Profit , Loss & Discount

1) Introduction

The SELLING PRICE (SP) and the COST PRICE (CP) of an item determine the profit or loss made on the particular transaction.

a) Cost Price (CP): The price at which an item is purchased .

b) Sale Price (SP): The price at which an item is sold.

c) Profit or Gain: If Sale Price (SP) is greater than Cost Price (CP) , the difference between SP and CP is said to be the profit or gain for the seller.

Example 1: A shopkeeper bought a chair for Rs 500 and sold it for Rs 600. What is his profit and profit percentage?

Given : Selling price of the chair = Rs 600

Given : Cost price of the chair = Rs 500

d) Loss: If Cost Price (CP) is greater than Sale Price (SP) , the difference between CP and SP is said to be the loss incurred by the seller.

**Imprtant Note : **

In the case of profit : Selling Price ( SP ) = (100% + Profit%) of CP

In the case of loss, Selling Price ( SP ) = (100% - Loss%) of CP

Example 2: A shopkeeper bought a chair for Rs 500 and sold it for Rs 400. What is his loss and loss percentage?

Given : Selling price of the chair = Rs 400

Given : Cost price of the chair = Rs 500

Example 3: A sells a suitcase to B at 10% profit. B sells it to C at 30% profit. If C pays Rs 2860 for it , then what is the price at which A bought the suitcase?

Assumption : Let the price at which A bought the suitcase be x.

CP of B = x (110/100) = 1.1 x

CP of C = (1.1x) (130/100) = (1.1x)(1.3)

Given : CP of suitcase for C = Rs 2860

So : (1.1x) (1.3) = Rs 2860

x = 2860/(1.1)*(1.3) = Rs 2000

So CP for A is Rs 2000

Example 4: Ajay sold his bag at a loss of 6 % . Had he sold it for Rs 42 more he would have made a profit of 8%.Find the cost price of the bag.

Assumption : Let the Cost Price (CP) of the bag = Rs 100

Given : Ajay sold his bag at a loss of 6 % . So the SP = Rs 94

Given : Had he sold the bag at 8% profit , SP would have been = Rs 108

Difference in the SP = 108 - 94 = Rs 14 when CP = Rs 100

Differebce in the SP = Rs 42 when CP = Rs x

So x i.e. CP = 42 * 100 / 14 = Rs 300

e) Marked Price (MP): The price that is indicated or marked on the item by seller is called marked price (MP). Generally marked price is the price we see as the M.R.P on the item.

Example 5: A trader marks his product 20% above the cost price and offers a discount of 30% . Find the loss percentge incurred by trader ?

Assumption : Let the Cost Price (CP) = Rs 100

Given : Marked Price (MP) is 20% above Cost Price i.e. Rs 120

Given : Selling Price (SP) is 30% discount on Marked Price = (70/100) * 120 = Rs 84

(f) Discounts:

The amount of interest deducted in the purchase or sale of or the loan of money on negotiable instruments is referred to as discount offered on that instrument.

Discounts and allowances are reductions to a basic price of goods or services.

**Note 1: Successive Discounts **

Certain discount is given on an item whose Marked Price is MP. If further discounts are given on this discounted price , such discounts are referred to as Successive Discount.

If the Successive Discount are p% , q% and r% on an item whose selling price is SP , then the effective price after all the discounts are as below :

Example 6: The price of Sony Laptop is marked at Rs.30,000. If successive discounts of 30%, 20% and 10% be allowed, then at what price does a seller sells it or a customer buy it?

Given : Marked Price (MP) of the Sony Laptop = Rs 30,000

Given : Successive Discount = 30 % (p) , 20 % (q) and 10% (r )

**Note 2 :**

When 2 items are SOLD at the same price (i.e. their SP is the same) , such that there is a PROFIT of p% on one item and a LOSS of p% on the other item (i.e. common profit and loss percentage) then , irrespective of what the SP actually is , then NET RESULT of transaction is LOSS .

This percentage LOSS is given by :

Example 7: A man sells two watches one at profit of 20% and another at loss of 20% but the SP of each watch is Rs 300. Find the Profit or Loss % ?

As above note , SP (Rs 300) of both the watches are same and profit and loss % is all same i.e 20% . So it will be definitely LOSS .

**Note 3 :**

A dealer purchases a certain number of items at x a rupee and the same number at y a rupee. He mixes them together and sells them at z a rupee , then :

Profit in above if positive sign & Loss in above if negative sign

Example 8: A fruit vendor buys 100 apples at Rs 50 each and buys 100 mangoes at Rs 30 each. He mixes them together and sells then at Rs 45 each. What is the profit or loss he makes ?

Given : No of apples and mangoes purchases = 100

Given : CP of Each Apple = Rs 50 (assume x)

Given : CP of each Mango = Rs 30 (assume y)

Given : SP of each item after mixing : Rs 45 (assume z)

**SOLVED EXAMPLES **

Example 9: Varun bought 12 kg. of sugar at the rate of Rs.15 per kg. and 20 kg. of sugar at the rate of Rs.17 per kg. He mixed the two varieties and sold the mixture at the rate of R.20 per kg. What was his total gain by doing so?

(1) < 25% gain (2) < 25% loss (3) > 25% gain (4) no loss, no gain

Given : Cost price of 1 kg. sugar of first type = Rs.15.

Then, Cost price of 12 kg. sugar of first type = 12 x 15 = Rs. 180.

Given : Cost price of 1 kg. sugar of second type = Rs.17.

Then, Cost price of 20 kg. sugar of second type = 20 x 17 = Rs. 340.

Total Cost price (CP) = Rs.180 + Rs.340 = Rs. 520.

Given : Selling price of 1 kg. sugar = Rs. 20

Total Selling Price (SP) = Selling price of 12 + 20 = 32 kg sugar = 32 x 20 = Rs. 640.

Gain = S.P. - C.P. = 640 - 520 = Rs. 120.

Gain % = Gain/CP x 100 = 120/520 x 100 = 23%.

Example 10: Ram sells a product at 20% profit. If it was bought at 10% less and sold it for Rs 21 less, he would have 30%. Find the cost price of the product (in Rs.)

(1) Rs 800 (2) Rs 700 (3) Rs 1400 (4) Rs 1200

Let the CP be Rs x

First SP = 120 % of x = (120/100) * x = 6x/5

Second CP = 90% of x = (90/100) * x = (9x/10)

Second SP = 130 % of (9x/10) = (130/100) * (9x/10) = 117x/100

Therefore, 6x/5 - 117x/100 = 21

Or 3x = 2100

Hence x = CP = Rs 700

Example 11: Ramesh buys apples at the rate of 3 kg for Rs 21 and sells at 5 kg for Rs 50. To earn Rs. 102 as profit he must sell?

(1) 51 kg (2) 34 kg (3) 102 kg (4) 17 kg

Given: CP of 3 kg = Rs 21

Given: SP of 5 kg = Rs 50

So, CP of 1 Kg apples = Rs 7

And SP of 1 Kg of apples = Rs 10

Thus, gain per kg = Rs 3

Hence, gain of Rs. 102 will be obtained by selling 102/3 = 34 kg.

Example 12: If the selling price of 12 paintings is same as the cost price of 15 paintings. What will be the gain percent of the shopkeeper?

(1) 20% (2) 30% (3) 25% (4) 40%

Let the Cost Price of each Painting = Rs 1

Then, Cost Price of 12 Paintings = Rs 12

So, the Sell Price of 12 Painting = Rs 15

As given that selling price of 12 Paintings = Cost Price of 15 Paintings

Gain or Profit = Rs 3

Therefore, Gain or Profit % = (3/12) x 100 = 25 %

Example 13: Varun buys a certain number of bananas at 8 for Rs 10 and an equal number at 10 for Rs 15. If he sells them at 15 for RS 20, does he gain or loss and by what percentage?

(1) No profit, No loss (2) Loss of 3 % (3) Loss of 5 % (4) Profit of 5 %

Let the number of bananas that Varun bought be 2x i.e. x at each of the two prices.

The Cost Price of x bananas at 8 for Rs 10 = x (10/8) = Rs 5x/4

The Cost Price of x bananas at 10 for Rs 15 = x (15/10) = Rs 3x/2

Total Cost Price of the bananas = 5x/4 + 3x/2 = 11x/4 Rs

Total Selling Price of the bananas = (2x) 20/15 = Rs 8x/3

As 8x / 3 < 11x/4, there is a loss.

Loss % = ((CP-SP)/CP)x 100 = ((11x/4 - 8x/3)/(11x/4))x100 = 100/33 %

Hence Varun incurred a loss of 100/33% = 3.03%

Example 14: If goods are purchased for Rs 120 and one-third of them are sold at a loss of 5%, then at what profit percentage should the rest be sold to obtain an overall profit percentage of 5%?

(1) 10 % (2) 12 % (3) 15 % (4) 5 %

Cost price of one-third of goods = 1/3 (120) = Rs 40

Selling Price of these goods at 5% Loss = 40* (95/100) = Rs 38

Let Selling Price of the rest of the goods be x

5% profit on Rs 120 gives SP as Rs 126

126 = x + 38

So, x= Rs 88

As Cost Price of the remaining goods is Rs 80, required profit % as below:

Profit % = ((SP-CP)/CP)x 100

= ((88-80)/80)x 100 =10 %

Example 15: If Varun sells an item at three-fourths of its Selling Price he incurs a loss of 4%. What will be the profit or loss percentage if he sells it at the actual selling price?

(1) 20 % Loss (2) 28 % loss (3) 28% Profit (4) 30 % Profit

Let the Cost Price be Rs 100

When sold at 3/4th of the Selling Price (SP) , the loss is 4 %

This means Selling Price in this case = Rs 96

= 3/4 times the actual selling price

Hence 96 = (3/4) [Actual Selling Price]

So, Actual Selling Price = 96 (4/3) Rs 128

If Varun sells at the actual SP then he makes a profit of Rs 28 on a Cost Price (CP) of Rs 100 i.e. 28 % Profit

Example 16: A trader marks his goods at a certain percentage over his cost price and then gives a 30% Discount, thereby making 5% Profit. What is the mark up percentage?

(1) 20 % (2) 25 % (3) 30 % (4) 50 %

Let Cost Price (CP) be Rs. 100

So, Selling Price will be Rs.105

Since 30% Discount was given, 70% of MP = Sale Price = 105, where MP is the marked Price

So, Marked Price = Rs 150

Hence Marked Price is Rs 50 above the Cost Price i.e. 50% above the Cost Price of Rs 100.

Example 17: The Profit percentage of the three articles A, B and C is 10%, 20% and 25% and the ratio of the cost price is 1: 2 :4. Also the ratio of number of articles sold of A, B and C is 2 : 5 : 2, then the overall profit percentage is

(1) 20 % (2) 21% (3) 22% (4) None of these

Let Cost Price (CP) of A, B and C = x , 2x and 4x Rs

And Number of Articles Sold = 2y, 5y and 2y

Hence, total Cost Price of A, B and C = x * 2y + 2x * 5y + 4x * 2y= 20 xy

Profit on A = 2xy * 10 / 100 = 0.2 xy

Profit on B = 10xy * 20/100 = 2xy

Profit on C = 8xy * 25 / 100 = 2xy

Hence Total Profit = 0.2 xy + 2xy + 2xy = 4.2 xy

Profit % = ((SP-CP)/CP)x 100

= ((SP-CP)/CP)x 100 = (Profit/CP)x 100

= ((4.2xy)/20xy)x 100 = 21%

Example 18: Even after a discount of y% on a marked price, a businessman gains by x %. What is the markup percentage over the Cost Price?

(1) ((x+y)/(y-x))*100 (2) ((x+y)/(100-x))*100 (3) ((x+y)/(100-y))*100 (4) Not Possible

Selling Price = Marked Price – Discount

SP = M - (y/100) x M

= 100M - yM/100

= M (100 - y)/100

Given: There is gain at x %.

Hence, As we know in the case of profit, SP = (100% + Profit%) of CP

CP = ((SP*100)/(100+x))

CP = ((M (100-y))/100)*100/(100+x) = M(100-y)/(100+x)

Marked Up Price = Marked Price – Cost Price i.e. amount which is marked or increased on Cost Price

= M - CP

= M - M(100-y)/(100+x) = (x+y)/(100-x)*100

Example 19: Sonal sold mobile at 96 Rs. in such a way that her percentage profit is same as the cost price of the mobile. If she sells it at twice the percentage profit at its previous percentage profit, then the new selling price will be?

(1) Rs 132 (2) Rs 140 (3) Rs 150 (4) Rs 156

Let CP of the mobile = x Rs.

Profit % = Cost Price, So Profit % = x/100

Selling Price = Rs. 96

In the case of profit: Selling Price (SP) = (100% + Profit%) of CP

SP = (100% + x/100) * x = x + x * x/100

96 = x + x * x/100

So, x = CP = Rs. 60

Need to find SP of twice profit.

So, Profit = 120%

SP of twice profit = 60 + 60 x (120/100) = 132 Rs