Commercial Math

Commercial math is the branch of mathematics that focus on the calculation of profits, discounts, taxes, percentages, and many others terms related to money. It was first developed in the early 3rd millennium BCE in the Mesopotamian and Egyptian civilizations for trade and commerce.

The use of commercial mathematics in today's world is mainly related to the banking and financial sectors. The term “commercial” itself refers to business, trade, or activities intended to generate a profit. 

Commercial math is mainly used in the banking and finance sectors, as it deals with trade and commerce-related subjects. Even though banking and financial sectors are major users of commercial math, it is not limited to them. The main functions of commercial math:

1. Calculating profit and losses 
    
2. Calculating monthly EMI, loans, and interest rates 
    
3. Accurate calculations of taxes 
    
4. Budgeting purposes 
    
5. Investments and savings 
    

Since we have already discussed the primary uses and the definition of commercial mathematics, let us discuss the essential topics in commercial mathematics:

• Profit and Loss
   
• Simple Interest
   
• Compound Interest
   
• Discounts
   
• Taxes
   
• Ratio and Proportion
   
• Partnerships
   
• Time and Work
   
• Time, Speed, and Distance

All businesses work on the basic principle of profit and loss. Business can be simply defined as the act of conducting trade, and in trade, there is either a profit or a loss.

What is Profit? 

Profit is the money a business retains after paying all its expenses. It is an important performance metric to understand the business’s financial gains. Profit is the value remaining after reducing the selling price from the cost price. The mathematical expression for profit is given by:

Profit = Selling Price - Cost Price
 

To find out how much profit was made compared to the cost, we use the formula,

Profit Percentage = \( \left( \frac{\text{Profit}}{\text{Cost Price}} \right) \times 100 \)

What is Loss? 

When a product is sold for an amount less than the original cost, it is said to be a loss. The formula for calculating loss is expressed as:

Loss = Cost Price - Selling Price

Let's understand these terms using an example.
 

Example: We bought a toy for ₹160 and sold it for ₹200. What percentage of profit did we make?

Solution:

Given:
 

Cost Price (CP) = ₹160

Selling Price (SP) = ₹200

1. Calculating Profit:
   
   Profit = SP - CP
   \(\text{​​​​​​​Profit} = 200 - 160 = ₹40\)
    
2. Now let's calculate the percentage of the profit we made:
   
   \(\text{Profit Percentage } = \left( \frac{40}{160} \right) \times 100 = 25 \% \)

So we made a profit of 25%.
 

Simple interest is the interest added or the extra money that is to be paid while returning borrowed money. It is only applicable on the original amount of money borrowed.

Simple interest depends on three factors:

1. Principal (P): It is the amount of money borrowed before any interest is added. 
    
2. Rate (R): The rate of interest is the percentage of the principal amount that is paid or earned as interest per unit of time, usually per year.
    
3. Time (T): The amount of time the money is borrowed or lent for in years or months.

So using these factors we have a formula to calculate the simple interest:

Simple interest (SI) = \( {{P \times T\times R} \over 100}\)

Now, let's use an example to understand what simple interest is and how it works.

Example: A person borrowed Rs 5000 and promised to pay the bank back in 3 years with a 5% interest per year. Calculate the simple interest.

• Step 1: We identify the values
  
  Principal = ₹5000
  Rate = 5%
  Time = 3 years

• Step 2: Calculate using the formula
  
  \(SI = \frac {5000 \space \times \space5 \space × \space3}{100}\)
  
  \(SI = {75000 \over 100} = ₹750\)
  
  Therefore, after 3 years you will earn, ₹750 as interest.

• So the total money you will earn is:
  
  \(5000 + 750 = ₹5750\)

The interest earned on both the original amount and the interest that has already been added is called compound interest. 

Compound interest can be calculated using:

• Principal (P): The initial amount of money
• Amount (A): Total amount of money you will earn
• Rate (R): The percentage of interest you would earn
• n: The total number of times the interest is compounded in a given year
• T: Time in years

So there are two formulas for calculating the amount:

1. Compounded Annually: 
   \(​ A = P (1 + \frac{R}{100})^T \)
    
2. Frequent Compounding (quarterly, monthly, etc.): 
   \(A = P(1 + \frac{r} {n})^{nT}\)
    

Now, after finding the amount, compound interest can be calculated by using formula:

\(\text{Compound Interest = Amount - Principal}\)

Here are a few examples that use these formulas:

Example 1: Rohan put ₹ 1500 in the bank at 15% interest per year, compounded quarterly, for two years. Find the compound interest.

• Step 1: Identify the values
  
  P = 1500
  n = 4 (because it's compounded quarterly)
  T = 2

• Step 2: Use the compound quarterly formula:
  
  \(A = P(1 + \frac{r} {n})^{nT}\)\(A = 1500(1 + \frac{0.15} {4})^{4 \times 2}\)

  \(A = 1500 (1.0375)^8\)
  \(A = 1500 (1.3425) = 2013.75\)

• Compound interest:
  
  \(\text{Compound Interest} = A - P \\=  2013.75 - 1500 = ₹ 513.75\)

Example 2: Now Rohan put ₹3000 in the bank at 12% interest per year, compounded annually, for three years. Find the compound interest.

• Step 1: Identify the values
  
  P = 3000 
  R = 12%
  T = 3

• Step 2: Calculating amount using the compounded annual formula:
  
  \(​ A = P (1 + \frac{R}{100})^T \)
  \(​ A = 3000 (1 + \frac{12}{100})^3 \)\(  A = 3000 (1.12)^3\)

  \( A = 3000 (1.404928)\)

  \(A = ₹4214.784\)

   

• Calculating compound interest:
  
  \( = 4214.784 - 3000 \\= ₹1214.784\)

   

Simple Interest Vs Compound Interest

The following table gives the differences between simple interest and compound interest.

The reduction in the price of an object or a set of objects is referred to as a discount. It is a widely used technique employed by businesspeople to attract both existing and new customers. A discount can be expressed in terms of a percentage reduction or as a flat money discount.

We use a following formula to calculate the discount:

• \(\text{Discount} = \text{Marked Price} - \text{Selling Price}\)

• \(\text{Discount percentage} = \frac{\text {Discount}} {\text{Marked Price}} \times 100\)

Calculating a discount is done in two cases:

When the marked price and selling price are both given, we use the discount formula, which is calculated by subtracting the selling price from the marked price. Another case is when the discount percentage is given, we use the discount percentage formula.

Let's use these formulas in real-life examples to get a better understanding:

Example: A PlayStation 5 costs around $450, but it is being sold for $360. Find the discount amount and the discount percentage.

Given:

Marked Price = $450
Selling Price = $360

• Step 1: We calculate the discount amount
  
  \(\text{Discount} = \text{Marked Price} - \text{Selling Price}\)
  \(\text{Discount}  = 450 - 360 \\\text{Discount} = $90\)

• Step 2: Now that we have the discount amount, we find the discount percentage.
  
  \(\text{Discount percentage} = \frac{{90}} {450} \times 100 \\ = 20 \%\)

So there is a 20% discount on the PlayStation 5.
 

Tax is the amount of money that is collected from eligible individuals and organizations that are liable to pay and is transferred to the government. The money is used for the development and welfare of the country. 

A formula used to calculate tax is:

\(\text{Tax amount} = \text {Selling Price} \times {{{\text{Tax Rate}} \over 100}}\)

There are two types of taxes :

Direct Tax - These are paid directly to the government, like when you get your salary a small portion of it goes to the government.

Indirect Tax - These taxes are paid to businesses, which then pays the government. Sales tax is an example of indirect tax.

What is GST? 

Goods and Services Tax (GST) is the price that is added to various products and services that are bought and sold. This is then collected by the government, which would be used for public.

GST has formulas of its own as well. The formulas are:

• \(\text{GST Amount}  = {\frac{\text{GST %} \space \times \space \text{Price}}{100} }\)

• \(\text{Final Price} = \text{Price} \times \big( {1 + \frac{\text{GST%}}{100} }\big)\)

Let's use these in some examples

Example 1: You buy a shirt that costs you around ₹500, the GST rate is 13%. Find the total amount after GST.

Given:

GST = 13%
Price =  ₹500
 

• Step 1: Find the GST amount:
  
  \(\text{GST Amount}  = {\frac{\text{GST %} \space \times \space \text{Price}}{100} }\)
  \(\text{GST Amount }= \frac{13 \space \times \space500}{100} \\\text{GST Amount }= ₹65\)

• Step 2: Find the Final price
  
  \(\text{Final price} = 500 + 65 = ₹565\)

So the price of the shirt after GST is  ₹565.
 

Ratio is a way of comparing two quantities, whereas proportion states the equality of two ratios. The concept of ratio and proportion is widely used in daily life as well as in commerce and science.

What is Ratio?

Two quantities compared with each other is what we call ratio. It tells us how much one thing is compared to another. 

We write ratio as: 

a:b (we read it is ‘a' is to b’)

Example 1: We mix 2 cups of milk with 3 cups of water. What is the ratio?

Solution: The ratio is 2:3 

This means that for every 2 cups of milk, there will be 3 cups of water.

What is Proportion?

Proportions is a concept of showing if two values are in ratio or not.

We write proportion as:

    \({{a} \over b}\) = \( {{c} \over d}\)

Example 2: If 2 cups of sugar is added to 4 cups of milk, then how many cups of sugar is required for 8 cups of milk?

Solution:  \({{2} \over 4}\) = \( {{x} \over 8}\)
 

Now we solve for x by cross multiplying:
 

\(​ 2 \times 8 = 4  \times x ​\\ \space \\​ 16 = 4x ​\\ \space \\ x = \frac{16}{4} ​\\ \space \\ x = 4  ​\)

So 4 cups of sugar is needed for 8 cups of milk.

Two people coming together and starting a business arrangement, sharing the profit and losses, is what we call a partnership. 

The amount of money a partner invests in the company decides how much profit or loss each partner gets. Usually, partnerships are formed among large companies like Ben & Jerry's or Apple.

How are the profits shared between partners?

The distribution of profits among partners according to the investment made by each business partner is called profit-sharing. 

To calculate profit-sharing, the formula is given by:

 \(\text{Partner's Share} = \frac{\text{Partner's Investment} \space \times \space \text{Total Profit}}{\text{Total Investment}} \)

Example 1: So two partners Pam and Tam start a business. Pam invests $3000 and Tam invests $2000. The total profit is, $5000. Find out each partner’s share.

• Step 1: Find the total investment 
  
  \(3000 + 2000 = $5000\)

• Step 2: Calculate the shares \(\text{Pam’s share } = \frac{3000 \space \times \space 5000}{5000} = $3000 \)

  
  \(\text{Tam’s share } = \frac{2000 \times 5000}{5000} = $2000 \)

Final profit: Pam gets ₹3000 and Tam gets ₹2000.
 

This topic is all about the amount of work done in a said time by a person or a group of people.

Here are some of the key points to remember:

1. Work: Any given task that must be completed.
    

2. Time: The number of minutes or hours taken to complete the said task.
    
3. Efficiency: The ability to complete the task correctly without spending too much time. 
    
4. Combined efficiency: The total efficiency of two or more people involved in a task. 

Some formulas for work and time are:
    

• \(\text{Work Done = Efficiency × Time}\)

• \(\text{Time} = \frac{\text{Work}}{\text{Efficiency}} \)

• \(\text{Efficiency} = \frac{\text{Work}}{\text{Time}} \)

• \(\text{Combined Efficiency} = \text{Efficiency}_1 + \text{Efficiency}_2 + ….. + \space \text{Efficiency}_n\)

• Time taken by multiple workers: \(\text{Time} = \frac{\text{Work}}{\text{Combined Efficiency}} \)

Let's understand this through an example

Example 1: Andy is a salesman who can complete his task in 10 days. How efficient is Andy in his work, and how much would he be able to complete in 4 days? 

Solution:
 

1. Calculating Efficiency:
   
   \(\text{Efficiency} = \frac{\text{Work}}{\text{Time}} \)
   
   ⇒ Total work = 1 job     
   ⇒ Time = 10 days
   
   \(\text{Efficiency} = \frac{\text{1}}{\text{10}} \)
   
   So, Andy completes \(\frac{1}{10} \) of the work per day
    
2. Calculating work:
   
   \(\text{Work Done = Efficiency × Time}\)
   
   ⇒ Efficiency  = \(\frac{1}{10} \)

   
   \(​ \text{Work} = \frac{1}{4} × 4 = 0.4 ​\)

So, Andy’s efficiency is \(\frac{1}{10} \) of the work per day, and in 4 days he will be able to complete only 40% of the work.

We use time, speed, and distance to understand how fast an object moves from point to another and how long it would take to get there.

1. Time: It is the measure of the duration that is required to cover the distance between two points.
   
   Formula: \(\text{Time} = \frac{\text{Distance}}{\text{Speed}} \)
    
2. Distance: the total length of path covered when travelling between two points.
   
   Formula: \(\text{Distance = Speed × Time}\)
    
3. Speed: The speed of an object is defined as the rate at which it travels a certain distance in a given period of time. Usually, speed is measured in kilometers per hour (km/h) or meters per second (m/s).
   
   Formula: \(\text{Speed} = \frac{\text{Distance}}{\text{Time}} \)

The following image provides a better visualization of time, distance and speed formula:

Let’s use these formulas in an example
    
Example 1: A car is travelling from point A to point B at a speed of 60 km/h. How much time will it take to travel 180kms?

Solution: We will use the time formula:

\(\text{Time} = \frac{\text{Distance}}{\text{Speed}} \)

Time = \(\frac{180}{60} \)

Time = 3 hours

To travel 180kms, the car will take 3 hours at a speed of 60 km/h
 

Learning commercial math can definitely feel overwhelming as there are a lot of formulas and concepts to learn. Here are some tips and tricks to know that will make studying these concepts much easier.

1. Memorize basic formulas. These formulas will help you avoid mistakes: 
   
   a. Profit and loss: 
   
   \(\text{Profit = Selling price - Cost price}\)
   \(\text{Loss = Cost price - Selling price}\)
   
   b. Simple interest: \(SI = \frac{P \times R \times T}{100} \)

   
   c. Time, Speed, and Distance:
   
   \(Distance = speed × time\)
   
   \(\text{Speed} = \frac{\text{Distance}}{\text{Time}} \)
   
   \(\text{Time} = \frac{\text{Distance}}{\text{Speed}} \)
   
    

2. Using percentage shortcuts for calculations can help you to solve problems faster: 
   
   10% of a number: You just need to move the decimal point one place to its left.
   Example: 10% of 500 = 50
   
   1% of a number: Over here, we have to just divide the number by 100.
   Example 1% of 3000 = 3000/100 = 30
   
    

3. Practice solving with real world examples: Using real world examples would definitely help you understand how formulas work. 
   
    

4. Make sure to understand and apply ratios and proportions: Go step by step, simplify the ratios or fractions first and then cross multiply when it comes to solving for proportions. 
   
    

5. Be careful about successive changes: In situations where there are multiple discounts or losses, it is essential to subtract or add accordingly based on the question. Not every time is the discount or loss applied to the original price.
    

We need commercial math to calculate profit, losses, or discounts. Every business also needs to pay tax to the government.

Here are some ways we apply Commercial Math in Real life:

1. Businesses or Startups: Commercial Math is very essential in running businesses. It is one of the first things that each business uses to calculate their profit and loss, their partnerships, basically anything needed to manage finances.
    
2. Banking: Banks use commercial math mainly to manage transactions with their customers, and they also use it for loans. Simple interest and compound interests are a few things that banks use.
    
3. Shopping and Retail: Commercial Math is quite widely used in shopping and retail, especially when calculating how much we save in a discount or how much tax you would end up paying during a purchase. 
    
4. Stocks and investing: We use compound interest, a concept in commercial math, to help us calculate our investments.
    
5. Saving money: Commercial math is used as a tool in personal finance management. Calculations related to the amount of money that needs to be saved for future expenses are done through commercial math.

These are some of the few areas in our daily lives where commercial math is widely used. 
 

From Numbers to Geometry and beyond, you can explore all the important Math topics by selecting from the list below: