## Chapter 7: PROFIT AND LOSS

### Introduction

Cost Price = cost of item
Selling price = price at which item is sold.
Gain = Selling price - Cost price (SP > CP)
Loss = Cost price - Selling price (CP > SP)

 SP={(100+GAIN%) /100}*CP SP={(100-LOSS%)/100}*CP CP=100/(100+GAIN%)}*SP CP=100/(100-LOSS%)}*SP

So when an article of Rs 100 is sold at 35% gain that means SP is Rs.135
So when an article of CP = 100 is sold for loss of 35% its SP is Rs. 65.

When a item is sold for x% gain and other for x% loss, the seller incurs a loss of

 Loss = ( X / 10) ^ 2

• Direct Costs or Variable Costs : This is the cost associated with direct selling of product/service. In other words, this is the cost that varies with every unit of the product sold. Hence, if the variable cost in selling a pen for  20 is  5, then the variable cost for selling 10 units of the same pen is 10 × 5 = Rs. 50.

• Indirect Costs (Overhead Costs) or Fixed Costs : There are some types of costs that have to be incurred irrespective of the number of items sold and are called as fixed or indirect costs. For example, irrespective of the number of units of a product sold, the rent of the corporate office is fixed. Now, whether the company sells 10 units or 100 units, this rent is fixed and is hence a fixed cost.

• Apportionment of indirect (or fixed) costs : Fixed Costs are apportioned equally among each unit of the product sold. Thus, if n units of a product is sold, then the fixed cost to be apportioned to each unit sold is given by : $$\frac{\text{Fixed cost}}{n}$$

• The Concept of the Break-even Point : The break-even point is defined as the volume of sale at which there is no profit or no loss. In other words, the sales value in terms of the number of units sold at which the company breaks even is called the break-even point. This point is also called the break-even sales.

• Since for every unit of the product the contribution goes towards recovering the fixed costs, as soon as a company sells more than the break-even sales, the company starts earning a profit. Conversely, when the sales value in terms of the number of units is below the break-even sales, the company makes losses. The entire scenario is best described through the following example.

Q. Let us suppose that a paan shop has to pay a rent of  1000 per month and salaries of  4000 to the assistants. Also suppose that this paan shop sells only one variety of paan for  5 each

• The direct cost (variable cost) in making one paan is  2.50 per paan, then the margin is  (5 – 2.50) =  2.50 per paan.

• Now, break-even sales will be given by: Break-even-sales = Fixed costs/Margin per unit = 5000/2.5 = 2000 paans

• Hence, the paan shop breaks-even on a monthly basis by selling 2000 paans

• Selling every additional paan after the 2000th paan goes towards increasing the profit of the shop. Also, in the case of the shop incurring a loss, the number of paans that are left to be sold to break-even will determine the quantum of the loss

• Profit = (Actual sales – Break-even sales) × Contribution per unit

• Loss = (Break-even sales – Actual sales) × Contribution per unit

• Profit Calculation on the Basis of Equating the Amount Spent and the Amount Earned

• A fruit vendor recovers the cost of 25 mangoes by selling 20 mangoes. Find his percentage profit.

• Since the money spent is equal to the money earned the percentage profit is given by

• % Profit = $$\frac{\text{Goods left}}{\text{Goods sold}} * 100$$ = 5* 100/20 = 25%

IF THE TRADER PROFESSES TO SELL HIS GOODS AT COST PRICE BUT USES FALSE WEIGHTS,THEN

• GAIN= $$\frac{\text{ERROR}}{\text{(TRUE VALUE)-(ERROR)}} * 100$$%

Q. A dishonest dealer professes to sell his goods at cost price but uses a weight of 960 gms for a kg weight . Find his gain percent.

• GAIN= $$\frac{\text{ERROR}}{\text{(TRUE VALUE)-(ERROR)}} * 100$$%

• $$\frac{40}{960} * 100$$%

• 4 1/6 %

Ans .

1333

1. Explanation :

The shopkeeper sells his items at a profit of 50%. This means that the selling price is 150% of
cost price (Since CP + % Profit = SP)
For short you should view this as SP = 1.5 CP.

Ans .

175

1. Explanation :

Let s = cost of a shirt
If s = 150, 1.12s will be got by increasing s by 12% i.e. 12% of 150 = 18. Hence the value of 1.12s = 150
+ 18 =168 and s + 1.12s = 318 is not equal to 371. Hence check the next higher option.
If s = 175, 1.12s = s + 12% of s = 175 + 21 = 196. i.e. 2.12 s = 371.
Hence, Option (c) is correct.

Ans .

1

1. Explanation :

A shopkeeper sells two items at the same price. If he sells one of them at a profit of
x% and the other at a loss of x%, find the percentage profit/loss. The result will always be a loss of [x/10]^2%. Hence, the answer here is [10/10]^2% = 1% loss.


Ans .

15

1. Explanation :

The percentage loss in this case will always be (20/10)^2 = 4% loss.
We can see this directly as 360 Æ 96% of the CP Æ CP = 360/0.96. Hence, by percentage change graphic
360 has to be increased by 4.166 per cent = 360 + 4.166% of 360 = 360 + 14.4 + 0.6 =  375.
Hence, the loss is  15.

Ans .

33.33

1. Explanation :

Here since the expenditure and the revenue are equated, we can use percentage profit = (goods left × 100)/goods sold = 5 × 100/15 = 33.33%

Ans .

11.11

1. Explanation :

Here again the money spent and the money got are equal. Hence, the percentage profit is got by goods left × 100/goods sold. This gives us 11.11%.


Ans .

6.15

1. Explanation :

For this problem, the first line gives us that the cost price of the TV for the manufacturer is 6000. Further, if you have got to the 6000 figure by the end of the first line, reading further you can increase this
Manufacturing cost increase by 30% so New manufacturing cost = 7800 and new selling price is 6900 + 20% of 6900 = 6900 + 1380 = 8280.
Hence, profit = 8280 – 7800 = 480 and profit percent = 480 × 100/7800 = 6.15%


Ans .

24.76 

1. Explanation :

Using percentage change graphic starting from 100: we get 100 Æ 88 Æ 83.6 Æ 75.24 (Note we can change percentages in any order).
Hence, the single discount is 24.76%

Ans .

4 

1. Explanation :

10% reduction in price Æ 11.11% increase in consumption.
But 11.11% increase in consumption is equal to 5 gallons. Hence, original consumption is equal to 45
gallons for $180. Hence, original price = 4$ per gallon.

Ans .

780 

1. Explanation :

Solve through options using percentage rule and keep checking options as you read. Try to finish the first option-check before you finish reading the question for the first time. Also, as a thumb rule
always start with the middle most convenient option. This way you are likely to be required lesser
number of options, on an average.

Ans .

22.22 

1. Explanation :

Assume that the businessman buys and sells 1 kg of items. While buying he cheats by 10%,
which means that when he buys 1 kg he actually takes 1100 grams. Similarly, he cheats by 10% while
selling, that is, he gives only 900 grams when he sells a kilogram. Also, it must be understood that since
he purportedly buys and sells the same amount of goods and he is trading at the same price while buying
and selling, money is already equated in this case. Hence, we can directly use: % Profit = (Goods left ×
100/Goods sold) = 200 × 100/900 = 22.22%

Ans .

8 

1. Explanation :

By selling 5 articles for  15, a man makes a profit of 20% so SP = 3. Hence, CP = 2.5, if he sells 8 articles for  18.4 so SP = 2.3. Hence percentage loss = 8%.

Ans .

10 

1. Explanation :

Since money spent and got are equated, use the formula for profit calculation in terms of goods left/goods sold.
This will give you percentage profit = 2/10 = 20%.
Alternatively, you can also equate the goods and calculate the percentage profit on the basis of money as
CP of 1 orange = 8.33 paise
SP of 1 orange = 10 paise
8.33 paise so 10 paise

Ans .

4000 

1. Explanation :

Use solving-while-reading as follows: Cost price (= 2500) + Bribe (= 10% of cost of article =
250) = Total cost to the shopkeeper (2500 + 250 = 2750).
He wants a profit of 9.09 percent on this value so Using fraction to percentage change table we get 2750
+ 9.09% of 2750 = 2750 + 250 =  3000. But this  3000 is got after a rebate of 25%. Since we do not have the value of the marked price on which 25% rebate is to be calculated, it would be a good idea to work reverse through the percentage change
graphic: Going from the marked price to  3000 requires a 25% rebate. Hence the reverse process will be got by
increasing  3000 by 33.33% and getting  4000

Ans .

9.2566

1. Explanation :

We have to calculate: (average CP – average SP)/average SP.
Here, the selling price is equal in all three cases. Since the maximum number of calculations are
associated with the SP, we assume it to be 100. This gives us an average SP of 100 for the three articles.
Then, the first article will be sold at 111.11, the second at 83.33 and the third at 133.33. (The student is
advised to be fluent at these calculations) Further, the CP of the three articles is 111.11 + 83.33 + 133.33
= 327.77. The average CP of the three articles is 327.77/3 = 109.2566.
Hence, (average CP – average SP)/average SP = 9.2566%. higher

Ans .

11.11

1. Explanation :

Profit percent = (100/900) × 100 = 11.11%

Ans .

550

1. Explanation :

0.9 × Price = 495 so Price 550

Ans .

32

1. Explanation :

The SP = 107.5% of the CP. Thus, CP = 34.4/1.075 =  32.

Ans .

4000 

1. Explanation :

1.15 × Price = 4600 so Price = 4000

Ans .

125 

1. Explanation :

A loss of 20% means a cost price of 100 corresponding to a selling price of 80. CP as a percentage of the SP would then be 125%

Ans .

6.25 

1. Explanation :

2400 = 1.25 × cost price Æ Cost price = 1920
Profit at 2040 =  120
Percentage profit = (120/2040) × 100 = 6.25%

Ans .

40

1. Explanation :

CP = 935/1.1 = 850. Selling this at 810 would mean a loss of 40 on a CP of  850

Ans .

5.625

1. Explanation :

The CP will be  5400. Hence at an S.P. of 5703.75 the percentage profit will be 5.625%

Ans .

66 

1. Explanation :

CP = 63/1.05 = 60. Thus, the required SP for 10% profit = 1.1 × 60 = 66

Ans .

33.33 

1. Explanation :

The buying price is  9 per dozen, while the sales price is  12 per dozen – a profit of 33.33%

Ans .

64 

1. Explanation :

Sales tax = 120/5 =24. Thus, the SP contains  24 component of sales tax. Of the remainder (120 – 24 = 96) 1/3rd is the profit. Thus, the profit = 96/3 = 32. Cost price = 96 – 32 = 64

Ans .

31.6%

1. Explanation :

100 becomes 80 (after 20% discount) becomes 72 (after 10% discount) becomes 68.4 (after 5% discount). Thus, the single discount which would be equivalent would be 31.6%

Ans .

8

1. Explanation :

C.P × 1.2 = 25 Æ CP = 20.833
At a selling price of  22.5, the profit percent 1.666/20.833 = 8%

Ans .

175 

1. Explanation :

Solve using options. Option (c) gives you  175 as the cost of the trouser. Hence, the shirt will cost 12% more i.e. 175 + 17.5 + 3.5 = 196.
This satisfies the total cost requirement of  371

Ans .

Loss of 4% 

1. Explanation :

The formula that satisfies this condition is:
Loss of a2/100% (Where a is the common profit and loss percentage). Hence, in this case 400/100
= 4% loss.

Ans .

8 

1. Explanation :

If marked price is 1200, then CP is 1000.  and the selling price is 1080. which means profit of (1080-1000)/1000 = 8%

Ans .

360000 

1. Explanation :

The cost per toffee = 75/125=  0.6 = 60 paise. Cost of 1 million toffees = 600000. But there is a discount of 40% offered on this quantity. Thus, the total cost for 1 million toffees is 60% of
600000 = 360000.

Ans .

60

1. Explanation :

On a marked price of  80, a discount of 10% would mean a selling price of  72. Since this represents a 20% profit we get:
1.2 × CP = 72 so CP = 60

Ans .

15%

1. Explanation :

180 × 0.9 × x = 137.7 becomes x = 0.85
Which means a 15% discount

Ans .

 20%

1. Explanation :

If you assume the cost price to be 100 and we check from the options, we will see that for Option (c) the marked price will be 120 and giving a discount of 12.5% would leave the shopkeeper with
a 5% profit

Ans .

11.11

1. Explanation :

Profit percent = (100/900) × 100 = 11.11%

Ans .

82.8

1. Explanation :

Cost per 100 apples = 60 + 15% of 60 =  69.
Selling price @ 20% profit = 1.2 × 69 =  82.8

Ans .

none

1. Explanation :

A cost price of  650 would meet the conditions in the problem as it would give us a loss of 140 (if sold at 510) and a profit of 70 (when sold at 720)

Ans .

53 

1. Explanation :

If the cost price is 100, a mark up of 80% means a marked price of 180. Further a 15% discount on the marked price would be given by:
180 – 15% of 180 = 180 – 27 =153. Thus, the percentage profit is 53%

Ans .

440 

1. Explanation :

Cost price to the watch dealer
= 250 + 10% of 250 =  275
Desired selling price for 20% profit
= 1.2 × 275 = 330
But 330 is the price after 25% discount on the marked price.
Thus,
Marked price × 0.75 = 330 so MP = 440
Hence, he should mark the item at  440.

Ans .

100/98 

1. Explanation :

Total cost = 50 × 10 + 40 × 12 = 980. Total revenue = 90 × 11 = 990. Gain percent = (10 × 100)/980 = 100/98 %.

Ans .

50 

1. Explanation :

goods left/Goods sold * 100 = = 10/20 × 100 = 50%

Ans .

5 (5/19) 

1. Explanation :

The profit percent would be equal to 50 × 100 /950 = 5000/950 = 100/19% = 5 (5/19)%

Ans .

50 

1. Explanation :

B sold the table at 25% profit at  75. Thus cost price would be given by: CPB × 1.25 = 75
B’s Cost price =  60.
We also know that A sold it to B at 20% profit.
Thus,
A’s Cost price × 1.2 = 60
so A’s cost price = 50.

Ans .

300 

1. Explanation :

300 (A buys at this value) Æ 345 (sells it to B at a profit of 15%) Æ 404 (B sells it back to A at a profit of 20% gaining  69 in the process). Thus, A’s original cost = 300

Ans .

1000 

1. Explanation :

The CP of the TV Æ CPTV × 0.8 =12000 Æ CPTV = 15000
The CP of the VCP Æ CPVCP × 1.2 = 12000 so CPVCP = 10000.
Total sales value = 12000 × 2 = 24000.
Total cost price = 15000 + 10000 = 25000. Loss = 25000 – 24000= 1000

Ans .

33.33 

1. Explanation :

The total manufacturing cost of the article = 60 + 45 + 30 = 135. SP = 180. Thus, profit =  45. Profit Percent = 45 × 100/135 = 33.33%

Ans .

From X

1. Explanation :

Assume marked price for both to be 100.
X’s selling price = 100 × 0.75 × 0.95 = 71.25
Y’s selling price = 100 × 0.84 × 0.88 = 73.92.
Buying from ‘X’ is more profitable

Q. A person incurs loss of 5% by selling a watch for Rs 1140. At what price should the watch be sold to earn a  5% profit ?

A. CP is 100/95 * 1140 as a loss of 5% on CP is 1140 so CP is 100/95% of 1140 [SP] as per formula above.

New selling price = (100 / 95) * ( 105 / 100) * 1140 = 1260

Q. By selling 33 metres of cloth , one gains the selling price of 11 metres . Find the gain percent .

A.

Selling price of 33m - CP of 33 m = SP of 11m [Gain].

SP of 22 m = CP of 33m i.e. SP of 2 m = CP of 3 m

Assume CP of 1 m = Re. 1 so 3 m is Rs. 3.

SP of 2 m = Rs. 3 and SP of 1 m = Rs. 1.5 so gain is 50%.

Q. A man brought toffees at 3 for a rupee. How many for a rupee must he sell to gain 50%?

A. Cost of a toffee is 3 toffees is re. 1. So he needs to sell them at Rs. 1.5 to make a gain of 50% so at 50 paise each or 2 for a rupee.

Q. A  grocer  purchased  80  kg  of  sugar  at Rs.13.50  per  kg  and mixed  it with 120kg sugar at Rs.16per kg. At what rate should he sell the mixer to gain 16%?

A. The cost of the mixture

= (weight1 * price1) + (weight2*price2) / (weight1+weight2)

= (80*13.5) + (120*16) / (120+80)

=(1080+1920)/200 = 3000/200 = Rs. 15 / kg

SP = 116/100 * 15 = 1.16*15=17.4 Rs

Q. Pure ghee cost Rs.100 per kg. After adulterating it with vegetable oil costing Rs.50 per kg,   A  shopkeeper  sells  the mixture at  the rate of Rs.96 per kg,  thereby making a profit of 20%. In What ratio does he mix the two?

A. SP of mixture = 96 ; CP of mixture = 80 as SP is 20% more than CP;

Rule of alligation:

(Quantity of oil : Quantity of ghee)

= ( CP of ghee - Mean price) / (Mean price - CP of oil)

= (100-80) / (80-50)

=20/30 = 2:3

Q. Monika purchased a pressure cooker at 9/10th of its selling price and sold it at 8% more than its S.P .find her gain percent.

A. Assume it was with a SP of Rs. 100 it was bought for Rs. 90 and sold for Rs. 108 thus making profit of Rs. 18 which is 20% more than Rs. 90.

Q. A  tradesman  sold  an  article  at  a  loss  of  20%. if  the  selling  price  has  been increased by Rs 100, there would have been a gain of 5%. what was the cost price of the article?

A. SP(old) = 80% of CP

SP(new) = 105% of CP; we know (105% of CP) - (80% of CP) = 100

i.e. 25% of CP = 100; Hence CP = Rs. 400

Q. A man sells an article at a profit of 25% if he had bought it 20% less and sold it for  Rs 10.50 less, he would have gained 30% find the cost price of the article.

A. SP1 = 1.25CP1 as it is sold for 25% profit. CP2 = (4/5)*CP1;

SP2 = (130/100)*CP2

SP2 = SP1 - 10.5 = 1.25CP1 - 10.5

so we get

1.25CP1 - 10.5 = (130/100)*(4/5)*CP1

1.25CP1 - 1.04CP1 = 10.5

0.21CP1 = 10.5

CP1 = Rs.50

Q. A dealer sold three-fourth of his article at a gain of 20% and remaining at a cost price. Find the gain earned by him at the two transaction.

A. assume he had 4 articles of CP = 100 so he sold three at 120 and 1 at 100. so he earned profit of Rs. 60. CP of total inventory is 400 so profit is 15%.

Q. A man bought a horse and a bull for Rs 3000.he sold the horse at a gain of 20% and the bull at a  loss of 10%,thereby gaining 2% on the whole. find the cost of the horse.

A. 1.2x + 0.9(3000-x) = 3600 is the equation as 'x' is price of horse. 1.2x is SP of horse at 20% profit and 0.9(3000-x) is SP of bull at 10% loss. rs 3600 is the SP of total transaction at 2% profit over Rs. 3000.

Q. find the single discount equivalent to a series discount of 20% ,10% and 5%

A. assume CP=100 so apply 20% discount to get CP=80 and then 10% to get 72 and then 5% to get 72-3.6 = 68.4 so total is 31.6%.

Q. A retailer marks all  its goods at 50% above  the cost price and thinking  that he will still make 25% profit,offers a discount of 25% on  the marked price.what is the actual profit on the sales?

A. Assume price is 100 so SP is 150 and 25% discount on SP gives new SP as Rs 112.5. So profit is 12.5%

Q. At what %  above C.P must  an  article be marked  so  as  to  gain  33%  after allowing a customer a discount of 5%?

A. We have to find the value of SP [assume as 'x'] whose 95% is 33% above CP. Assume CP to be Rs 100 and SP(new) will be Rs. 133.

0.95x = 133

x = 133 * 100 / 95 = 133 * 20 / 19 = 140

Q. A merchant sold his goods for Rs.75 at a profit percent equal to C.P. The C.P was :

A. SP = (100+Gain%) * CP /100 we get below equation by substituting these values

75 = ( 100 + x) * x / 100

### Quiz

Score more than 80% marks and move ahead else stay back and read again!