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Profit and Loss In this section we are going to learn an interesting and very important topic of Quant section Profit and Loss. This is a very wide topic of the quant section in which you will learn some real-life solutions to problems and it will increase your aptitude dramatically.

So let’s start by taking a simple example and try to understand what profit/loss in a trade is.

E.g. A man buys an article for 300 rs. and sells it for 900 rs. Find profit/loss.

We saw he buys at 300 rs. and sells that article at 900 rs. so he got additional

⇒ 900 – 300 = 600 that is the extra money he got this is called profit for him.

Let’s take another scenario of loss;

E.g. Aman buys an article at 900 rs. and sells it for 300 rs. Find profit/loss.

He buys at 900 rs. and sells at 300 rs. so he losses 900 – 300 = 600 rs.

Whenever a purchased article is sold, then either profit is earned or loss is incurred.

Let’s learn some terminologies;

Selling price(SP): The price at which article is sold.

Cost price(CP): The price at which article is manufactured or purchased.

Profit(SP – CP): When an article is sold at a price more than its cost price, then profit is earned.

Loss(CP – SP): When an article is sold at a price less than its cost price, then the loss is incurred.

In day-to-day life we don’t talk about absolute loss/gain we talk about profit/loss in percentage term, eg. 15 % off, 20 % sale. what are they, let’s talk about them.

We learn percentage is the fraction with denominator 100

Profit % & Loss%:

Profit/Loss (%) = (Selling price – Cost price) × 100% / Cost price

= (SP – CP) × 100% / CP

In formula if

Selling price – Cost price > 0 Then there is profit

Selling price – Cost price < 0 Then there is loss

TIP- Profit and loss are always calculated on cost price unless otherwise stated in the question.

E.g. A person buys a toy for 50 and sells it for 75. What will be his gain percent?

So according to our question

Selling Price = 75

Cost Price = 50

Selling Price – Cost Price = 75 – 50 = 25 > 0

So there would be profit

Profit (%) = (75 – 50) / 50 × 100

Profit (%) = 25 / 50 × 100

Profit (%) = 1 / 2 × 100

Profit (%) = 50%  (Ans.)

So he would get 50% profit.

E.g. Find the SP, when CP is 80 rs. and the gain is 20%.

He gains 20 % for selling at ‘x’ rs. and buying it for 80 rs.

So, Gain (%) = (SP – CP) / CP × 100

⇒ 20 = (x – 80) / 80 × 100

⇒ (20 × 80) / 100 = x – 80

⇒ 16 = x – 80

⇒ x = 96 rs.

Rearranging above formula we would get

⇒ SP = ((100 + gain%) / 100) × CP

⇒ SP = ((100 – Loss%) / 100) × CP

⇒ CP = (100 / (100 + gain%)) × SP

⇒ CP = (100 / (100 – Loss%)) × SP

We don’t have to remember these formulas; they are just derived from the main formula.

E.g. Find the SP, when CP is 80 rs. and the gain is 20%.

We know the direct formula for SP when there is gain

⇒ SP = ((100 + Gain%) / 100) × CP

⇒ SP = ((100 + 20) / 100) × 80

⇒ SP = 1.2 × 80

⇒ SP = 96 rs.

Multiplying factor:

So we got CP × MF = SP no matter what we have profit/loss, just simple.

E.g. Find the SP, when CP is 80 rs. and the gain is 20%.

So multiplying factor would be  = 1 + 20 / 100 = 1 + 0.2 = 1.2

So by multiplying factor method

⇒ 80 × 1.2 = SP

⇒ 96 = SP

This is the same question which we had solved above with two methods.

Let’s take some variety of questions related to profit and loss;

E.g. A shopkeeper buys 100 eggs at Rs. 1.20 per piece Unfortunately 4 eggs got spoiled during transportation. The shopkeeper sells the remaining eggs at rs. 15 a dozen. Find his profit or loss.

The cost price of one egg = 1.2 rs.

Cost Price of 100 eggs = 100 × 1.2 = 120 rs.

The selling price of one egg = 15 / 12 = 1.25 rs.   [One Dozen = 12 units]

The selling price of (100 – 4) eggs = 96 × 15 / 12 = 120rs. (Ans.)

⇒ SP = CP

So there is neither profit nor loss incurred.

E.g. A grocer buys 160 kg of rice at 27 per kg and mixes it with 240kg of rice available at 32 per kg. At what rate per kg should he sell the mixture to gain 20% on the whole?

Total rice =160 + 240 = 400kg

Total cost price of rice = 160 × 27 + 240 × 32 = 12000 rs.

The multiplying factor would be = 1 + 20 / 100 = 1 + 0.2 = 1.2

By multiplying factor

⇒ SP = CP × MF

⇒ SP = 12000 × 1.2 = 14000 rs.

E.g. A man sold two radios for 2000 each. At first, he gains 16%, and on the other he losses 16%. Find his gain or loss percent in the whole transaction.

Let’s talk about the first case

⇒ SP = 2000 rs.

⇒ MF = 1 + 16 / 100 = 1.16

⇒ SP = MF × CP

⇒ CP = SP / MF

⇒ CP = 2000 / 1.16 = 1724 (Approx)

Now, see the second case

⇒ SP = 2000 rs.

⇒ MF = 1 – 16 / 100 = 0.84

⇒ SP = MF × CP

⇒ CP = SP / MF

⇒ CP = 2000 / 0.84 = 2381 (Approx)

So total SP = 2000 + 2000 = 4000

So total CP = 2381 + 1724 = 4104

Profit% = (SP – CP) / CP × 100

Profit% = (4000 – 4104) / 4104 × 100

Profit% = -104 / 4104 × 100 = 2.56% (Ans.)

NOTE: If a person sells two similar articles, one at a gain of a% and another at a loss of a%, Then the seller always incurs a loss which is given by

Loss% = (x / 10)2

E.g. A man sold two radios for 2000 each. At first, he gains 16%, and on the other he losses 16%. Find his gain or loss percent in the whole transaction

By applying formula

⇒ Loss(%) = (x / 10)2

⇒ Loss(%) = (16 / 10)2

⇒ Loss(%) = 2.56% (Ans.)

Discount is defined as the amount of rebate given on a fixed price (called a marked price) of an article.

So we got to know

Marked price = CP + Markup, So we got to know

Goods are sold at market price, if there is no discount then

⇒ Marked price = CP

So we got to know how marked price is calculated but in case of discount product it sold at d% discount,

Selling price = Marked price – Discount

Selling price = Marked price(MP) – discount on MP(d%)

NOTE- discount is always calculated on the marked price, not on cost price and markup is calculated on the basis of CP.

MP(1 – d / 100) = SP = CP(1 + a / 100)

Where

SP = selling price

MP = Marked price

CP = Cost price

d = Discount %

a = Gain%

E.g. A trader offers his consumer 10% discount still makes a profit of 26% what is the actual cost of an article marked at rs 280

By putting the values in the formula

⇒  MP(1 – d / 100) = SP = CP(1 + a / 100)

⇒  280(1 – 10 / 100) = CP(1 + 26 / 100)

⇒  280 × 0.9 = CP × 1.26

⇒  CP = (280 × 0.9) / 1.26

⇒  CP = 200 rs. (Ans.)

E.g. if the marked price of an article is Rs.660 and the discount is 10%, then what is the selling price of the article?

Putting the values in formula

⇒  MP(1 – d / 100) = SP = CP(1 + a / 100)

⇒ MP(1 – d / 100) = SP

⇒ 660(1 – 10 / 100) = SP

⇒ 660(1 – 0.1) = SP

⇒ 660 ×  0.9 = SP

SP=600Rs. (Ans.)

E.g. A trader markup the goods by 10% and then gives a discount of 10%. What is the profit or loss percentage?

Let the cost price of the good be 100

As we know the markup is calculated on CP

MP = 100(1 + 10 / 100)

MP = 110

We got a marked price and we know the discount is always calculated on the marked price so the selling price would be

SP = 110 × (1 – 10 / 100)

SP = 110 × 0.9

SP = 99

So there is a loss of 1% (Ans.)

E.g. The marked price of a bicycle is 1100. A shopkeeper allows a discount of 10% and gets a profit of 10%. Find the cost price of the bicycle.

As we have given gain%,MP and discount% so we will use our general formula

⇒ MP(1 – d / 100) = SP = CP(1 + a / 100)

⇒ MP(1 – d / 100) = CP(1 + a / 100)

⇒ 1100(1 – 10 / 100) = CP(1 + 10 / 100)

⇒ 1100 × 0.9 = CP × 1.1

⇒ CP = (1100 × 0.9) / 1.1

⇒ CP = 900 rs. (Ans.)

Successive discount

When two or more discounts are allowed one after the other, then such discounts are known as successive discounts.

Let r1%, r2%, r3% … be the series of discounts on an article with marked price P then the selling price of the article is given by

SP = P × (1 – r­/ 100)(1 – r/ 100)(1 – r/ 100)…

E.g. An item is sold for 680 rs. by allowing a discount of 15% on its marked price. What is the marked price of an item?

⇒  SP = P × (1 – r / 100)

⇒ P = SP / (1 – r / 100)

⇒ P = 680 / (1 – 0.15)

⇒ P = 680 / 0.85

⇒ P = 800 rs. (Ans.)

E.g. Which is a better bargain for the customer?

1. Successive discounts of 20% and 15%

2. Successive discounts of 10% and 25%

Let’s calculate the total discount for the first case, as we don’t have given marked price so we assume MP=100

⇒ SP = 100 × (1 – 20 / 100)(1 – 15 / 100)

⇒ SP = 100 × 0.8 × 0.85

⇒ SP = 68

So the discount would be 100-68=32%

In the second scenario

⇒ SP = 100 × (1 – 10 / 100)(1 – 25 /  100)

⇒ SP = 100 × 0.9 × 3 / 4

⇒ SP = 67.5

The discount would be 100 – 67.5 = 32.5%

So the second bargain would be better. (Ans.)

E.g. Successive discount of 10%,15%,20%, and 25% is equivalent to?

Let the marked price of the item be 100 rs.

Then the selling price would be

⇒ SP = P × (1 – r1 / 100)(1 – r2 / 100)(1 – r3 / 100)(1 – r4 / 100)

⇒ SP = 100 × (1 – 10 / 100)(1 – 15 / 100)(1 – 20 / 100)(1- 25 / 100)

⇒ SP = 100 × 0.9 × 0.85 × 0.8  × 0.75

⇒ SP = 45.9 rs.

So the discount would be = 100 – 45.9 = 54.1%(Ans.)

In the case of two successive discounts

Total discount = x + y – xy / 100

Where x, y are the successive discounts.

E.g. which is a better bargain for the customer?

1. Successive discounts of 20% and 15%

2. Successive discounts of 10% and 25%

In the first case, the total discount would be

⇒ 20 + 15 – (20 × 15) / 100

⇒ 35 – 300 / 100

⇒ 35 – 3 = 32%

In the second case

⇒ 10 + 25 – (10 × 25) / 100

⇒ 35 – 2.5

⇒ 32.5%

The bargain would be higher in the second case. (Ans.)

NOTE- If a shopkeeper wants a profit of R% after allowing a discount of r% then

Marked Price = CP((100 + R) / (100 – r)}

Or

Cost Price = MP{(100 – r) / (100 + R)}

E.g. the marked price of a bicycle is 1100. A shopkeeper allows a discount of 10% and gets a profit of 10%. Find the cost price of the bicycle.

So by formula

⇒ Cost Price = M{(100 – r) / (100 + R)}

⇒ Cost Price = 1100 ×{(100 – 10) / (100 + 10)}

⇒ Cost Price = 1100 × 90 / 110

Cost Price = 900 rs. (Ans.)

E.g. if the cost price of 15 apples is the same as the selling price of 20 apples. What is the gain or loss percent?

⇒ CP of 15 apples = SP of 20 apples

⇒ CP × 15 = SP × 20

⇒ CP / SP = 20 / 15 = 4 / 3

Let SP = 3 and CP = 4 as SP – CP < 0, so there would be a loss.

⇒ Loss(%) = (SP – CP) / CP × 100

⇒ Loss(%) = (3 – 4) / 4 × 100

⇒ Loss(%) = -1 / 4 × 100

⇒ Loss(%) = -25% (Ans.)

Here negative sign signifies that loss is incurred.

E.g. by selling 8 bananas, a fruit vendor would gain a selling price of 1 banana. Calculate the gain percent.

Let the SP of one banana =1 rs.

SP of 8 bananas = 8 rs.

Profit = 1 rs.

So CP would be 8 – 1 = 7 rs.

Profit(%) = (SP – CP) / CP × 100

Profit(%) = (8 – 7) / 7 × 100

Profit(%) = 1 / 7 × 100 = 14.28% (Ans.)

Some shopkeepers use different ways to gain profit illegally by cheating with costumers. He might use false weights, report the lower weight of his weighing instrument, etc. This will be profitable for the shopkeeper but would cause loss to the buyer.

Use of false weight for selling an article

A shopkeeper uses a false scale to sell his goods to the customer. The value of a false scale would be lower than the true scale, so the shopkeeper gains profit by selling a lesser quantity of goods to the customer.

So, the profit percentage would be

Profit(%) = (true weight – false weight) / (false weight) × 100%

Here, while calculating gain or profit percent, we have taken false weight as a base. Because CP is what is paid when an item is purchased or manufactured. Here, in this case, the dishonest shopkeeper is telling false weight to be the CP and he is gaining only when sells at the false weight.

E.g. A dishonest dealer professes to sell his goods at cost price but he uses a weight of 930g for 1kg weight. Find his gain percent.

This dealer sells his goods at CP but uses false weight so

Profit(%) = (true weight – false weight) / (false weight) × 100%

Profit(%) = (1000 – 930) / 930 × 100%

Profit(%) = 70 / 930 × 100%

Profit(%) = 7 / 93 × 100%

Profit(%) = 7.53% (Ans.)

We discussed the case where the dealer sells his good at cost price what would happen if he not sell his product at cost price and uses a false scale too. Let him not sell his product at CP and gains/loss of x%.

By combining these two effects the dealer would get a gain of G% in selling his good to the consumer by using false weight.

E.g. A cloth merchant says that due to a slump in the market, he sells the cloth at 10% loss, but he uses a false-meter scale and gains 15%. Find the actual length of the scale.

Let the actual length of scale be lcm and true be 100cm

Final gain (%) = G = 15%

x (%) = -10%

So

⇒ (100 + G) / (100 + x) = (true length) / (false length)

⇒ (100 + 15) / (100 – 10) = 100 / l

⇒ 115 / 90 × 1 / 100 = 1 / l

⇒ l = 90 / 115 × 100

⇒ l = 78.25cm (Ans.)

The actual length would be 78.25cm instead of 1m

E.g. A man sells rice at 10% profit and uses weight 30% less than the actual measure. His gain percent is?

Let the weight be 1000g and if he uses 30% less weight then the false scale would be 1000×0.7=700g(multiplying factor)

According to formula

⇒ (100 + G) / (100 + x) = (true value) / (false value)

⇒ (100 + G) / (100 + 10) = 1000 / 700

⇒ 100 + G = 10 / 7 × 110

⇒ 100 + G = 157.1428

⇒ G = 57.1428% Profit (Ans.)

SPECIAL CASE: If a shopkeeper sells his goods at a% loss on cost price but uses ‘b’ g instead of ‘c’ g, then his percentage profit or loss is

[(100 – a) × c / b – 100]%

E.g. A dealer sells goods at a 6% loss on cost price but uses 14g instead of 16g. What is his percentage profit or loss?

By comparing with the formula we got

a = 6

b = 14

c = 16

So putting the values in the formula

⇒ [(100 – 6) × 16 / 14 -100)]

⇒ 94 × 8 / 7 – 100

⇒ (752 – 700) / 7

⇒ 52 / 7% Or 7.43% gain (Ans.)

E.g. A milkman makes a profit of 20% on the sale of milk. If he were to add 10% water to the milk, by what percentage his profit increase?

Let the quantity of milk be 100g  and if he will add 10% water then it will become 100×1.1=110g so the true value would be 110 and false would be 100

So according to formula

⇒ (100 + G) / (100 + x) = (true value) / (false value)

⇒ (100 + G) / (100 + 20) = 110 / 100

⇒ (100 + G) / 120 = 11 / 10

⇒ 100 + G = 11 / 10 × 120

⇒ 100 + G = 11 × 12

⇒ G = 32% (Ans.)
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