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Important Formulas and Shortcuts to solve Profit, Loss and Discount

Cost Price Cost Price is the price at which an article is purchased, abbreviated as C.P.
Selling PriceSelling Price is the price at which an article is sold, abbreviated as S.P.
ProfitIf the Selling Price exceeds the Cost Price, then there is Profit.
Profit or gain = SP – CP
Profit % = Profit/(C P)×100
S P = (100+gain % )/100  ×C P
C P = 100/(100+gain %)×S P
Loss
If the overall Cost Price exceeds the selling price of the buyer then he is said to have incurred loss.
Loss = C P – S P
Loss % = LOSS/(C P)×100
S P = (100-loss %)/100×C P
C P = 100/(100-loss %)×S P
Profit and Loss Based on Cost Price
To find the percent gain or loss, divide the amount gained or lost by the cost price and multiply it by 100. Example: A toy that cost 80 rupees is sold at a profit of 20 rupees. Find the percent or rate of profit.
Answer:
Gain/cost × 100 = % profit.
20/80 × 100 = 25%. - Answer
To find the loss and the selling price when the cost and the percent loss are given, multiply the cost by the percent and subtract the product from the cost.
Example: A damaged chair that cost Rs.110 was sold at a loss of 10%. Find the loss and the selling price.
Answer:
Cost x percent loss = loss.
110 x 1/10 = 11, loss.
Cost - loss = selling price.
110 - 11 = 99, selling price.
Profit and Loss Based on Selling Price
To find the profit and the cost when the selling price and the percent profit are given, multiply the selling price by the percent profit and subtract the result from the selling price.
Example: A toy is sold for Rs. 6.00 at a profit of 25% of the selling price. Separate this selling price into cost and profit.
Answer :
Selling price x % profit = profit.
Selling price = profit + cost.
6.00 x .25 = 1.50, profit.
6.00 - 1.50 = 4.50, cost.
To find the loss and the cost when the selling price and the percent loss are given, multiply the selling price by the percent loss and subtract the result from the selling price.
Example: At a sale, neckties selling at Rs. 50.00 are sold at a loss of 60% of selling price. What is the loss and the original cost?
Selling price x % loss = loss.
Selling price + loss = cost.
50.00 x .60 = 30.00, loss.
50.00 - 30.00 = 20.00, cost.
To find the selling price when the cost and the percent loss are given, add the percent loss to 100% and divide the cost by this sum.
Example: Socks that cost 7.00 per pair were sold at a loss of 25% of selling price. What was the selling price?
Answer: Cost / (100% + % loss) = selling price.
7.00 / 1.25 = 5.60, selling price.
To find the selling price when the profit and the percent profit are given, or to find the selling price when the loss and the percent loss are given, divide the profit or loss by the percent profit or loss.
Note: This rule should be compared with the one under Profit and Loss Based on Cost. The two rules are exactly similar except that in one case 100% represents cost while in the other case 100% represents selling price.
Example: A kind of tape is selling at a profit of 12% of selling price, equal to 18 per yard. What is the selling price of the tape?
Answer: Profit / % profit = selling price.
18 /.12 = 1.50 selling price.
To find the percent profit or loss, divide the amount gained or lost by the selling price.
Example: A candy bar sells for 1.30 at a profit of 65. What percent of profit on selling price does this represent?
Answer: Gain / selling price = % profit.
65 / 1.30 = .5 or 50% profit.
Mark-up Price
Generally the SP is less than the marked price (MP) the difference MP – SP is known as discount, D.
Discount = M P – S P
Discount %, D% = (Discount) / (M P) ×100
To reduce percent loss on cost to percent loss on selling price, divide percent loss on cost by 100% minus percent loss on cost.
 Example: 20% loss on cost is what percent loss on selling price?
Answer:
% loss on cost / (100% - % loss on cost) = % loss on selling price.
0.20 / 80 = .0025 or 25% loss on selling price
To reduce percent loss on selling price to percent loss on cost, divide percent loss on selling price by 100% plus percent loss on selling price.
Example: 20% loss on selling price is what percent loss on cost?
Answer:
% loss on selling price / (100% + % loss on selling price) = % loss on cost.
.20 / 1.20 = .16666 or .16.67% loss on cost.
To reduce percent mark-up (percent profit on cost) to percent profit on selling price, divide percent mark-up by 100% plus percent mark-up.
Example: A coat marked up 60% carries what percent of profit on selling price?
Answer : % profit on cost / ( 100% + % profit on cost ) = % profit on selling price.
.60 / 1.60 = .375 or 37.5% on selling price.

Here we are providing you all the types of questions that have been asked in SSC Exams and How to solve it in an Easy way with  Grade Stack methods:-

Type 1: 
The cost price of 40 articles is the same as the selling price of 25 articles. Find the gain per cent. (CGL-2012)
(a) 65%                                
(b) 60%
(c) 15%                                
(d) 75%
Answer: (b) Gain per cent
=(40-25)/25×100
=15/25×100=60%
Grade Stack methods
In Above question We take x = 40 , y = 25
Then Gain % = (x –y) x 100/ y
Type2: 
Bananas are bought at the rate of 6 for Rs. 5 and sold at the rate of 5 for Rs. 6. Profit per cent is:  (CGL-2004)
(a) 36%                                
(b) 42%
(c) 44%                                
(d) 48%
Answer : (c) To avoid fraction, let the number of bananas bought
LCM of 5 and 6 = 30
CP of 30 bananas
= 5 x 5 = Rs. 25
SP of 30 Bananas = 6 x 6
= Rs. 36
Profit = Rs. (36-25) = Rs. 11
Profit %
= 11/25×100=44%
Grade Stack Method
[(6 x 6 -5x 5)/ (5 x 5)] x 100 = 44%
Type 3:
A man bought oranges at the rate of 8 for Rs 34 and sold them at the rate of 12 for Rs. 27. How many oranges should be sold to earn a net profit of Rs 45? (CGL-2011)
(a) 90                                    
(b) 100
(c) 135                                  
(d) 150
Answers: (a) Let the man buy 24 (LCM of 8 and 12) oranges.
C.P. of 24 oranges  = 34/8  ×24 = Rs. 102
S.P. of 24 oranges = 27/12×24=  Rs. 114
Gain = 114 – 102 = Rs. 12
Rs. 12 = 24 oranges
Rs. 45 =  24/12×45= 90 oranges
Type 4: 
A shopkeeper earns a profit of 12% on selling a book at 10% discount on printed price. The ratio of the cost price to printed price of the book is ?(CGL-2013)
(a) 45 : 56                                
 (b) 50 : 61
(c) 90 : 97                                 
(d) 99 : 125
Answer:  (a) C.P. of the book = Rs. x
Printed price = Rs. y
(y×90)/100=x × 112/100
x/y=90/112=45/56
Type 5: 
A dealer sold two types of goods for Rs 10,000 each. On one of them, he lost 20% and on the other he gained 20%. His gain or loss per cent in the entire transaction was (CGL-2012)
(a) 2% loss                          
(b) 2% gain
(c) 4% gain                         
(d) 4% loss
Answers:  (d) Here, S.P. is same, Hence there is always a loss. Loss per cent =(20×20)/100=4%
 Gradestack Trick
Loss % = (n^2)/100= (20)^2/100= 4%
Where n= 20
Type 6: 
On selling an article for Rs170, a shopkeeper loses 15%. In order to gain 20%, he must sell that article at rupees: (CGL-2013)
(a) 215.50                           
(b) 212.50
(c) 240                                
(d) 210
Answer ; (c) C.P. of article = (200×120)/100 = Rs. 240
Type 7:
An article is sold at a loss of 10%. Had it been sold for Rs. 9 more, there would have been a gain of 12 1/2% on it. The cost price of the article is (CGL – 2002)
(a) Rs. 40                                             
(b) Rs. 45
(c) Rs. 50                                             
(d) Rs. 35
Answers: (a) Let the cost price of the article = Rs. x
S.P. at 10% loss
= x×90/100= Rs. 9x/10
  1. P. at 12 1/2 % gain
x  × (100+12 1/2)/100 = Rs. 225x/200
According to the question
9x/10 + 9 = 225x/200
180x + 1800 = 225x
x = Rs. 40
Type 8: 
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 the price at which a bought it is (CGL-2013)
(a) 1000                                               
(b) 1600
(c) 2000                                               
(d)  2500
Answer:  (c) If the C.P. of the suitcase for A be Rs. x, then
x ×110/100×130/100=2860
x=(2860×100×100)/(110×130) = Rs. 2000
Type 9: 
Arun marks up the computer he is selling by 20% profit and sells them at a discount of 15%. Arun’s net gain percent is 
(CGL-2013)
(a) 4                                      
(b) 2
(c) 3.5                                   
(d) 2.5
Answer (b)
Gradestack method:
r1 = 20 , r2 = 15
Formula = r1 – r2 – (r1 x r2)/100
(20-15-(20×15)/100)
= 20 -18 = 2%
Type10:
A tradesman sold an article at a loss of 20%. If the selling price had been increased by Rs. 100, there would have been a gain of 5%. The cost price of the article was: (CGL-2004)
(a) Rs. 200                                          
(b) Rs. 25
(c) Rs. 400                                          
(d) Rs. 250
Answer  (c) Let the C.P. of article be Rs. x.
105% of x - 80% of x = Rx. 100
25% of x = Rx. 100
x = Rs. (100×100)/25
= Rs. 400
That’s how Profit and loss questions are solved easily.



1. What is the MRP of novel?
MRP: Marked Retail Price is the price which is printed on an object. So MRP of the novel is 250.
2. What is the discount?
Discount is calculated on MRP, Raunak and vendor both agreed at 20% discount.
So, Discount = 20% of MRP = 20% of 250
   = (20×250)/100 = 50
3. What is the Selling Price of novel for Vendor?
Selling Price (SP) is the price at which an object is sold.
SP = MRP – discount
SP = 250 – 50 = Rs.200
4. What is the Cost Price of novel for Raunak?
Cost Price is the price at which an object is purchased.
Raunak purchased this novel at Rs.200.
5. What is the Selling Price of novel for Raunak?
Raunak sold it at the MRP, so the SP for Raunak is 250.
6. What is the Cost Price of novel for Manish?
Manish purchased it at the MRP. So, CP for Manish is Rs.250.
7. What is the profit for Raunak?
Raunak purchased it at Rs.200 and sold it at Rs. 250
So, Profit = SP – CP = 250 -200 = Rs.50
8. What is the SP for Manish?
He sold it to Navneet at Rs. 150. So, SP for Manish is Rs.150
9. How much discount Manish give to Navneet?
Discount = CP of Manish – SP of Manish
 = 250 – 150 = 100
10. What is the discount% given to Navneet by Manish?
discount% = (Discount/ CP for Manish)× 100
= (100/250)× 100
= 40%
11. What is the Loss for Manish?
Loss = CP for Manish – SP for Manish
Loss = 250 – 150 = 100 Rs.
Now, we will discuss concepts by solving the questions based on this topic.
Example 1:
If a man purchases 12 toys for Rs.10 and sells 10 toys for Rs.12. How much profit or loss does he make?
Approach: 
You can see that the man purchases more number of toys at less price than selling less number of toys at more price. So definitely, we can say that he makes a profit. In the exam, the options which have the loss, you can easily eliminate those options.
Solution: CP of 12 toys = Rs.10
SP of 10 toys =Rs. 12
So, SP of 12 toys = (12/10)× 12 = 14.4
Profit% = ((SP-CP)/CP)× 100 = (4.4/10)= %
Tricks for this type of questions:
Purchases: 12 toys for Rs.10
Sells:         10 toys for Rs.12
For profit% or loss% Cross multiply
profit% or loss% = ((12× 12 – 10× 10)/(10× 10))× 100
= 44%
Example 2:
If a man purchases 12 toys for Rs.10 and sells 10 toys for Rs.8. How much profit or loss does he make?
Solution
There will be loss using the same approach.
profit% or Loss% = ((12×8-10×10)/(10× 10))× 100
 = -4% (Don’t be confused with negative sign, it represents loss)
So loss is 4%.
SP = [(100±profit or loss)/100]× CP
Example 3:
A person sells an article for ₹890 at a loss of 11%. What will be the price of the article when sold at a profit of 10%?
Solution:SP1 = 890 , loss 11%
We know that loss% = [(CP-SP)/CP]×100
11CP =(CP-890)×100
11CP = 100CP – 890×100
89CP = 890×100
CP = 1000
If he had sold it at 10% profit,
then new SP = CP + 10%CP
New SP = 1000+100 = 1100
Approach: Loss is 11% and it is calculated on CP, So we can say loss = 11% CP
New SP = [(100±new profit or loss %)/100]× CP….(1)
Old SP = [(100±old profit or loss%)/100]×CP….(2)
CP = Old SP/ [(100±old profit or loss%)/100]….(3)
Now, putting eqn 3 it in eqn 1.
New SP = Old SP [(100±new profit or loss %)/(100±new profit or loss %)]
Important: Put + sign for profit and – sign for negative.

New SP = 890 [(100+10)/(100-11)] = 1100
Example 4:
A shopkeeper sold an item for Rs.6080 after giving 20% discount on the labelled price and made 18% profit on the cost price. What would have been the percentage profit if he had not given the discount?
Solution: 
SP = MP (100-discount)%
6080 = MP(80%)
MP = 7600
Also, SP = CP [(100+profit%)/100]
SP = CP(118/100)
CP = (6080×100)/118
CP = 5125.54
He sold it at MP. So New SP = MP
profit% = [(MP-CP)/CP]×100
= [(7600-5125.24)/5125.24]×100
=47.5%
Another approach: 
Reqd. profit% = [(Discount±Profit or loss)/(100-discount%)]× 100
Reqd. profit% = [(20+18)/(100-20)]× 100
Reqd.profit% = 380/8 = 47.5% 
Example5: A shopkeeper sells an object at a profit of 25% after giving a discount of 20%. Find the ratio of Cost price, selling price and Marked price.
Solution: Let Cost price of object is Rs.100.
SP = (125/100)×100 = 125
SP = [(100-discount%)/100]MP
SP = (80/100)MP
MP= (125×100)/80
MP=  625/4
ratio CP : SP :MP
     100 : 125 : (625/4)
       4 : 5 : (25/4)
       16: 20 :25
Another approach: Let MP is 100 Rs
MP = 100
SP = 80
then, SP =125% of CP
So, CP = 64
ratio of CP:SP:MP = 64:80:100
 = 16:20:25
Dishonest Shopkeeper Concept
Example 6:
A dishonest dealer professes to sell his goods at cost price, but he uses a weight of 900gm of weight for the kg. Find his gain percent.
Solution: Let price of 1kg is Rs.100 then, price for 900gm will be Rs.90
Hence, he sells 900gm instead of 1kg for Rs.100 but cost price of it is only Rs.90.
So he earns a profit of Rs.10 on Rs.90 not on Rs.100  
So, profit% = (10/90)×100
 = 11(1/9)%
You can also use formula:
gain% = [Error/(true value-error)]×100
gain% = [100/(1000-100)]×100
  = 100/9 = 11(1/9)%
Example7:
A dishonest dealer professes to sell his goods at cost price, but he earns a profit of 25%.Find the weight he has used instead of 800gm?
Solution:Let cost price of 800gm goods is Rs. 100
He sells goods at cost price i.e. Rs.100 but earns 25% profit.
So, CP of goods he sold = [SP/(100+profit)]× 100
  CP of goods = (100/125)× 100 = 80
100Rs. costs for 800gm
80Rs. costs for (800/100)×80 = 640 gm.
He used 640gm instead of 800gm. 
Example8:
A machine is sold for Rs.5060 at a gain of 10%. What would have been the gain or loss if it had been sold for Rs.4370?
Solution:SP = CP × [(100+10)/100]
SP = CP×(11/10)
CP = 4600
New SP = 4370
Loss% =(230/4600)×100 = 5%
Example9:
Ashish sold a pen at 5% loss and a book at 15% profit. In the whole business, he earned Rs.7. If he had sold a pen at 5%profit and a book at 10% profit then he has earned Rs.6 more. What is the cost price of a pen and a book?
Solution:Let CP of a book is B and a pen is P. We know that profit or loss is calculated on CP.
So, In case 1: loss for pen = 5%P, profit for book = 15%B
In case 2: profit for book = 10%B, profit for pen = 5%P
Use sign (-) for loss and (+) for profit.
In case 2 he earned Rs. 13(6 more than previous one)
15%B-5%P =7…..(1)
10%B+5%P =13…(2)
Adding eqn (1) and (2)
25%B = 20
B = 20×(100/25)
B = 80Rs.
Putting B=80 in eqn(2)
5%P = 13-8
5%P = 5
P = Rs. 100