(Note: Before you read this lesson Please read percentages lesson to understand this lesson better Click here )
Profit loss and Discount is an application of percentages.
Cost Price: The rate at which a merchant buys goods. This is his investment
Selling Price: The rate at which a merchant sells his goods.
Marked Price: The rate at which a merchant rises his price above the cost price (may be anticipating some hagglers)
Profit Case:
Loss Case:
Key Formulas:
Profit or Gain = Selling Price  Cost Price = SP  CP
Loss = Cost price  Selling price = CP  SP
Profit = ${\rm{CP \times \text(Profit\% )}}$
Loss = $CP \times (Loss\% )$
Profit % = $\displaystyle\frac{{SP  CP}}{{CP}} \times 100 = \frac{{Profit}}{{CP}} \times 100$
Loss % = $\displaystyle\frac{{CP  SP}}{{CP}} \times 100 = \frac{{Loss}}{{CP}} \times 100$
Important: Profit or Loss always calculated on Cost price Only.
Discount = Marked price  Selling Price = MP  SP
Discount % = $\displaystyle\frac{{MP  SP}}{{MP}} \times 100 = \frac{{Discount}}{{MP}} \times 100$
Calculating Selling price from Cost price:
In the profit case selling price is greater than cost price, and this case we gain some profit. That is we are increasing the cost price by some percentage to get the selling price. This can be done in several ways
In profit case
$CP + CP \times (\Pr ofit\% )$ = SP
$CP \times (100 + \Pr ofit)\% $ = SP
(V.imp: To get the selling price we have to multiply the cost price by (100+x)% if profit percent = x%)
In loss case
$CP  CP \times Loss\% $=SP
$CP \times (100  Loss)\% $ = SP
(V.imp: To get the selling price we have to multiply the cost price by (100x)% if loss percent = x%)
Calculating Selling Price from Marked Price:
$MP  MP \times discount\% $ = SP
$MP \times \left( {100  discount} \right)\% $ = SP
Calculating Cost price from Selling Price:
This is the reverse operation of the above
In profit case: $\displaystyle\frac{{SP}}{{(100 + \Pr ofit)\% }} = CP$
In loss Case: $\displaystyle\frac{{SP}}{{(100  Loss)\% }} = CP$
(V.imp: To get the cost price we have to divide the selling price by (100+x)% in profit case and by (100x)% in loss case)
Ans: Profit percent = $\dfrac{{sp  cp}}{{cp}} \times 100 = \dfrac{{600  500}}{{500}} \times 100$ = 20%
2. John made a profit of 25% while selling a book for Rs.250. Find the cost price of the book.
Ans: From the above formulas, $cp \times \left( {\dfrac{{100 + 25}}{{100}}} \right) = 250$
$ \Rightarrow cp = \dfrac{{100}}{{125}} \times 250 = 200$
3. Nivas sold a pen for Rs.900 thus making 10% loss. Find the cost price.
Ans: From the above formulas, $cp \times \left( {\dfrac{{100  10}}{{100}}} \right) = 900$
$ \Rightarrow cp = \dfrac{{100}}{{90}} \times 900 = 1000$
4. A trader buys oranges at 7 for a rupee and sells them at 40% profit. How many oranges does he sell for a rupee?
Ans: Cost price of 1 apple = $\dfrac{1}{7}$
Selling price of 1 apple at 40% profit is equal to 140% of its cost price.
Therefore, Selling price of 1 apple = $\dfrac{1}{7} \times 140\% = \dfrac{1}{7} \times \left( {\dfrac{{140}}{{100}}} \right) = \dfrac{1}{5}$
Therefore, 5 oranges are sold are 1 rupee.
5. On selling mangoes at 36 for a rupee, a shopkeeper loses 10%. How many mangoes should he sell for a rupee in order to gain 8%?
Ans: Selling price of 1 apple = Rs.$\dfrac{1}{{36}}$
Cost price of 1 apple = $\dfrac{1}{{36}} \times \dfrac{1}{{90\% }}$
Therefore, New selling price = $\left( {\dfrac{1}{{36}} \times \dfrac{1}{{90\% }}} \right) \times 108\% $ = $\dfrac{{108}}{{36 \times 90}} = \dfrac{1}{{30}}$
Therefore, For one rupee, he should sell 30 mangoes.
6. A boy buys eggs at 10 for Rs.1.80 and sells them at 11 for Rs. 2. What is his gain or loss per cent?
Ans: To avoid fractions, let the number of eggs purchased be, LCM of 10 and 11 = 110
CP of 110 eggs = $\dfrac{{110 \times 1.80}}{{10}}$ = Rs. 19.80
SP of 110 eggs = $\dfrac{{110 \times 2.0}}{{11}}$ = Rs. 20.00Profit loss and Discount is an application of percentages.
Cost Price: The rate at which a merchant buys goods. This is his investment
Selling Price: The rate at which a merchant sells his goods.
Marked Price: The rate at which a merchant rises his price above the cost price (may be anticipating some hagglers)
Profit Case:
Loss Case:
Key Formulas:
Profit or Gain = Selling Price  Cost Price = SP  CP
Loss = Cost price  Selling price = CP  SP
Profit = ${\rm{CP \times \text(Profit\% )}}$
Loss = $CP \times (Loss\% )$
Profit % = $\displaystyle\frac{{SP  CP}}{{CP}} \times 100 = \frac{{Profit}}{{CP}} \times 100$
Loss % = $\displaystyle\frac{{CP  SP}}{{CP}} \times 100 = \frac{{Loss}}{{CP}} \times 100$
Important: Profit or Loss always calculated on Cost price Only.
Discount = Marked price  Selling Price = MP  SP
Discount % = $\displaystyle\frac{{MP  SP}}{{MP}} \times 100 = \frac{{Discount}}{{MP}} \times 100$
Calculating Selling price from Cost price:
In the profit case selling price is greater than cost price, and this case we gain some profit. That is we are increasing the cost price by some percentage to get the selling price. This can be done in several ways
In profit case
$CP + CP \times (\Pr ofit\% )$ = SP
$CP \times (100 + \Pr ofit)\% $ = SP
(V.imp: To get the selling price we have to multiply the cost price by (100+x)% if profit percent = x%)
In loss case
$CP  CP \times Loss\% $=SP
$CP \times (100  Loss)\% $ = SP
(V.imp: To get the selling price we have to multiply the cost price by (100x)% if loss percent = x%)
Calculating Selling Price from Marked Price:
$MP  MP \times discount\% $ = SP
$MP \times \left( {100  discount} \right)\% $ = SP
Calculating Cost price from Selling Price:
This is the reverse operation of the above
In profit case: $\displaystyle\frac{{SP}}{{(100 + \Pr ofit)\% }} = CP$
In loss Case: $\displaystyle\frac{{SP}}{{(100  Loss)\% }} = CP$
(V.imp: To get the cost price we have to divide the selling price by (100+x)% in profit case and by (100x)% in loss case)
Practice Problems
1. Kathy buys a watch for Rs.500 and sells it to Jake for Rs.600. Find the profit percent.Ans: Profit percent = $\dfrac{{sp  cp}}{{cp}} \times 100 = \dfrac{{600  500}}{{500}} \times 100$ = 20%
2. John made a profit of 25% while selling a book for Rs.250. Find the cost price of the book.
Ans: From the above formulas, $cp \times \left( {\dfrac{{100 + 25}}{{100}}} \right) = 250$
$ \Rightarrow cp = \dfrac{{100}}{{125}} \times 250 = 200$
3. Nivas sold a pen for Rs.900 thus making 10% loss. Find the cost price.
Ans: From the above formulas, $cp \times \left( {\dfrac{{100  10}}{{100}}} \right) = 900$
$ \Rightarrow cp = \dfrac{{100}}{{90}} \times 900 = 1000$
4. A trader buys oranges at 7 for a rupee and sells them at 40% profit. How many oranges does he sell for a rupee?
Ans: Cost price of 1 apple = $\dfrac{1}{7}$
Selling price of 1 apple at 40% profit is equal to 140% of its cost price.
Therefore, Selling price of 1 apple = $\dfrac{1}{7} \times 140\% = \dfrac{1}{7} \times \left( {\dfrac{{140}}{{100}}} \right) = \dfrac{1}{5}$
Therefore, 5 oranges are sold are 1 rupee.
5. On selling mangoes at 36 for a rupee, a shopkeeper loses 10%. How many mangoes should he sell for a rupee in order to gain 8%?
Ans: Selling price of 1 apple = Rs.$\dfrac{1}{{36}}$
Cost price of 1 apple = $\dfrac{1}{{36}} \times \dfrac{1}{{90\% }}$
Therefore, New selling price = $\left( {\dfrac{1}{{36}} \times \dfrac{1}{{90\% }}} \right) \times 108\% $ = $\dfrac{{108}}{{36 \times 90}} = \dfrac{1}{{30}}$
Therefore, For one rupee, he should sell 30 mangoes.
6. A boy buys eggs at 10 for Rs.1.80 and sells them at 11 for Rs. 2. What is his gain or loss per cent?
Ans: To avoid fractions, let the number of eggs purchased be, LCM of 10 and 11 = 110
CP of 110 eggs = $\dfrac{{110 \times 1.80}}{{10}}$ = Rs. 19.80
Profit per cent = $\dfrac{{20  19.80}}{{19.80}} \times 100$ = $\dfrac{1}{{99}} \times 100$ = 1.01%
7. A woman buys apples at 15 for a rupee and the same number at 20 a rupee. She mixes and sells them at 35 for 2 rupees. What is her gain per cent or loss per cent?
Ans: Suppose the woman buys two lots of (LCM of 15, 20 and 35) = 420 apples.
One apple cost from the first lot = $\dfrac{1}{{15}}$.
420 apples cost = $\dfrac{1}{{15}} \times 420 = 28$
One apple cost from the second lot = $\dfrac{1}{{20}}$.
420 apples cost = $\dfrac{1}{{20}} \times 420 = 21$
Total cost for 840 apples = 28 + 21 = Rs. 49
Selling price of 1 apple = $\dfrac{2}{{35}}$.
840 apples selling price = $\dfrac{2}{{35}} \times 840 = 48$
Loss per cent = $\dfrac{1}{{49}} \times 100 = 2.04\% $
8. Some quantity of coffee is sold at Rs. 22 per kg, making 10% profit. If total gain is Rs. 88, what is the quantity of coffee sold?
Ans: Cost price of the coffee = $\dfrac{{22}}{{110\% }} = 22 \times \dfrac{{100}}{{110}} = 20$
Profit on one kg of coffee = 22  20 = Rs.2.
Let $x$ kgs of coffee sold. So total profit = $2x$.
$ \Rightarrow 2x = 88$
$ \Rightarrow $ x = 44
9. A manufacturer sells a scooter at 10% profit to wholesaler who in turn sells it to a retailer at 20% profit. If the price paid by the retailer is Rs. 13200, how much the scooter costs to the manufacturer?
Ans: Let the manufacturer bought the scooter for $x$ rupees.
Given, $x \times 110\% \times 120\% = 13200$
$ \Rightarrow x = \dfrac{{13200}}{{110\% \times 120\% }}$ = $\dfrac{{13200}}{{\dfrac{{11}}{{10}} \times \dfrac{{12}}{{10}}}}$ = $\dfrac{{13200 \times 10 \times 10}}{{11 \times 12}}$ = 10000
10. A man bought some oranges at the rate of 3 oranges for one rupee and equal number of oranges at the rate of 2 oranges for one rupee. What is his profit, if he sells 2 oranges for one rupee.
Ans: Let us take LCM of quantities purchased and sold i.e. LCM of 3, 2 and 2 is 6.
Now, cost price of 6 oranges @ 3 oranges for a rupee = $\displaystyle\frac{{\rm{1}}}{{\rm{3}}}$ × 6 = Rs. 2
And, cost price of 6 oranges @ 2 oranges for a rupee = $\displaystyle\frac{{\rm{1}}}{{\rm{2}}}$ × 6 = Rs. 3
Therefore, Cost price of 12 (i.e. 6 + 6) oranges = Rs. 2 + Rs. 3 = Rs. 5
Selling price of 12 oranges = $\displaystyle\frac{{\rm{1}}}{{\rm{2}}}$ × 12 = Rs. 6
Profit = Rs. 6  Rs. 5 = Re. 1 on Rs. 5
Therefore, Profit percent = $\displaystyle\frac{{\rm{1}}}{{\rm{5}}}$ = 20%
11. If the cost price of 11 oranges is equal to selling price of 10 oranges. Find profit per cent.
Ans: Given, 11cp = 10sp $ \Rightarrow \dfrac{{cp}}{{sp}} = \dfrac{{10}}{{11}}$
Therefore, cp = 10, sp = 11.
Therefore, Profit % = $\dfrac{{\rm{1}}}{{{\rm{10}}}}$ = 10%
12. On selling 10 articles, a merchant loses equal to cost price of 2 articles. Find his loss per cent.
Ans: Let the merchant bought 10 articles and sold 10 articles.
$ \Rightarrow $ 10cp  10sp = 2cp
$ \Rightarrow $ 8cp = 10sp
$ \Rightarrow \dfrac{{cp}}{{sp}} = \dfrac{{10}}{8}$
Therefore, cp = 10, sp = 8.
Loss % = $\frac{2}{{10}} \times 100 $ = 20%
13. Sumit buys 9 books for Rs. 100 but sells 8 for Rs. 100. What is the net per cent profit?
Ans: SP of 8 books = Rs. 100
SP of 1 book = $\displaystyle\frac{{100}}{8}$ = Rs. 12.50
SP of 9 books = 12.50 × 9 = Rs. 112.50
Profit per cent = $\dfrac{{112.5  100}}{{100}} \times 100$ = 12.5%
Alternative method:
CP of 9 books = SP of 8 books $ \Rightarrow $ 9cp = 8sp
$ \Rightarrow \dfrac{{cp}}{{sp}} = \dfrac{8}{9}$
Let cp = 8, sp = 9
Profit percentage = $\dfrac{{sp  cp}}{{cp}} \times 100$ = $\dfrac{{9  8}}{8} \times 100$ = 12.5%
14. Goods are purchased for Rs. 450 and onethird is sold at a loss of 10%. At what profit per cent should the remainder be sold so as to gain 20% on the whole transaction?
Ans: Assume Total cost price of goods = Rs. 450
Our target SP of total goods = $\dfrac{{120}}{{100}} \times 450 = Rs.540$
Onethird of the goods costs = 450/3 = Rs.150
Onethird of the goods costs = 450/3 = Rs.150
SP of onethird goods = $\dfrac{{90}}{{100}} \times 150 = Rs.135$
SP of the remaining goods = 540 – 135 = Rs. 405
CP of remaining (twothirds) goods = Rs. 300
Hence, profit per cent = $\dfrac{{105}}{{300}} \times 100 = 35\% $
Alternate method:
Applying weighted average, one part of quantity sold at a loss of 10% (or a profit of –10%) and balance two units are to be sold at x% to give a overal profit of 20%.
Hence, overall profit is given by , $\dfrac{{mx + ny}}{{m + n}}$ = $\dfrac{{1 \times (  10\% ) + 2 \times (x\% )}}{{1 + 2}} = 20\% $15. A retailer buys goods at 10% discount on its marked price and sells them at 20% higher than the marked price. What is his profit per cent?
Ans: Let, marked price of the article = Rs. 100, Then, its cost price = Rs. 100  Rs. 10 = Rs. 90, And selling price = Rs. 100 + Rs. 20 = Rs. 120
Therefore, Profit = Rs. 30 on Rs. 90 i.e. $\displaystyle\frac{{\rm{1}}}{{\rm{3}}}$ = 33.33%
16. A dishonest merchant professes to sell his goods at cost price, but uses a weight of 900 grams for one kg. weight. What is his profit per cent?
Ans: Assume 1gm costs 1 rupee. The merchant gives 900 grams charging the price of 1000 grams.
His gain is 100 grams on every 900 grams. i.e., for Rs.900 investment his gain is Rs.100.
Therefore, Profit percent = $\displaystyle\frac{{{\rm{100}}}}{{{\rm{900}}}}{\rm{ \times 100 = 11}}\displaystyle\frac{{\rm{1}}}{{\rm{9}}}$%
17. A merchant professes to sell goods at 20% profit but uses weight of 900 grams in place of a kilogram. What is his actual profit per cent?
Ans: Assume 1gm costs 1 rupee. The merchant gives 900 grams charging the price of 1200 grams.
Therefore, His gain is 300 grams on every 900 grams or on investment of Rs.900 his gain is Rs.300.
Therefore, Profit per cent = $\displaystyle\frac{{{\rm{300}}}}{{{\rm{900}}}}$ x 100 = 33.33%
18. A shopkeeper buys some pens. If he sells them at Rs. 13 per pen, his total loss in Rs. 150 but on selling them at Rs. 15 per pen, his total gain is Rs. 100. How many pens did he sell?
Ans: Difference in sales amount due to change in selling price = Rs. 150 + Rs. 100 = Rs. 250
Difference in selling price per pen = Rs. 15  Rs. 13 = Rs. 2
Therefore, On selling 1 pen, sales amount is increased by Rs. 2 in second case.
Therefore, Total pens sold = $\displaystyle\frac{{{\rm{250}}}}{{\rm{2}}}$ = 125 pens
19. A man sold an article at 10% profit. Had it been sold for Rs. 50 more, he would have gained 15%. Cost Price of the article is:
Ans: Let us assume cost price of the article is Rs.100x. then selling price = 110x. But if he sold the product for Rs.50 more his profit is 15%. In this case his selling price is 115x. But the difference in the selling prices were gives as Rs.50. So 115x  110x = 50, therefore x = 10. Substituting in cost price, CP = Rs.1000
Alternate method:
Difference in two selling prices = 15%  10% = 5% of cost price
Actual difference in two selling price = Rs. 50 (i.e. 10 times of 5)
Therefore, Cost Price = 10 × Rs. 100 = Rs. 1000
20. A machine is sold at a loss of 10%. Had it been sold at a profit of 15%, it would have fetched Rs. 50 more. The cost price of the machine is:
Ans: Let us assume cost price of the article is Rs.100x. then selling price = 90x. But if he sold the product for Rs.50 more his profit is 15%. In this case his selling price is 115x. But the difference in the selling prices were gives as Rs.50. So 115x  90x = 50, therefore x = 2. Substituting in cost price, CP = Rs.200
Alternate method:
Difference in two selling prices = 10%  (15%) = 10% + 15% = 25% of cost price
Actual difference in two selling price = Rs. 50 (i.e. 2 times of 25)
Therefore, Cost Price = 2 × Rs. 100 = Rs. 200
21. A bicycle is sold at 10% profit. Had it been sold for Rs. 10 less, the profit would have been 5% only. What is the cost price of the bicycle?
Ans: Difference in two selling prices = 10%  5% = 5% of cost price
Actual difference in two selling price = Rs. 10 (i.e. 2 times of 5)
Therefore, Cost Price = 2 × Rs. 100 = Rs. 200
22. A man sells an article at a profit of 25%. If he had bought it at 20% less and sold it for Rs. 10.50 less, he would have gained 30%. Find the CP of the article.
Ans: Let the cost price = x ; Selling Price = 125%(x)
New Cost Price = 80%(x) ; New SP = 125%(x) – 10.50
But new SP = 130% of new CP =130%(80%x)
Therefore, 130%(80%x) = 125%(x) – 10.50 $ \Rightarrow 104\% (x) = 125\% (x)  10.50$
$ \Rightarrow 21\% (x) = 10.50$ $ \Rightarrow x = 10.50 \times \displaystyle\frac{{100}}{{21}} \Rightarrow Rs.50$
Alternate Method:
Let the CP be Rs. 100. So, SP is Rs. 125.
The new CP is Rs. 80. So the new SP =130%(80)= Rs. 104
So the difference of SP’s = Rs. 21.
Now, if the difference is 21, CP is 100
So, when the difference is 10.5, the CP is = $ \Rightarrow \displaystyle\frac{{10.5}}{{21}} \times 100 \Rightarrow Rs.50$.
Ans: Let marked price of the goods = Rs. 100. Then cost price = Rs. 100 x $\displaystyle\frac{{{\rm{19}}}}{{{\rm{20}}}}$ = Rs. 95
Selling price = Rs. 100 + 14% of Rs. 100 = Rs. 114
Therefore, Profit = Rs. 114  95 = Rs. 19 on Rs. 95
Therefore, Profit = $\dfrac{{{\rm{19}}}}{{{\rm{95}}}}{\rm{ = }}\dfrac{{\rm{1}}}{{\rm{5}}}$ = 20%
24. A merchant fixed selling price of his articles at Rs. 700 after adding 40% profit to the cost price. As the sale was very low at this price level, he decided to fix the selling price at 10% profit. Find the new selling price.
Ans: New Selling price = Rs. 700 × $\displaystyle\frac{{{\rm{100 + 10}}}}{{{\rm{100 + 40}}}}$ = Rs. 700 × $\displaystyle\frac{{{\rm{110}}}}{{{\rm{140}}}}$ = Rs. 550
25. A shopkeeper bought some apples at the rate of Rs. 16 per dozen. Due to harsh climate 20% of the apples bought were rotten during the transportation. At what rate of per dozen should he sell the remaining apples so as to gain 30% on the total cost price?
Ans: Since, 20% of the quantity is spoiled, selling the apples at cost price will result in 20% loss.
Therefore, We are to find the selling price which gives him 30% profit instead of a loss of 20%.
Therefore, Selling Price = Rs. 16 × $\displaystyle\frac{{{\rm{130}}}}{{{\rm{80}}}}$ = Rs. 26 per dozen.
26. A Watch is sold at 10% discount on its marked price of Rs. 480. If the retailer makes 20% profit on the cost price, find the cost price of the watch.
Ans: If marked price is Rs. 100, selling price = 100  10 = Rs. 90
If cost price is Rs. 100, selling price = 100 + 20 = Rs. 120
Therefore, Cost price = 480 × $\displaystyle\frac{{{\rm{90}}}}{{{\rm{100}}}}{\rm{ \times }}\displaystyle\frac{{{\rm{100}}}}{{{\rm{120}}}}$ = Rs. 360
27. A shopkeeper allows 25% discount on the marked price of his articles and hence gains 25% of the Cost Price. What is the marked price of the article on selling which he gains Rs. 120?
Ans: Marked price of the article = Rs. 120 × $\dfrac{{125}}{{25}} \times \dfrac{{100}}{{75}}$ = Rs. 800
Hint: If profit is Rs. 25, then Selling price = Rs. 100 + Rs. 25 = Rs. 125.
If marked price is Rs. 100, then Selling price = Rs. 100  Rs. 25 = Rs. 75.
28. A man purchased two articles for Rs. 10000 each. On selling first, he gains 20% and on the other, he loses 20%. What is profit/loss in the transaction?
Ans: Here, the cost price of both the articles are same. Therefore, Profit made on one item is exactly equal to loss suffered on the other.
Therefore, No profit, no loss.
29. A man sold two articles for Rs. 10000 each. On selling first, he gains 10% and on the other, he loses 10%. What is profit/loss in the transaction.
Ans: Loss % = $\displaystyle\frac{{\left( {{\rm\text{Common Gain and Loss}}} \right)^{\rm{2}} }}{{{\rm{100}}}}{\rm{ = }}\displaystyle\frac{{{\rm{10}}^{\rm{2}} }}{{{\rm{100}}}}$ = 1%
30. Two tables are purchased for the total cost of Rs. 5000. First table is sold at 40% profit and second at 40% loss. If selling price is same for both the tables, what is the cost price of the table that was sold at profit?
Ans: 140% of cost price of first table = 60% of cost price of second table.
Therefore, Cost price of first table : Cost price of second table
= 60 : 140 = 3 : 7
Therefore, Cost price of first table = $\displaystyle\frac{{\rm{3}}}{{{\rm{10}}}}$ × 5000 = Rs. 1500.
31. A reduction of 10% in the price of sugar enables a man to buy 25 kg more for Rs. 225. What is the original price of sugar (per kilogram)?
Let the original price be x. Then Original quantity = $\displaystyle\frac{{{\rm{225}}}}{{\rm{x}}}$
New price = 90%(x) (if a number reduced by 10% it becomes 90% of the original number)
New quantity = $\;\displaystyle\frac{{225}}{{0.9{\kern 1pt} x}}$
Equating $\displaystyle\frac{{225}}{{0.9{\kern 1pt} x}}\;  \;\frac{{225}}{x}\; = 25$
$ \Rightarrow x = Rs.1$
Alternate method:
When the price of the sugar got reduced by 10%, Now we could pay 10% less than the actual expenditure. But used the savings to take extra 25 kgs of sugar so
CP of 25 kg = $\displaystyle\frac{{10}}{{100}}\,{\kern 1pt} \times \,\,225\,\, = \,\,Rs.\,\,22.5\,;$
Reduced CP of 1 kg = $\displaystyle\frac{{22.5}}{{25}}\, = \,{\mathop{\rm Re}\nolimits} .\,\,0.90$
We got this reduced price after we reduced the original cost price by 10%. To calculate the original cost price we need to divide =Original price of sugar (per kg) can b = $\displaystyle\frac{{0.90}}{{90}}\, \times \,100\,\, = \,\,{\mathop{\rm Re}\nolimits} .\,1\,.$
Since the price is reduced by 10% (i.e.$\displaystyle\frac{1}{{10}}$) the new price has become $\displaystyle\frac{9}{{10}}$ the original. So, the consumption becomes the $\displaystyle\frac{{10}}{9}$, i.e. an increase of $\displaystyle\frac{1}{9}$
Therefore by unitary method
Now, 225 kg is worth Rs. 225. So original price of 1 kg is Rs. 1.Since the price is reduced by 10% (i.e.$\displaystyle\frac{1}{{10}}$) the new price has become $\displaystyle\frac{9}{{10}}$ the original. So, the consumption becomes the $\displaystyle\frac{{10}}{9}$, i.e. an increase of $\displaystyle\frac{1}{9}$
Therefore by unitary method
MCQ's
1. A dealer marks his goods 20% above cost price. He then allows some discount on it and makes a profit of 8%. The rate of discount is :
a. 12%  b. 10% 
c. 6%  d. 4% 
Explanation:
Let C.P.=Rs.100
Marked price = Rs.120, S.P. = Rs.108
Discount = $\left[ {\displaystyle\frac{{12}}{{120}} \times 100} \right]$% = 10%
2. A cloth merchant has announced 25% rebate in prices. If one needs to have a rebate of Rs.40, then how many shirts, cash costing Rs.32, he should purchase ?
a. 6  b. 5 
c. 10  d. 7 
Explanation:
Suppose the number of shirts = x.
Then, rebate = $\left[ {\displaystyle\frac{{25}}{{100}} \times 32x} \right] = 8x$
8x=40 or x = 5.
3. The price of an article was increased by p%. Later the new price was decreased by p%. If the latest price was Rs.1, the original price was :
a. Rs. 1  b. $\left[ {\displaystyle\frac{{1  {p^2}}}{{100}}} \right]$ 
c. $\left[ {\displaystyle\frac{{10000}}{{10000  {p^2}}}} \right]$  d. $\left[ {\sqrt {\displaystyle\frac{{1  {p^2}}}{{100}}} } \right]$ 
Explanation:
Let the original price = Rs.x
Price after P% increase = (100+P)% of x
=$\displaystyle\frac{{(100 + P)x}}{{100}}$
New price after P% decrease
= (100P)% of $\left[ {\displaystyle\frac{{(100 + P)x}}{{100}}} \right]$
= $\displaystyle\frac{{(100  P)}}{{100}} \times \displaystyle\frac{{(100 + P)}}{{100}} \times x$
= $\displaystyle\frac{{(100  P)(100 + P)}}{{100 \times 100}} \times x = 1$
or x = $\displaystyle\frac{{100 \times 100}}{{(100  P)(100 + P)}} = \displaystyle\frac{{10000}}{{10000  {P^2}}}$
4. The difference between the selling prices after a discount of 40% on Rs.500 and two successive discount of 36% and 4% on the same amount is :
a. 0  b. Rs.2 
c. Rs.1.93  d. Rs.7.20 
Explanation:
Sale after 40% discount = 60% of Rs.500
=Rs.300. Price after 36% discount = 64% of Rs.500=Rs.320.
Price afdter next 4% discount = 96% of Rs.320 = Rs.307.20
Difference in two prices = Rs.7.20
5. Tarun bought a T.V with 20% discount on the labelled price. Had he bought it with 25% discount, he would have saved Rs.500. At what price did he buy the T.V ?
a. Rs.5000  b. Rs.10,000 
c. Rs.12000  d. Rs.6000 
Explanation:
Let the labelled price be Rs.100, S.P in 1st case = Rs.80, S.P in 2nd case = Rs.75. If saving is Rs.5, labelled price
= Rs.$\left[ {\displaystyle\frac{{(100}}{5} \times 500} \right]$
= Rs.10000
6. A man purchases an electric heater whose printed price is Rs.160. If he received two successive discounts of 20% and 10%; he paid :
a. Rs.112  b. Rs.129.60 
c. Rs.119.60  d. Rs.115.20 
Explanation:
Price after Ist discount = 100% of Rs.160 = Rs. 128
Price after 2nd discount = 90% of Rs.128 = Rs.115.20
7. The marked price is 10% higher than the cost price. A discount of 10% is given on the marked price. In this kind of sale, the seller
a. bears no loss, no gain  b. gains 
c. losses  d. None of these 
Explanation:
Let C.P = Rs.100
Marked price = Rs.110
S.P = 90 % of Rs.110 = Rs.99
Loss = 1%
8. A trader lists his articles 20% above C.P and allows a discount of 10% on cash payment. His gain percent is :
a. 10%  b. 8% 
c. 6%  d. 5% 
Explanation:
Let C.P = Rs.100
Then, marked price = Rs.120
S.P = 90% of Rs.120= Rs.108
Gain = 8%
9. While selling a watch, a shopkeeper gives a discount of 5%. If he gives a discount of 7%, he earns Rs.15 less as profit. The marked price of the watch is :
a. Rs.697.50  b. Rs.712.50 
c. Rs.787.50  d. None of these 
Explanation:
Let the marked price be Rs. x
Then (7% of x )  15 = 5% of x
or $\displaystyle\frac{{7x}}{{100}}  \displaystyle\frac{{5x}}{{100}} = 15$ or x =750
10. A shopkeeper earns a profit of 12% on selling a book at 10% discount on the printed price. The ratio of the cost price and the printed price of the book is :
a. 45 : 56  b. 50 : 61 
c. 99 : 125  d. None of these 
Explanation:
Let the printed price of the book be Rs.100. After a discount of 10% S.P= Rs.90 Profit earned = 12%
C.P. of the book =
Rs. $\left[ {\displaystyle\frac{{100}}{{112}} \times 90} \right]$=Rs.$\displaystyle\frac{{1125}}{{14}}$
Hence, (C.P) : (printed price)=$\displaystyle\frac{{1125}}{{14}}$:100
or 45:56
11. A retailer buys a sewing machine at a discount of 15% and sells it for Rs.1955. Thus he makes a profit of 15%. The discount is :
Correct Option : Ca. Rs.270  b. Rs.290 
c. Rs.300  d. None of these 
Explanation:
Cost price for the retailer = $\displaystyle\frac{{100}}{{(100 + 15)}} \times 1955 = 1700$
But this price is what retailer got after having got a discount of 15%.
Let the marked price be Rs.x . Purchase price by the retailer = (10015)% of Rs.x.
So $\displaystyle\frac{{85}}{{100}} \times x = 1700 \Rightarrow x = 2000$
Discount received by retailer
= (15% of Rs.2000) = Rs.300
12. An umbrella marked at Rs.80 is sold for Rs.68. The rate of discount is :
a. 15%  b. 12% 
c. $17\displaystyle\frac{{11}}{{17}}$%  d. 20% 
Explanation :
Discount = $\left[ {\displaystyle\frac{{12}}{{80}} \times 100} \right]$%=15%
13. Kabir buys an article with 50% discount on its marked price. He makes a profit of 10% by selling it at Rs.660. The marked price is :
a. Rs.600  b. Rs.700 
c. Rs.800  d. 685 
Explanation:
Cost price for Kabir = $\displaystyle\frac{{100}}{{100 + 10}} \times 660 = 600$
But this price is what he got after having a discount of 50%. Let the marked price be x.
Then (100  25)% of x = 600 $ \Rightarrow $ x = Rs.800
Alternatively:
Let the original price be Rs.x
C.P = (x  50% of x ) = $\displaystyle\frac{{3x}}{4}$
S.P = $\left[ {\displaystyle\frac{{3x}}{4} + 10\% {\rm\text{ of }}\displaystyle\frac{{3x}}{4}} \right] = \displaystyle\frac{{33x}}{{40}}$
$\displaystyle\frac{{33x}}{{40}}$=660$ \Rightarrow $x=800
14. The ratio of the prices of three different types of cars is 4:5:7. If the difference between the costliest and the cheapest cars is Rs.60000, the price of the car of modest price is :
a. Rs.80000  b. Rs.100000 
c. Rs.140000  d. Rs.120000 
Explanation:
Let the price be 4x, 5x and 7x rupees.
Then, 7x4x=60000 $ \Rightarrow $ x=20000.
Required price = 5x=Rs.100000.
15. A discount series of 10%, 20% and 40% is equal to a single discount of :
a. 50%  b. 56.8% 
c. 60%  d. 70.28% 
Explanation:
Let original price = Rs.100. Price after first discount = Rs.90. Price after second discount
= Rs. $\left[ {\displaystyle\frac{{80}}{{100}} \times 90} \right]$=Rs.72
Price after third discount = Rs.$\left[ {\displaystyle\frac{{60}}{{100}} \times 72} \right]$
= Rs.43.20
Single discount = (10043.20)=56.8%
16. Subhash purchased a tape recorder at $\displaystyle\frac{9}{{10}}th$ of its selling price and sold it at 8% more than its S.P. His gain is :
a. 8%

b. 10%

c. 18%

d. 20%

Correct Option : D
Explanation:
Let the S.P be Rs.x
Then, C.P paid by Subhash = Rs.${\displaystyle\frac{{9x}}{{10}}}$
S.P. received by Subhash = (108% of Rs.x)
= Rs.${\displaystyle\frac{{27x}}{{25}}}$
Gain = Rs.$\left[ {\displaystyle\frac{{27x}}{{25}}  \displaystyle\frac{{9x}}{{10}}} \right]$=Rs.$\displaystyle\frac{{9x}}{{50}}$
Hence Gain % =$\left[ {\displaystyle\frac{{9x}}{{50}} \times \displaystyle\frac{{10}}{{9x}} \times 100} \right]$% = 20%
Alternatively:
Assume Selling price is 100. So he gets it for 90. and sold it for 108. His profit is 18. Profit percentage is 18/90 x 100 = 20%
Easy. is it not!!
17. At what price must Kantilal sell a mixture of 80kg. Sugar at Rs.6.75 per kg. with 120 kg. at Rs.8 per kg. to gain 20% ?
a. Rs.7.50 per kg

b. Rs.8.20 per kg

c. Rs.8.85 per kg

d. Rs.9 per kg.

Correct Option : D
Explanation:
Total C.P of 200 kg of sugar
= Rs.$(80 \times 6.75 + 120 \times 8)$=Rs.1500
C.P of 1 kg = Rs.$\left[ {\displaystyle\frac{{1500}}{{200}}} \right]$=Rs.7.50
Gain required = 20%
S.P of 1 kg = (120% of Rs.7.50)
= Rs.$\left[ {\displaystyle\frac{{120}}{{100}} \times 7.50} \right]$=Rs.9 per kg.
18. A person bought an article and sold it at a loss of 10%. If he had bought it for 20% less and sold it for Rs.55 more, he would have had a profit of 40%. The C.P.of the article is :
a. Rs.200

b. Rs.225

c. Rs.250

d. None of these

Correct Option : C
Explanation:
Let C.P = Rs.x. Then S.P = Rs.$\left[ {\displaystyle\frac{{90}}{{100}} \times x} \right]$ = Rs.$\left[ {\displaystyle\frac{9}{{10}}x} \right]$
New C.P = Rs.$\left[ {\displaystyle\frac{{80}}{{100}} \times x} \right]$=Rs.$\left[ {\displaystyle\frac{{4x}}{5}} \right]$
Now gain = 40%
New S.P = $\left[ {\displaystyle\frac{{140}}{{100}} \times \displaystyle\frac{{4x}}{5}} \right]$=Rs.$\left[ {\displaystyle\frac{{28}}{{25}}x} \right]$
${\displaystyle\frac{{28}}{{25}}x  \displaystyle\frac{9}{{10}}x = 55}$ or x = 250
Hence, C.P = Rs.250
Alternatively:
Assume Cost price is 100x. Then initial selling price is 90x (Why? 10% loss!)
Had he bought it for 20% less, then his cost price be 80x
Now on this 80x, he got a profit percentage of 40%. So new selling price is 140% (80x) = 112x
But the difference in selling prices is 55. So 112x  90x = 22x = 55 $ \Rightarrow $ x = 5/2
Substituting this value in 100x we get cost price = Rs.250
19. The cost price of an article, which on being sold at a gain of 12% yields Rs.6 more than when it is sold at a loss of 12% is :
a. Rs.30

b. Rs.25

c. Rs.24

d. Rs.20

Correct Option : B
Explanation:
Let C.P = Rs.x. Then ${\displaystyle\frac{{112}}{{100}}x  \displaystyle\frac{{88}}{{100}}x = 6}$
or 24x = 600 or x =${\displaystyle\frac{{600}}{{24}} = 25}$
C.P = Rs.25
20. A man sells a car to his friend at 10% loss. If the friend sells it for Rs.54000 and gains 20%, the original C.P.of the car was :
a. Rs.25000

b. Rs.37500

c. Rs.50000

d. Rs.60000

Correct Option : C
Explanation:
S.P = Rs.54,000. Gain earned = 20%
C.P = Rs.$\left[ {\displaystyle\frac{{100}}{{120}} \times 54000} \right]$=Rs. 45000
This is the price the first person sold to the second at at loss of 10%.
Now S.P = Rs.45000 and loss = 10%
C.P. Rs.$\left[ {\displaystyle\frac{{100}}{{90}} \times 45000} \right]$= Rs.50000.
21. Bhajan Singh purchased 120 reams of paper at Rs.80 per ream. He spent Rs.280 on transportation, paid octroi at the rate of 40 paise per ream and paid Rs.72 to the coolie. If he wants to have a gain of 8% , what must be the selling price per ream ?
a. Rs.86

b. Rs.87.48

c. Rs.89

d. Rs.90

Correct Option : D
Explanation:
C.P of 120 reams = Rs.$(120 \times 80 + 280 + 72 + 120 \times 0.40)$= Rs.10000.
C.P. of 1 ream = ${\displaystyle\frac{{10000}}{{120}}}$= Rs.${\displaystyle\frac{{250}}{3}}$
S.P. of 1 ream = Rs.${\displaystyle\frac{{108}}{{100}} \times \displaystyle\frac{{250}}{3} = }$Rs.90
22. Of two mixers and one T.V cost Rs.7000, while two T.Vs and one mixer cost Rs.9800, the value of one T.V is :
Explanation:
C.P of 120 reams = Rs.$(120 \times 80 + 280 + 72 + 120 \times 0.40)$= Rs.10000.
C.P. of 1 ream = ${\displaystyle\frac{{10000}}{{120}}}$= Rs.${\displaystyle\frac{{250}}{3}}$
S.P. of 1 ream = Rs.${\displaystyle\frac{{108}}{{100}} \times \displaystyle\frac{{250}}{3} = }$Rs.90
22. Of two mixers and one T.V cost Rs.7000, while two T.Vs and one mixer cost Rs.9800, the value of one T.V is :
a. Rs.2800

b. Rs.2100

c. Rs.4200

d. Rs.8400

Correct Option : C
Explanation:
2x+y = 7000 ............ (i)
x+2y= 9800 ..............(ii)
Solving (i) and (ii), we get y = 4200
23. Profit after selling a commodity for Rs.425 is same as loss after selling it for Rs.355. The cost of the commodity is :
Explanation:
2x+y = 7000 ............ (i)
x+2y= 9800 ..............(ii)
Solving (i) and (ii), we get y = 4200
23. Profit after selling a commodity for Rs.425 is same as loss after selling it for Rs.355. The cost of the commodity is :
a. Rs.385

b. Rs.390

c. Rs.395

d. Rs.400

Correct Option : B
Explanation:
Let C.P = Rs.x. Then.
425x= x355 or 2x = 780 or x = 390.
24. A merchant sold his goods for Rs.75 at a profit percent equal to C.P. The C.P was :
Explanation:
Let C.P = Rs.x. Then.
425x= x355 or 2x = 780 or x = 390.
24. A merchant sold his goods for Rs.75 at a profit percent equal to C.P. The C.P was :
a. Rs.40

b. Rs.50

c. Rs.60

d. Rs.70

Correct Option : B
Explanation:
Let C.P=Rs.x
x + x% of x = 75 or x + ${\displaystyle\frac{{{x^2}}}{{100}}}$=75 or
${{x^2} + 100x  7500 = 0}$ or (x + 150)(x50)=0
x = 50 (Neglecting x =  150)
25. A horse and cow were sold for Rs.12000 each. The horse was sold at a loss of 20% and the cow at a gain of 20%. The entire transaction resulted in :
Explanation:
Let C.P=Rs.x
x + x% of x = 75 or x + ${\displaystyle\frac{{{x^2}}}{{100}}}$=75 or
${{x^2} + 100x  7500 = 0}$ or (x + 150)(x50)=0
x = 50 (Neglecting x =  150)
25. A horse and cow were sold for Rs.12000 each. The horse was sold at a loss of 20% and the cow at a gain of 20%. The entire transaction resulted in :
a. No loss or gain

b. Loss of Rs.1000

c. Gain of Rs.1000

d. Gain of Rs.2000

Correct Option : B
Explanation:
In the special case of profit and loss percentages are equal and Selling price is same, then the transaction always results in Loss. This loss percentage is given by a simple formula $  {\left( {\displaystyle\frac{x}{{10}}} \right)^2}$
So in this case, Profit% = Loss% = 20. So x = 20
Loss percentage = $  {\left( {\displaystyle\frac{{20}}{{10}}} \right)^2} =  4$
Total S.P = Rs.24000
Cost price = $24000 \times \displaystyle\frac{{100}}{{96}}$ = 25000
Loss = Rs.1000
Explanation:
In the special case of profit and loss percentages are equal and Selling price is same, then the transaction always results in Loss. This loss percentage is given by a simple formula $  {\left( {\displaystyle\frac{x}{{10}}} \right)^2}$
So in this case, Profit% = Loss% = 20. So x = 20
Loss percentage = $  {\left( {\displaystyle\frac{{20}}{{10}}} \right)^2} =  4$
Total S.P = Rs.24000
Cost price = $24000 \times \displaystyle\frac{{100}}{{96}}$ = 25000
Loss = Rs.1000