Hello, and welcome to today's mathematics lesson. Today, we are going to explore profit, loss and discount. This is a practical and important topic that you will encounter in everyday life — whether you are buying something from a shop or selling something yourself. We will learn how to calculate profit and loss, how to find selling price and cost price when certain information is given, and finally, we will understand what discount means and how it affects the final price you pay.
Let us begin with the fundamental concepts of profit and loss. Every transaction involves two important prices. The cost price, written as C.P., is the price at which an article is purchased. The selling price, written as S.P., is the price at which that article is sold.
Now, when the selling price is greater than the cost price, the seller makes a profit. Profit equals selling price minus cost price. That is, Profit = S.P. − C.P.. From this, we can also say that S.P. = C.P. + Gain, or simply S.P. = C.P. + Profit.
On the other hand, when the selling price is less than the cost price, the seller suffers a loss. Loss equals cost price minus selling price. That is, Loss = C.P. − S.P.. And therefore, S.P. = C.P. − Loss.
Here is a crucial point you must always remember. Profit percent and loss percent are always calculated on the cost price only, never on the selling price. The formulas are: Profit % = (Profit/C.P.) × 100% and Loss % = (Loss/C.P.) × 100%.
Remember, both are calculated on cost price only.
Let us work through an example to make this clear. Suppose an article is bought for 120 rupees and sold for 150 rupees. The gain is 150 minus 120, which equals 30 rupees. The gain percent is 30 divided by 120, times 100 percent, which equals 25 percent.
Now consider another case: a bicycle bought for 600 rupees is sold for 550 rupees. Here, the loss is 600 minus 550, which equals 50 rupees. The loss percent is 50 divided by 600, times 100 percent, which equals 25 over 3 percent, or 8 1/3 percent.
Next, let us learn how to find the selling price when we know the cost price and the profit or loss percent. There are two ways to approach this.
First, the step-by-step method. Calculate the actual profit or loss amount using the percentage, then add to or subtract from the cost price. For example, if a watch costs 450 rupees and you want a 10 percent gain, the gain is 10 percent of 450, which is 45 rupees. So the selling price is 450 plus 45, equals 495 rupees.
Second, the direct method, which is faster. If gain is 10 percent, the selling price is 110 percent of the cost price. So S.P. = (110/100) × C.P., which gives (110/100) × 450, equals 495 rupees.
Similarly, for a loss of 25 percent on an article costing 800 rupees, the selling price is 75 percent of the cost price. That is, S.P. = (75/100) × 800, which equals 600 rupees.
Remember these shortcuts: if gain is 20 percent, selling price is 120 percent of cost price. If loss is 20 percent, selling price is 80 percent of cost price.
Sometimes, the cost price and selling price are given for different quantities of articles. In such cases, you must first find the cost price and selling price of one article, or of an equal number of articles, before calculating profit or loss percent.
For instance, if a shopkeeper buys 50 pencils for 80 rupees and sells them at 40 pencils for 90 rupees, we first find the unit prices. The cost price of one pencil is 80 divided by 50, which equals 1 rupee 60 paise. The selling price of one pencil is 90 divided by 40, which equals 2 rupees 25 paise. The gain per pencil is 65 paise, and the gain percent is 0.65 divided by 1.60, times 100 percent, which equals 40 5/8 percent, or 40.625 percent.
Now, let us turn to finding the cost price when the selling price and profit or loss percent are known.
Suppose by selling an article for 550 rupees, a profit of 10 percent is made. We want to find the cost price.
Using the unitary method: if cost price is 100 rupees, profit is 10 rupees, so selling price is 110 rupees. When selling price is 110 rupees, cost price is 100 rupees. Therefore, when selling price is 1 rupee, cost price is 100/110 rupees, which is 10/11 rupees. And when selling price is 550 rupees, cost price is (100/110) × 550, which simplifies to (10/11) × 550, which equals 500 rupees.
Alternatively, using algebra: let cost price equal x rupees. Then profit equals 10 percent of x, which is x over 10. Since cost price plus gain equals selling price, we have x + x/10 = 550. This gives 11x/10 = 550, so x equals 500 rupees.
For a loss situation: if an article is sold for 270 rupees at a loss of 10 percent, we can find the cost price similarly. With 10 percent loss, selling price is 90 percent of cost price. So 270 = (90/100) × C.P., giving C.P. = 270 × (100/90), which equals 300 rupees.
Once we have the cost price, we can also find what the new selling price should be to gain a different percentage. For example, to gain 12 percent on a cost price of 300 rupees, the new selling price would be 112% of 300, which equals 336 rupees.
An important practical consideration: sometimes extra expenses are incurred after purchase, such as transportation or repairs. These must be added to the cost price to get the total cost price.
For example, Rohit buys an article for 2400 rupees and spends 100 rupees on transportation. His total cost price is 2500 rupees. If he sells it for 3000 rupees, his profit is 500 rupees, and his profit percent is (500/2500) × 100%, which equals 20 percent.
Now we come to the final section: Discount. When you shop, you often see price tags on articles. This tagged price is called the Marked Price, or M.P., also known as list price or advertised price.
Shopkeepers often offer a discount — a reduction in the marked price. Discount is always calculated on the marked price, not on the cost price. The fundamental formula is: S.P. = M.P. − Discount.
Let us see how this works. A dealer marks a television set for 9000 rupees but agrees to give 20 percent discount. The discount amount is 20% of 9000, which equals 1800 rupees. Therefore, the selling price is 9000 minus 1800, equals 7200 rupees.
We can also find discount percent when we know the marked price and selling price. If an article marked at 155 rupees is sold for 124 rupees, the discount is 155 minus 124, equals 31 rupees. The discount percent is (31/155) × 100%, which equals 20 percent.
Sometimes problems involve multiple steps: a shopkeeper buys an article, marks it up by a certain percentage, then offers a discount. For example, buying for 300 rupees, marking up by 20 percent, then giving 10 percent discount on the marked price. The marked price becomes 360 rupees, the discount is 36 rupees, so selling price is 324 rupees. The actual profit is 24 rupees, and the profit percent is (24/300) × 100%, which equals 8 percent.
Let us now recap the key takeaways from this chapter.
First, profit equals selling price minus cost price, and loss equals cost price minus selling price. Profit or loss percent is always calculated on cost price.
Second, to find selling price from cost price and profit or loss percent: if profit, selling price equals (100 + Profit%)/100 × C.P.; if loss, selling price equals (100 − Loss%)/100 × C.P..
Third, to find cost price from selling price and profit or loss percent: if profit was made, cost price equals 100/(100 + Profit%) × S.P.; if loss was made, cost price equals 100/(100 − Loss%) × S.P..
Fourth, always include additional expenses like transportation and repairs in the total cost price.
Fifth, discount is a reduction on the marked price, and selling price equals marked price minus discount.
Sixth, discount percent equals (Discount/M.P.) × 100%. This is always calculated on the marked price, not the cost price.
That brings us to the end of our lesson on Profit, Loss and Discount. I hope you now feel confident in tackling these calculations. Remember, mathematics becomes powerful when you apply it to real situations — so notice these concepts the next time you go shopping or help with a family business. Keep practicing, stay curious, and I look forward to seeing you in the next lesson. Goodbye!