Commercial Mathematics

The One Thing

Commercial mathematics depends on identifying the correct reference amount before applying a percentage. Profit and loss use the cost price, discount uses the marked price, simple interest uses the principal, and tax uses the taxable value.

Definitions and Results

• Cost Price (CP): The price paid to buy an article, including any directly related expenses when specified.
• Selling Price (SP): The price at which an article is sold.
• Profit: The gain made when the selling price is greater than the cost price.
• Loss: The amount lost when the selling price is less than the cost price.
• Profit Percent: Profit expressed as a percentage of the cost price.
• Loss Percent: Loss expressed as a percentage of the cost price.
• Marked Price (MP): The price printed or displayed on an article before allowing a discount.
• Discount: The reduction offered on the marked price.
• Discount Percent: The discount expressed as a percentage of the marked price.
• Principal (P): The original sum of money borrowed, invested, or deposited.
• Rate of Interest (R): The percentage charged or earned per year on the principal.
• Time (T): The period for which money is borrowed or invested, usually measured in years.
• Simple Interest (SI): Interest calculated only on the original principal throughout the given time.
• Amount (A): The total money obtained by adding simple interest to the principal.
• Tax: A compulsory charge collected by the government on goods, services, income, or transactions.
• Taxable Value: The value on which tax is calculated.
• Final Bill: The total amount payable after adding tax and applying any stated discount.
• Profit and loss conditions:

$$\text{Profit}=SP-CP$$ when $SP>CP$, and $$\text{Loss}=CP-SP$$ when $CP>SP$.

• Profit percentage:

$$\text{Profit percent}=\frac{\text{Profit}}{CP}\times 100.$$

• Loss percentage:

$$\text{Loss percent}=\frac{\text{Loss}}{CP}\times 100.$$

• Selling price with profit:

$$SP=CP+\text{Profit}=CP\times \frac{100+\text{Profit percent}}{100}.$$

• Selling price with loss:

$$SP=CP-\text{Loss}=CP\times \frac{100-\text{Loss percent}}{100}.$$

• Cost price when there is a profit:

$$CP=SP\times \frac{100}{100+\text{Profit percent}}.$$

• Cost price when there is a loss:

$$CP=SP\times \frac{100}{100-\text{Loss percent}}.$$

• Discount:

$$\text{Discount}=MP-SP.$$

• Discount percentage:

$$\text{Discount percent}=\frac{\text{Discount}}{MP}\times 100.$$

• Selling price after discount:

$$SP=MP-\text{Discount}=MP\times \frac{100-\text{Discount percent}}{100}.$$ For successive discounts, apply each discount one after another rather than usually adding the percentages directly.

• Simple interest:

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

• Amount:

$$A=P+SI.$$ Simple interest remains constant each year because it is calculated only on the original principal, not on accumulated interest.

• Time conversion: If time is given in months,

$$T=\frac{\text{number of months}}{12}.$$ If time is given in days, use the convention stated in the problem; commonly, $$T=\frac{\text{number of days}}{365}$$ for one year.

• Tax:

$$\text{Tax}=\frac{\text{Tax rate}}{100}\times \text{Taxable value}.$$

• Final price including tax:

$$\text{Final price}=\text{Taxable value}+\text{Tax}.$$

• Price before tax: When the final price already includes tax,

$$\text{Price before tax}=\text{Final price}\times \frac{100}{100+\text{Tax rate}}.$$

• Tax and discount order: If tax is charged on the discounted price, first apply the discount and then calculate tax on the discounted price as the taxable value.
• Percentage bases: Profit and loss percentages are based on $CP$, discount percentage is based on $MP$, and simple interest is based on $P$.
• Percentage changes: An equal percentage increase and decrease do not generally cancel because they are calculated on different base amounts.
• Units and notation: Money and time must use consistent units, the currency unit should be written in the final answer, and percentages may be expressed as fractions or decimals provided the chosen form is used consistently.

Worked Methods

Profit and loss

• Identify the cost price $CP$ and selling price $SP$.
• Compare the two prices:

- If $SP>CP$, calculate profit using $$\text{Profit}=SP-CP.$$ - If $CP>SP$, calculate loss using $$\text{Loss}=CP-SP.$$

• If required, calculate the percentage using $CP$ as the base:

$$\text{Profit percent}=\frac{\text{Profit}}{CP}\times 100,\text{Loss percent}=\frac{\text{Loss}}{CP}\times 100.$$

• For a known profit percentage, use

$$SP=CP\times \frac{100+\text{Profit percent}}{100}.$$

• For a known loss percentage, use

$$SP=CP\times \frac{100-\text{Loss percent}}{100}.$$

• If the selling price and profit percentage are known, find the cost price using

$$CP=SP\times \frac{100}{100+\text{Profit percent}}.$$

• If the selling price and loss percentage are known, use

$$CP=SP\times \frac{100}{100-\text{Loss percent}}.$$

Discount

• Identify the marked price $MP$.
• Calculate the discount:

$$\text{Discount}=MP-SP.$$

• If the discount percentage is required, divide by the marked price:

$$\text{Discount percent}=\frac{\text{Discount}}{MP}\times 100.$$

• To find the selling price after a discount, use

$$SP=MP-\text{Discount}$$ or $$SP=MP\times \frac{100-\text{Discount percent}}{100}.$$

• For successive discounts, apply the first discount to the marked price and then apply the next discount to the reduced price. The discount percentages should not usually be added directly.

Simple interest

• Identify the principal $P$, annual rate $R$, and time $T$.
• Convert the time into years:

$$T=\frac{\text{number of months}}{12}$$ when time is given in months.

• When time is given in days, use the convention stated in the question; commonly,

$$T=\frac{\text{number of days}}{365}.$$

• Substitute into

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

• Find the total amount using

$$A=P+SI.$$

• Since simple interest is calculated only on the original principal, the interest remains constant each year rather than being calculated on accumulated interest.

Tax

• Identify the taxable value and the tax rate.
• Calculate the tax:

$$\text{Tax}=\frac{\text{Tax rate}}{100}\times \text{Taxable value}.$$

• Add the tax to the taxable value:

$$\text{Final price including tax}=\text{Taxable value}+\text{Tax}.$$

• If a discount is also present and the question states that tax is charged on the discounted price, apply the discount first.
• Use the discounted price as the taxable value, then calculate and add the tax.
• If the final price already includes tax, recover the price before tax using

$$\text{Price before tax}=\text{Final price}\times \frac{100}{100+\text{Tax rate}}.$$

General percentage procedure

• Identify the quantity that forms the base: $CP$, $MP$, $P$, or taxable value.
• Identify the relevant operation: profit, loss, discount, simple interest, or tax.
• Select the corresponding formula.
• Substitute the values with consistent units.
• Check whether the result is reasonable and include the currency unit where appropriate.

Where It Goes Wrong

• Using $SP$ instead of $CP$ as the base for profit or loss percentage.
• Using $SP$ instead of $MP$ as the base for discount percentage.
• Calculating simple interest on accumulated interest rather than only on the original principal $P$.
• Forgetting to convert months into years, using $T=\text{number of months}/12$, or failing to apply the stated convention for days.
• Adding successive discount percentages directly instead of applying them one after another.
• Applying tax to the wrong value when both discount and tax are present; if tax is charged on the discounted price, the discount must be applied first.

What Gets Asked

This material supports questions requiring students to:

• Calculate profit, loss, profit percentage, or loss percentage from $CP$ and $SP$.
• Find $SP$ from $CP$ and a stated profit or loss percentage.
• Find $CP$ from $SP$ and a stated profit or loss percentage.
• Calculate a discount, discount percentage, or selling price after discount from $MP$.
• Apply successive discounts in the correct order.
• Calculate simple interest and the amount from $P$, $R$, and $T$.
• Convert time given in months or days into years before using the simple-interest formula.
• Calculate tax from a tax rate and taxable value.
• Find the final bill after applying a discount and adding tax.
• Recover the price before tax when the final price already includes tax.
• Identify the correct base amount and explain why an equal percentage increase and decrease do not generally cancel.
• Check calculations for appropriate units, currency, order of operations, and reasonable results.