Commercial Math

Commercial math focuses on dealing with the economical world, and in our everyday life whether it's buying groceries, selling items or even toys. It involves the concepts of profit and loss, interests, taxes, ratios, etc. Anything related to money comes under commercial mathematics.

So let's learn more about commercial math and all its subcategories in detail. In early 3000 BCE, in Egypt and Mesopotamia, commercial mathematics was used for trading and taxation. The commercial math as we know it today was developed by the Ancient Greeks and Romans.

Commercial maths is mostly used in banking and finance sectors. Apart from banking and business operations, in daily life also we use commercial mathematics. Why is it important? It is used for many functions like:

• For calculating profit and losses 
   
• For calculating monthly EMI, loans, interest rates 
   
• Accurate calculations of taxes 
   
• For budgeting purposes 
   
• For investments and savings 

So far, we’ve learned about commercial math and its importance. Now, let’s focus on the key topics in commercial math. Following are the topics that we are going to discuss.

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

Every business has its share of profit and loss. When the business is making more money than the investment made, we call it profit. If the profit is less than the investment, then the business is said to be under loss.

What is profit? 

Profit is the money that a business has retained after paying all its expenses. It is an important performance metric to understand the business’s financial gains. Profit is calculated by subtracting cost price from selling price. 
Profit = Selling Price - Cost Price

What is loss? 

When a product is sold for an amount lesser than the original cost, it is said to be a loss.

Loss = Cost Price - Selling Price

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

Profit Percentage = (Profit/cost price) x 100

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

Cost Price (CP) = $160
Selling Price (SP) = $200
Profit = SP - CP 
Profit = 200 - 160 = $40

Now let's calculate the percentage of the profit we made:

Profit Percentage = (40/160) x 100 = 25%

So we made a profit of 25%!

Simple interest is the extra money you earn when you lend someone money. It also works, when you borrow money, and you have to pay a bit more than you originally borrowed.

Simple interest depends on three factors:

• Principal (P): It is the amount of money borrowed before any interest is added. 

• Rate (R): The percentage of interest or money you earn or have to pay every month or year.

• 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 X R X T/100

Now let's use this formula in an example to better understand

Example: So a friend came and asked you to lend him Rs5000 and promised to pay you in 3 years with a 5% interest per year.

Step 1: We identify the values

Principal = Rs5000

Rate = 5%

Time = 3 years

Step 2: Calculate using the formula

    SI = (5000 × 5 × 3) / 100

    SI = (75000) / 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 compound interest

Compounded Annually:

A = P (1 + R/100)T

Frequent Compounding (quarterly, monthly, etc):

A = P(1 +r/n)^nt- P

Here are a few examples that use these formulas.

Example 1: Rohan put $1500 in the bank at 15% interest per year, compounded quarterly ( 4 times a year), for two years. Find the compound interest.

Step 1: Identify the values
    
    P = 1500
    R = 15% (convert to decimal for calculations)
    n = 4 (because it's compounded quarterly)
    T = 2

Step 2: Use the compound quarterly formula

 A = P(1 +r/n)^nt- P

A = 1500(1 +0.15/4)⁴⁽²⁾

A = 1500 (1.0375)⁸

A = 1500 (1.3425) = 2013.75

Compound interest = 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: Use the compounded annual formula

    A = P (1 + R/100)^T

    A = 3000 (1 + 12/100)³

    A = 3000 (1.12)³

    A = 3000 (1.404928)

    A = $4214.784

Compound Interest = 4214.784 - 3000 = $1214.784

Simple Interest VS Compound Interest

The price of any service or product can be reduced to entice customers. This is called a discount. It is a very common technique used by business people to attract old and new customers. 

We use a formula to calculate the discount:

• Discount = Marked Price - Selling Price
   
• Discount percentage = (Discount/ Marked Price) x 100

Calculating discount is done in two cases:

• When the marked price and selling price are both given. We use the discount formula, which is marked price - selling price.
   
• Another case is when the discount percentage is given we would use the discount percentage formula.

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

Example 1: A playstation 5 costs around $45000, but it is being sold for $36000. Find the discount amount and the discount percentage.

Step 1: We calculate the discount amount
    
            Discount = Marked price - Selling Price
    
            Discount  = 45000 - 36000 = $9000

Step 2: Now that we have the discount amount we find the discount percentage.

           Discount percentage = 9000/45000 x 100 = 20%

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

Taxes are money that people pay to the government to help fund things like building of roads, schools, hospitals, etc. 

A formula we used to calculate tax is:

Tax amount = Selling Price x Tax rate/100

There are two types of taxes that we know of:

• 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 you can buy. This is then collected by the government which would be used for any public facilities like schools, parks or even hospitals.

GST has formulas of its own as well:

• GST Amount:
  GST Amount  = (GST% x Price)/100

• Final Price:
  Final Price = Price x (1 + GST% / 100)

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.

Step 1: Find the GST amount:

    GST Amount = (13 x 500) / 100
            = $65

Step 2: Find the Final price

    Final price = 500 + 65 = $565

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

We use ratio and proportion everyday while we cook or share money. Ratios and proportions are two ways to compare quantities and very useful in our daily lives.

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 show whether two ratios are equal or not. 

We write proportion as:
    a/b = c/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:  2/4 = x/8
    Now we solve for x by cross multiplying:
        2 x 8 = 4 × x
        16 = 4x
        x = 16/4 = 4
        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:
Partner’s Share = Partner’s Investment x Total Profit/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

        Pam’s share = (3000 x 5000) / 5000 = $3000

        Tam’s share = (2000 x 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:

• Work: Any given task that must be completed.
   
• Time: The number of minutes or hours taken to complete the said task.
   
• Efficiency: The ability to complete the task correctly without spending too much time. 
   
• Combined efficiency: The total efficiency of two or more people involved in a task. 

Some formulas for work and time are:
    

• Work Done = Efficiency x Time
   
• Time = Work / Efficiency 
   
• Efficiency = Work / Time
   
• Combined Efficiency = Efficiency1 + Efficiency2 + ….. + Efficiency_n
   
• Time taken by multiple workers: Time = Work / 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: Efficiency = Work / Time
    Total work = 1 job 
    Time = 10 days

    Efficiency = 1/10 

    So Andy completes 1/10 of the work per day

    Work = Efficiency x Time
    Efficiency  = 1/10
    Time = 4 days

    Work = 1/10 x 4 = 0.4

So Andy’s efficiency is 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.

• Time: It tells us long it takes to go from point A to point B. Usually measured in hours, minutes or seconds
  Formula: Time = Distance / Speed

• Distance: Distance is how far you have travelled from point A to point B. It is measured in kms, meters or miles.
  Formula: Distance = Speed x Time

• Speed: Speed tells us how fast we have been travelling a certain distance to get from point A to point B. Usually, speed is measured in kilometers per hour (km/h) or meters per second (m/s).
  Formula: Speed = Distance / Time

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:

    Time = Distance / Speed

    Time = 180 / 60

    Time = 3 hours

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

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:

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.

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.

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. 

Stocks and investing

We use compound interest, a concept in commercial math, to help us calculate our investments.

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. 

It definitely does feel overwhelming when learning commercial math. As there are a lot of formulas and concepts to learn. So here are some tips and tricks to learn that would make studying these concepts way easier.

• There are a few basic formulas that can be memorized easily. These formulas will help you avoid mistakes.
  
  Here are some basic formulas to remember: 
  a. Profit and loss: 
  Profit = Selling price - Cost price
  Loss = Cost price - Selling price
  
  b. Simple interest: SI = (P x R x T) / 100
  Time, Speed, and Distance: 
  Distance = speed x time
  Speed = distance / time
  Time = distance / speed

• 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

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

• 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.