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Chapter 04 : Percentages, Profit and Loss> Profit & Loss



Basic concepts in profit and loss

1. Cost price (C.P): The price at which an article is bought

2. Selling price (S.P): The price at which an article is sold

3. Marked Price: The price listed on the label

4. Discount: The reduction offered on the list prize, it may be a value or a percent

5. Mark-up: The increment over the cost price

6. Profit or loss = SP – CP (the negative value indicates the loss)

7. Profit or loss percentage = (SP-CP)/CP x 100 % (the negative value indicates the loss percentage). This formula is the most important formula, if you use only this formula in the entire profit and loss chapter, you will never make an error, so try following it as much as possible.

8. To calculate gain/loss percentage, it is not required to have all the values of cost price and selling price, you can assume the values to be x, or even 10 or 100.

Example: A shopkeeper puts the marked price on his goods 25 % above cost price and gives discount of 12.5 % on marked price. What is his profit %?
Let the CP be 10, then marked price is 1.25*10 = 12.5
Now discount to the cash purchaser = 12.5 %, which is 0.125*12.5 = 1.56
Net gain = 2.5(excess on CP) – 1.56 = 0.94
gain % = 0.94/10*100 = 9.4 %

Important Relations

1. If two items are sold each at rupees R, one at a gain of X % and other at a loss of X %, there is always an overall loss given by ( X2 / 100 ) % and the value of loss is given by ( 2X2S )/( 1002 – X2 ). In case of the cost price of both the items is the same and percentage loss and gain are equal, then net loss or profit is zero. The difference between the two cases is the cost price, in first case it is not same, in the second it is same.

Example: Amar sells two watches for Rs 1200, one at a profit of 10 % and other at a loss of 10 %. Find his gain or loss percentage and the actual gain or loss.
Using the formula, Loss % = (X2/100) % = 10 x 10/ 100% = 1 %

The value of loss = (2 x X2 x S) / (1002 – X2)
= (2 x 10 x 10 x 1200) / (1002 – 102) = Rs 24.24.

Alternatively, SP of both watches = 1200, Total SP = 2400 (2 watches)
One watch was sold at a profit of 10%, using our basic formula:
Profit or loss percentage = (SP-CP)/CP x 100 %
10 = (1200 –CP)/CP x 100
110CP =120000
CP = 1090.90

One watch was sold at a loss of 10%, so its CP is
-10 = (1200 –CP)/CP x 100
90CP =120000
CP = 1333.33
Total CP = 1333.3333 + 1090.9090 = 2424.24
Total SP = 2400
Loss = 2424.24 – 2400 = 24.24
Loss % = 24.24/2424.24*100 = 1 %

So it is important to understand both formula and basic logic for the question, as explained earlier, if the basic formula of profit and loss is used, you will never go wrong.

Students should learn to do the above calculations in one step, using the formula and concept of percentage, if an item is sold at 10% profit, then:
CP = SP
CP = SP/1.1

2. A dishonest shopkeeper claims to sell goods at cost price, but uses a lighter weight , then his Gain % = [ 100 x excess / ( original value – excess ) ]

Example: A shopkeeper sells rice to a customer, using false weight and gains 100/8 % on his cost. What weight has he substituted for a kilogram?

Using the formula, Gain % = [100 x excess / (original value – excess)]
100/8 = [100 x excess/(1 – excess)]
From here, Excess = 0.111.. Kg, which is 111.11 grams
Weight used by shopkeeper = 1000 – 111.11 = 888.89 grams

Alternatively, if he is selling 1 gram for Re. 1, then 1000 grams are for Rs. 1000, so once a shopkeeper sells 1 kg, he makes Rs. 1000, using the basic profit and loss formula, Profit or loss percentage = (SP-CP)/CP x 100
100/8 = (1000 – CP)/CP x 100
100 CP = 800000 – 800CP
CP = 8000/9 = 888.89
Since here grams is equal to rupees, the weight used is 888.89 grams

Again we establish that the basic formula works well everywhere.

Miscellaneous Examples:

Q11. Ravi sells two tables at same price, one at a profit of 10 % and other at a loss of 10 %. Find his gain or loss percent

Ans. As per the formula, there will be loss and it will be given by (X2/100) %
= 10 x 10/ 100% = 1 %

Q12. A bookshop sells an old book for Rs. 49.35, making a 6% loss on the cost. What was the cost price of the book, and what is the cash value of the loss?
Ans. Since loss is 6%, and assuming cost of the book is ‘A’
0.94 A = 49.35
A = Rs. 52.50
Value of the loss = 52.50 – 49.35 = Rs. 3.15

Q13. If the cost price of 20 articles is equal to the selling price of 25 articles, what is the % profit or loss made by the merchant?

Ans. CP of 20 Articles = SP of 25 articles
Which is CP of 20 Articles = SP of 20 articles - SP of 5 articles
Now SP of 5 articles = SP of 20 articles - CP of 20 Articles
SP of 5 articles = Loss (Since CP is greater than SP in this case)
Also SP of 5 articles = CP of 4 articles (as per our first equation)
Loss Percent = (SP of 5 articles)/( CP of 20 Articles) x 100
= (CP of 4 articles)/( CP of 20 Articles) x 100
= 1/5 x 100 = 20%

Alternatively, assume the CP = 100, CP of 20 articles will be 2000
And SP of 25 articles will also be 2000, therefore SP of one article is 80, since CP is 100, loss is 20%

Q14. If SP of 10 articles is equal to CP of 12 articles. What is the gain percent?

Ans. Assume the CP = 100, CP of 12 articles will be 1200
And SP of 10 articles will also be 1200, therefore SP of one article is 120, since CP is 100, gain is 20%

Q15. If Kapil had sold a stereo for Rs. 6000, he would have made a 20% profit. Instead, he sold it for a 40% loss. At what price was the stereo sold?

Ans. Here at 6000 he was making 20% profit, therefore 1.20 CP = 6000, CP = 5000
And since he sold at a 40 percent loss, it is 0.60 CP
Which is 0.60 x 5000 = Rs. 3000

Q16. Rajan sold an article at a profit of 30 %; and had he sold it for Rs 3 more, the profit would have been 40 %. Find the cost price.

Ans. Suppose of the CP = 100. To Sell at gain of 30%, SP = 1.3 x 100 = 130
Now, To Sell at gain of 40%, SP = 1.4 x 100 = 140
Here when the difference in 30% and 40% is 10 (140-130) then CP is 100
When the difference is 1 the CP would be 100/10 = 10
When the difference is 3 the CP would be 10 x 3 = 30

Alternatively, it is given 40 % - 30 % of CP = Rs 3
Therefore 10% of CP = Rs. 3
CP = 3 x 100/10 = Rs. 30

Q17. A Vegetable vendor buys 240 kilograms of potato for Rs. 380. If 20 percent of the potato is unusable, at what average price per kilogram must he sell the rest of the potato in order to make a profit of 25 percent?

Ans. Cost price = Rs. 380, Total potato bought = 240 kg
Potato unusable = 20% = .2 x 240 = 48 kg
Potato left = 192 kg
To gain 25% on CP (380), the total SP should be 1.25 x 380 = 475
Selling Price per kg = 475/192 = Rs. 2.47 per kg.

Q18. By selling an article at 80% of its marked price, a merchant makes a loss of 12%. What will be the % profit made by the merchant if he sells the article at 95% of its marked price?

Ans. Let the marked price be 100. When the SP is 80%, which is 80, he is at a loss of 12 %, therefore by the basic formula (80 – CP)/CP x 100 = -12
8000 – 100CP = -12CP
CP = 8000/88 = 90.90
If the article is sold at 95%, which is 95 here, profit = 95 – 90.9 = 4.1
Profit % = 4.1/90.9 x 100 = 4.51 %


Q19. A shopkeeper marks his goods at 30 % above cost price and allows discount of 15 % for cash payment. What profit % does he make?

Ans. Assume the cost price = 100, so the marked price (30% more) = 1.3 x 100 = 130
A discount of 15% on the same is 0.15 x 130 = 19.50
Actual SP = 130 – 19.50 = 111.50
Gain percent = 111.50 – 100 = 11.50 %( since base is 100)

Q20. Krishna sold his pen for Rs 24 and got a % of profit equal to the cost price; find the cost price.
Ans. By the basic formula, Gain % = (SP-CP)/CP x 100 %
Here gain % = CP and SP = 24
Therefore, CP = (SP-CP)/CP x 100
(CP)2 = 2400 - 100CP
(CP)2 +100CP - 2400 = 0
Solving CP = 20

Other topics covered

Number Theory