangeable.
  • If more than one like quantities are expressed in a ratio format, the resultant is termed as a Continued Ratios or a Compassion between the quantities. It can simply be represented as a : b : c : d…

Besides the above-mentioned properties of ratios, there are some particular types of ratios you do need to know about before you begin with problems –

Compound Ratio Formula

When we compound/merge two or more ratios with each other through multiplication, the result is simply a compound ratio.
Consider two known ratios – a : b and c : d. Then the Compound Ratio of the two mentioned ratios is ac : bd.

If we compound a ratio a : b with itself once, it results in a Duplicate Ratio, which we give as a2 : b2. Similarly, we can write a triplicate ratio as a3 : b3. So, you can extend the concept to any exponent.

Since we talked about the powers of the terms involved in ratios, it logically follows that we must talk about the roots as well. By taking the roots of the terms in ratios, we can, therefore, form Sub-ratios.

We give A Sub-Duplicate ratio of a : b as √a : √b. Similarly, you can also form Sub-Triplicate ratios by taking a cube root and so on. These ratios don’t come straightaway under the category of Compound Ratios but are analogous to the self-compounded ratios.

Inverse Ratio Formula

By taking the reciprocal of the fractional form of the given ratios, we get inverse ratios. The inverse ratio of a : b = b : a. Another property of the inverse ratios which is simple to understand is –

Ratio × Its Inverse Ratio = 1

Commensurate Ratios

Ratios whose antecedent and consequent both, are only integers, is a Commensurate Ratios. For example, in the solved example that we solved above resulted in a commensurate ratio 2:3. Instead of such a simple ratio, if we get something like 3:Π or √2:5, then it is not a commensurate ratio.

NOTE: A ratio such as 0.75:2 may seem like commensurate ratios at first, but actually if you just multiply its fractional form by 4, you’ll get the resultant ratio 3:8. This indeed is a commensurate ratio. Thus, you have to be careful with some cases.

Solved Examples on Ratio Formula

Question 1: What number do we need to subtract from each of the terms in the ratio 19:31 to reduce it to the ratio 1 : 4?

Solution: Let the required number be x. Writing the modified ratio (after subtracting x from each of the terms in 19 : 31) in fractional form and expressing it equal to 1 : 4 – $$\frac{19 – x}{31 – x} = \frac{1}{4}$$
$$ 4\times (19 – x) = 1\times (31 – x)$$ $$ 76 – 4x = 31 – x $$ $$ 45 = 3x $$ $$ x = 15 $$

Question 2: The ratio componded of 4 : 9, the duplicate ratio of 3 : 4, the triplicate ratio of 2 : 3, and 9 : 7 is ?

Solution: We need to compound four different ratios here –

  • 4 : 9
  • 32 : 42 = 9 : 16
  • 23 : 33 = 8 : 27
  • 9 : 7

By the definition of the compounded ratios, the final result would be – $$ 4 \times 9 \times 8 \times 9 : 9 \times 16 \times27 \times 7 $$ Writing it in fractional form – $$\frac{4 \times 9 \times 8 \times 9}{9 \times 16 \times27 \times 7}$$ $$ = \frac{2}{21}$$ Thus, the required ratio is 2 : 21.

Question 3: If Rs 782 is to be divided in the ratio 6 : 8 : 9, then what value would the first share correspond to?

Solution: Consider a unit share to be equivalent to Rs x. Then the first share would be equivalent to Rs 6x. The second share ⇒ 8x. The third share ⇒ 9x. By common sense, the sum of all shares should be equal to the total amount. Thus – $$6x + 8x + 9x = 782$$ $$ 23x = 782$$ $$ x = 34 $$ Clearly then, the value of the first share ⇒ Rs 34*6 ⇒ Rs 204

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