Hello, and welcome to your mathematics lesson. Today, we are diving into Profit, Loss, and Discount. We will also explore overhead expenses and tax, including the modern Goods and Services Tax system. By the end of this lesson, you will understand how to calculate profit and loss percentages, handle additional costs, work with discounts, and compute taxes on transactions. Let us begin.
First, let us review the fundamental concepts of profit and loss. When you sell an article for more than what it cost you, you earn a profit. When you sell it for less, you incur a loss.
Here are the precise definitions. Profit equals selling price minus cost price. That is, Profit = S.P. − C.P. Loss equals cost price minus selling price. That is, Loss = C.P. − S.P.
From these, we derive useful rearrangements. If you know the cost price and profit, the selling price equals cost price plus profit. If you know the selling price and profit, the cost price equals selling price minus profit. Similarly, for loss: selling price equals cost price minus loss, and cost price equals selling price plus loss.
Now, percentages are crucial in business mathematics. Profit percentage equals profit divided by cost price, multiplied by one hundred. Loss percentage equals loss divided by cost price, multiplied by one hundred. Remember this golden rule: profit and loss percentages are always calculated on the cost price, never on the selling price.
Let us work through an example to solidify these ideas. Suppose articles are bought at ten for eight rupees and sold at eight for ten rupees. To find the gain percent, first find the cost price and selling price of one article.
The cost price of one article is eight tenths of a rupee, which is eighty paise, or zero point eight rupees. The selling price of one article is ten eighths of a rupee, which is one rupee and twenty-five paise, or one point two five rupees. The profit per article is forty-five paise. The profit percentage is forty-five paise divided by eighty paise, multiplied by one hundred, giving fifty-six and one-fourth percent, or fifty-six point two five percent.
Next, we turn to overhead expenses. When you purchase goods at one location and transport them elsewhere, you incur additional costs. These include transportation, labour, packing, and other incidental charges. These are called overhead expenses or simply overheads.
Here is the key principle. Overhead expenses are added to the actual cost price to obtain the total cost price. Profit or loss is then calculated on this total cost price, not on the original purchase price alone.
Consider this example. Raju buys an article in Delhi for six thousand five hundred rupees and sells it in Agra for eight thousand rupees. He spends seven hundred rupees on travel and food. His total cost price is six thousand five hundred plus seven hundred, which is seven thousand two hundred rupees. His gain is eight thousand minus seven thousand two hundred, which is eight hundred rupees. His gain percentage is eight hundred divided by seven thousand two hundred, multiplied by one hundred, which equals eleven and one-ninth percent, or approximately eleven point one one percent.
Now, let us learn how to find the selling price when cost price and gain or loss percent are given.
Suppose Bhanu buys a fountain pen for twelve rupees and wants to gain fifteen percent. Her gain is fifteen percent of twelve rupees, which is one rupee and eighty paise. Therefore, her selling price is twelve plus one rupee and eighty paise, which is thirteen rupees and eighty paise.
Alternatively, use the direct formula. Selling price equals (100 + gain%) / 100 × C.P. For a loss, selling price equals (100 − loss%) / 100 × C.P.
Similarly, to find cost price when selling price and gain or loss percent are known, use these formulas. Cost price equals 100 / (100 + gain%) × S.P. when there is a gain. Cost price equals 100 / (100 − loss%) × S.P. when there is a loss.
Here is a powerful technique for complex problems: the unitary method. Assume the cost price is one hundred rupees, calculate the corresponding selling price, then scale appropriately.
For instance, if selling an article for three hundred sixty rupees yields a twenty percent gain, what is the cost price? Assume cost price is one hundred rupees. Gain is twenty rupees, so selling price would be one hundred twenty rupees. When selling price is one hundred twenty, cost price is one hundred. When selling price is one, cost price is one hundred over one hundred twenty. When selling price is three hundred sixty, cost price is one hundred over one hundred twenty multiplied by three hundred sixty, which equals three hundred rupees.
Now we move to discount, a concept you encounter every time you shop. Discount is a reduction offered on the marked price of goods. The marked price, also called list price or printed price, is the price tag on the article.
The fundamental relationship is: selling price equals marked price minus discount. Equivalently, discount equals marked price minus selling price.
If discount is given as a percentage, selling price equals (100 − d) / 100 × M.P. Conversely, marked price equals 100 / (100 − d) × S.P.
Consider a tradesman who marks goods thirty-five percent above cost price, then allows fifteen percent discount. Let cost price be one hundred rupees. Marked price becomes one hundred thirty-five rupees. Discount is fifteen percent of one hundred thirty-five, which is twenty rupees and twenty-five paise. Selling price is one hundred thirty-five minus twenty point two five, which is one hundred fourteen rupees and seventy-five paise. Profit is fourteen point seven five rupees, giving a profit percentage of fourteen point seven five percent.
Successive discounts deserve special attention. When two discounts are applied one after another, the second discount applies to the reduced price, not the original.
To find a single discount equivalent to successive discounts of ten percent and twenty percent, assume marked price is one hundred rupees. After first discount: one hundred minus ten equals ninety rupees. Second discount is twenty percent of ninety, which is eighteen rupees. Final selling price is ninety minus eighteen, which is seventy-two rupees. Total discount is one hundred minus seventy-two, which is twenty-eight rupees. Thus, the single equivalent discount is twenty-eight percent.
The general formula for successive discounts is: selling price equals marked price multiplied by (100 − d₁) / 100 multiplied by (100 − d₂) / 100.
Now we address tax, an essential aspect of modern commerce. Governments levy taxes on goods and services to fund public facilities, infrastructure, healthcare, education, and administrative expenses.
Tax is calculated on the sale price. Tax equals rate of tax multiplied by sale price, divided by one hundred. The total amount paid by customer equals sale price plus tax.
Alternatively, if tax rate is x percent, the price paid equals sale price multiplied by (100 + x) / 100
For example, if shoes cost eight hundred fifty rupees and tax rate is six percent, tax is fifty-one rupees. Total payment is nine hundred one rupees.
Conversely, if total price including eight percent tax is seven hundred two rupees, the sale price before tax is seven hundred two multiplied by one hundred over one hundred eight, which equals six hundred fifty rupees. This uses the direct method: sale price equals total price multiplied by one hundred over hundred plus tax rate.
Finally, we explore the Goods and Services Tax, or GST, introduced in India on July first, two thousand seventeen. This revolutionary system replaced multiple indirect taxes with one unified tax, embodying the principle of one nation, one tax.
GST comprises three components. Central GST, or CGST, is levied by the central government on intra-state transactions. State GST, or SGST, is levied by the state government on intra-state transactions. Integrated GST, or IGST, is levied by the central government on inter-state transactions and imports.
Intra-state means supply within the same state. Inter-state means supply from one state to another.
For intra-state transactions, CGST and SGST are equal, each being half the total GST rate. If GST is twelve percent, CGST is six percent and SGST is six percent. For inter-state transactions, the full GST rate applies as IGST.
Consider an intra-state purchase of goods worth twenty thousand rupees with twenty-eight percent GST. CGST is fourteen percent of twenty thousand, which is two thousand eight hundred rupees. SGST is also two thousand eight hundred rupees. Total bill amount is twenty thousand plus two thousand eight hundred plus two thousand eight hundred, which equals twenty-five thousand six hundred rupees.
For an inter-state transaction of fifteen thousand rupees with twenty percent discount and five percent GST: first, taxable amount is fifteen thousand minus three thousand, which is twelve thousand rupees. IGST is five percent of twelve thousand, which is six hundred rupees. Total bill is twelve thousand plus six hundred, which equals twelve thousand six hundred rupees.
Let us recap the key takeaways from this lesson.
First, profit equals selling price minus cost price, and loss equals cost price minus selling price. Profit and loss percentages are always calculated on cost price.
Second, overhead expenses are added to the actual cost price to get total cost price, on which profit or loss is calculated.
Third, selling price can be found using (100 ± %) / 100 × C.P., and cost price can be found using 100 / (100 ± %) × S.P.
Fourth, discount is reduction on marked price, and successive discounts apply sequentially, not simultaneously.
Fifth, tax is calculated on sale price, and total payment equals sale price plus tax.
Sixth, GST unifies indirect taxes, with CGST and SGST for intra-state transactions and IGST for inter-state transactions.
You have now mastered the essential concepts of profit, loss, discount, and taxation. These skills will serve you well, whether you are analyzing business transactions, planning a purchase, or simply understanding the world of commerce around you. Keep practicing, stay curious, and remember that mathematics is a powerful tool for real-world decision making. Until next time, happy learning.