Commercial Mathematics

The One Thing

Commercial mathematics applies percentages, interest, and algebra to taxation, banking, and share transactions. Correct answers depend on identifying the appropriate base value: GST is calculated on taxable value, interest on principal or deposited instalments, and dividends on face value.

Definitions and Results

• Goods and Services Tax (GST): An indirect tax charged on the supply of goods and services.
• Input Tax Credit: The GST already paid on purchases that can be set off against GST collected on sales, subject to applicable rules.
• CGST: The Central Goods and Services Tax charged on an intra-state supply.
• SGST: The State Goods and Services Tax charged on an intra-state supply.
• IGST: The Integrated Goods and Services Tax generally charged on an inter-state supply.
• Taxable Value: The value on which GST is calculated, usually the selling price after applicable discounts.
• GST amount:

$$\text{GST}=\frac{\text{GST rate}}{100}\times \text{taxable value}.$$

• Price including GST:

$$\text{Inclusive price}=\text{taxable value}+\text{GST}=\text{taxable value}\left(1+\frac{\text{GST rate}}{100}\right).$$

• Taxable value from an inclusive price:

$$\text{Taxable value}=\frac{\text{inclusive price}\times 100}{100+\text{GST rate}}.$$

• Intra-state supply: GST is divided equally:

$$\text{CGST}=\frac{\text{GST}}{2},\text{SGST}=\frac{\text{GST}}{2}.$$

• Inter-state supply: The applicable tax is generally IGST.
• Bank Deposit: Money placed in a bank account, such as a savings, fixed, or recurring deposit, to earn interest.
• Principal ($P$): The original amount invested, deposited, or borrowed.
• Rate of Interest ($R$): The percentage charged or earned on the principal during a stated period.
• Maturity Value: The total amount received when a deposit or investment becomes due, including principal and interest.
• Simple Interest:

$$\text{SI}=\frac{P\times R\times T}{100},$$ where $T$ is time in years and $R$ is the annual rate percentage.

• Amount under simple interest:

$$A=P+\text{SI}.$$

• Compound Amount: For annual compounding,

$$A=P{\left(1+\frac{R}{100}\right)}^n,$$ where $n$ is the number of years or compounding periods as applicable.

• Compound Interest:

$$\text{CI}=A-P.$$

• Recurring Deposit: A bank deposit in which a fixed amount is deposited regularly, usually every month, for a specified period.
• Recurring-deposit interest: For monthly instalment $p$, $n$ months, and annual rate $R\%$,

$$I=p\times \frac{n(n+1)}{2\times 12}\times \frac{R}{100}.$$

• Recurring-deposit maturity value:

$$M=pn+I.$$

• Share: A unit representing ownership in the share capital of a company.
• Face Value: The nominal value printed on a share, such as ₹10 or ₹100.
• Market Value: The current price at which a share is bought or sold in the market.
• Dividend: The portion of a company’s profit distributed to shareholders, usually expressed as a percentage of face value.
• Brokerage: A commission charged by a broker on the purchase or sale of shares.
• Investment: The total amount paid to purchase shares, including applicable brokerage.
• Return: The income earned from an investment, such as dividend received from shares.
• Yield: The percentage return on the actual amount invested:

$$\text{Yield}=\frac{\text{annual dividend}}{\text{actual investment}}\times 100.$$

• Dividend per share:

$$\text{Dividend per share}=\frac{\text{dividend rate}}{100}\times \text{face value per share}.$$

• Total dividend:

$$\text{Total dividend}=\text{number of shares}\times \text{dividend per share}.$$

• Share profit or loss: This depends on the difference between the selling price and purchase price, after considering brokerage and other charges.

Worked Methods

1. Calculating GST on a taxable value

• Identify the taxable value and GST rate.
• Calculate GST using

$$\text{GST}=\frac{\text{GST rate}}{100}\times \text{taxable value}.$$

• Add the GST to the taxable value:

$$\text{Price including GST}=\text{taxable value}+\text{GST}.$$

If a discount is given before GST:

• Calculate the reduced taxable value:

$$\text{Taxable value}=\text{marked price}-\text{discount}.$$

• Calculate GST on this reduced value, not on the marked price.
• Add the GST to the reduced taxable value.

For an intra-state sale, divide the GST equally into CGST and SGST: $$\text{CGST}=\frac{\text{GST}}{2},\text{SGST}=\frac{\text{GST}}{2}.$$

For an inter-state sale, apply IGST generally.

2. Finding the taxable value from a price including GST

• Identify the inclusive price and GST rate.
• Remove GST by division:

$$\text{Taxable value}=\frac{\text{inclusive price}\times 100}{100+\text{GST rate}}.$$

• If required, calculate the GST as the difference between the inclusive price and the taxable value.

The inclusive price must not be reduced by simply subtracting the GST percentage, because the GST percentage is applied to the taxable value rather than directly to the inclusive price.

3. Calculating simple interest and amount

• Identify the principal $P$, annual rate $R\%$, and time $T$ in years.
• If time is given in months, convert it into years.
• Substitute into

$$\text{SI}=\frac{P\times R\times T}{100}.$$

• Calculate the amount:

$$A=P+\text{SI}.$$

4. Calculating compound interest

For annual compounding:

• Identify $P$, $R$, and the number of years or periods $n$.
• Calculate the compound amount:

$$A=P{\left(1+\frac{R}{100}\right)}^n.$$

• Calculate compound interest:

$$\text{CI}=A-P.$$

For half-yearly compounding:

• Use the rate for half a year:

$$\frac{R}{2}.$$

• Use $2n$ as the number of periods.
• Substitute these values into the compound-amount formula.

For quarterly compounding:

• Use the rate for one quarter:

$$\frac{R}{4}.$$

• Use $4n$ as the number of periods.
• Apply the compound-amount formula using the adjusted rate and number of periods.

5. Calculating recurring-deposit interest and maturity value

• Identify the monthly instalment $p$, number of months $n$, and annual interest rate $R\%$.
• Calculate the recurring-deposit interest:

$$I=p\times \frac{n(n+1)}{2\times 12}\times \frac{R}{100}.$$

• Calculate the total instalments:

$$pn.$$

• Calculate the maturity value:

$$M=pn+I.$$

The formula reflects that each monthly instalment remains deposited for a different length of time.

6. Calculating investment, dividend, and yield on shares

• Identify the number of shares, market value per share, face value per share, dividend rate, and brokerage.
• Calculate the purchase amount:

$$\text{Number of shares}\times \text{market price per share}.$$

• Add brokerage when applicable:

$$\text{Investment}=\text{purchase amount}+\text{brokerage}.$$

• Calculate dividend per share using face value:

$$\text{Dividend per share}=\frac{\text{dividend rate}}{100}\times \text{face value}.$$

• Calculate total dividend:

$$\text{Total dividend}=\text{number of shares}\times \text{dividend per share}.$$

• Calculate yield on the actual investment:

$$\text{Yield percentage}=\frac{\text{annual dividend}}{\text{actual investment}}\times 100.$$

• For profit or loss, compare the selling price with the purchase price and include brokerage and other charges.

Where It Goes Wrong

• GST is calculated on the taxable value, not usually on the marked price when a discount has already been allowed.
• In reverse GST calculations, the taxable value is found by division:

$$\frac{\text{inclusive price}\times 100}{100+\text{GST rate}},$$ rather than by simply subtracting the GST percentage.

• For an intra-state sale, GST must be divided equally into CGST and SGST; for an inter-state sale, IGST is generally used.
• Simple interest requires time in years when the rate is annual; months must therefore be converted into years.
• Half-yearly and quarterly compound interest require both the adjusted rate and the adjusted number of compounding periods: $R/2$ with $2n$, or $R/4$ with $4n$.
• Dividend rates are calculated on face value, whereas yield is calculated using the actual investment based on market value and applicable brokerage.

What Gets Asked

This material supports questions requiring students to:

• Calculate GST, the final price including GST, and the taxable value from an inclusive price.
• Determine CGST and SGST for an intra-state sale and identify IGST for an inter-state sale.
• Apply a discount before calculating GST.
• Calculate simple interest and amount from principal, annual rate, and time.
• Calculate compound amount and compound interest under annual, half-yearly, and quarterly compounding.
• Calculate recurring-deposit interest and maturity value from the monthly instalment, number of months, and annual rate.
• Find the total investment in shares, including brokerage where applicable.
• Calculate dividend per share and total dividend from face value and dividend rate.
• Calculate yield from annual dividend and actual investment.
• Determine profit or loss on shares after considering purchase price, selling price, brokerage, and other charges.
• Distinguish among taxable value, principal, face value, market value, investment, interest, dividend, brokerage, yield, and maturity value.