Concept: Profit & Loss

Example:
(a) If C.P. = 125, S.P. = 96, then loss =?
(b) If C.P. = 112, S.P. = 132, then Gain =?
(c) If C.P. = 120, S.P. = 90, then Loss % =?
(d) If C.P. = 80, S.P. = 100, then Gain % =?
(e) If C.P. = 90, Gain % = 10, then S.P. =?
(f) If C.P. = 20, loss % = 25, then S.P. =?
(g) If S.P. = 84, Gain % = 20, then C.P. =?

Solution:
(a) Loss = CP - SP = 125 - 96 = 29

(b) Gain = SP – CP = 132 – 112 = 20

(c) Loss = CP – SP = 120 – 90 = 30
Loss % = Loss/(C.P.) × 100% = 30/120 × 100 = 25%

(d) Gain % = Gain/(C.P.) × 100 = (SP - CP)/CP × 100
= 20/80 × 100 = 25%

(e) SP = CP × mf = 90$\left(1+\frac{10}{100}\right)$ = 99

(f) SP = CP × mf = 20$\left(1-\frac{25}{100}\right)$ = 15

(g) CP = $\frac{SP}{multiplication factor}$ = $\frac{84}{\left(1+{\displaystyle \frac{20}{100}}\right)}$ = 70

Example: A sells a bicycle to B at a profit of 30% and B sells it to C at a loss of 20%. If C pays Rs. 520 for it, at what price did A buy?

Solution:
B sells the bicycle to C for Rs. 520 at a loss of 20%.

∴ Cost price for B = $\frac{520}{\left(1-{\displaystyle \frac{20}{100}}\right)}$ = 650

⇒ A's selling price is Rs. 650 and his profit % is 30% (given).

∴ Cost price for A = $\frac{650}{\left(1+{\displaystyle \frac{30}{100}}\right)}$ = 500

Alternate Method:
Price of the bicycle first increases by 30% and then decerases by 20% and finally becomes 520.

∴ 520 = (Initial Price) × $\left(1+\frac{30}{100}\right)\left(1-\frac{20}{100}\right)$

⇒ Initial Price = $\frac{520}{{\displaystyle \frac{13}{10}}\times {\displaystyle \frac{8}{10}}}$ = Rs. 500

∴ A bought the bike for Rs. 500.

Example: A man sold a horse at a loss of 7%. Had he been able to sell it at a gain of 9%, it would have fetched Rs. 64 more than it did. What was the cost price?

Solution:
Let the cost price of hourse be Rs. 'x'

It is initially sold at a loss of 7%, i.e. SP = 0.93x

If the horse were sold for Rs. 64 more there would have been a gain of 9% i.e. SP = 1.09x

⇒ 0.93x + 64 = 1.09x

⇒ 0.16x = 64

⇒ x = Rs. 400

Alternately:
Due to Rs. 64 increase, there is a change of 16% (7% loss to 9% profit)

∴ 16% of CP = 64

⇒ CP = Rs. 400

Example: An article is sold at 20% profit. If its cost price is increased by Rs. 50 and at the same time if its selling price is also increased by Rs. 30, the percentage of profit decreases by 3$\frac{1}{3}$%. Find the cost price.

Solution:

Let the original cost price be Rs. x

∴ original selling price is Rs. 1.2x

⇒ (1.2x + 30) = (x + 50) × $\left(1+\frac{20-3{\displaystyle \frac{1}{3}}}{100}\right)$

⇒ 1.2x + 30 = (x + 50) × $\frac{7}{6}$

⇒ 7.2x + 180 = 7x + 350

∴ x = Rs. 850

Example: I sell 16 articles for the same money as I paid for 20. What is my gain per cent?

Solution:
16 × SP = 20 × CP

$\frac{\mathrm{SP}}{\mathrm{CP}}$

∴ % profit = $\left(\frac{SP}{CP}-1\right)\times 100\%$  = $\left(\frac{5}{4}-1\right)\times 100\%$ = 25%

Example: If goods be purchased for Rs. 840, and one-fourth be sold at a loss of 20%, at what gain per cent should the remainder be sold so as to gain 20% on the whole transaction?

Solution:
Cost of goods = Rs. 840

CP of one-fourth of these goods = 1/4 × 840 = Rs. 210

SP of these one-fourth goods = 0.8 × 210 = 168

Overall gain % = 20%

∴ Total revenue = 1.2 × 840 = Rs. 1008

∴ Revenue for the remaining three-fourth goods = 1008 - 168 = 840

CP of these three-fourth of these goods = 3/4 × 840 = Rs. 630

∴ profit on remaining three-fourth goods = 840 - 630 = Rs. 210

⇒ % Profit = $\frac{210}{630}$ × 100 = 33$\frac{1}{3}$%

Example: If profit is 20% on selling price, calculate actual profit%?

Solution:
Let the sellign price be Rs. 100

∴ Profit = 20% of 100 = Rs. 20

∴ Cost price = 100 - 20 = Rs. 80

⇒ Actual profit % = 20/80 × 100 = 25%

Example: Three articles are bought at same cost price and sold at 40% profit, 30% profit and 10% loss respectively. Calcular the over profit/loss percentage?

Solution:
Let the cost price of each article be Rs. 100

Profit on first article = 40% of 100 = Rs. 40

Profit on second article = 30% of 100 = Rs. 30

Loss on third article = 10% of 100 = Rs. 10

Overall Profit on these three articles = 40 + 30 - 10 = 60

Total cost price of these three articles = 3 × 100 = Rs. 300

∴ overall profit % = 60/300 × 100 = 20%

Example: Two articles are sold at same price, such that one is sold at 20% profit and the other at 20% loss. Calcular the over profit/loss percentage?

Solution:
Let the selling price of each article be Rs. 960

Cost price of first article = 960/1.2 = Rs. 800

Cost price of second article = 960/0.8 = Rs. 1200

Total cost price of these articles = 800 + 1200 = Rs. 2000

Total selling price of these articles = 2 × 960 = Rs. 1920

Since total cost price is more than total selling price, there is a loss of 2000- 1920 = Rs. 80

∴ overall loss % = 80/2000 × 100 = 4%

Example: A shopkeeper marks up his goods by 25% and then sells them at a discount of 20%. Find his profit/loss percetange?

Solution:
Method 1:
Let the cost price of each article be Rs. 100

The articles is marked up by 25%, hence its marked price = 100 × 1.25 = Rs. 125

Now the article is sold by giving a discount of 20%, hence the selling price = 125 × 0.8 = Rs. 100

Since cost price = selling price = Rs. 100, the shopkeeper earns no profit no loss

∴ % profit = 0%

Alternately:
Price of the article is first increased by 25% and then decreased by 20%.

⇒ overall % change in price = 25 - 20 - $\frac{\mathrm{25\; \times \; 20}}{100}$ = 5 - 5 = 0%

There overall no profit no loss

Alternately:
Price of the article is first increased by 25% and then decreased by 20%.

Overall multiplication factor = $\left(1+\frac{25}{100}\right)\left(1-\frac{20}{100}\right)$ = $\frac{5}{4}$ × $\frac{4}{5}$ = 1

∴ overall percentage change = (1 - 1) × 100 = 0% i.e. no profit no loss.

Example: A shopkeeper claims to sell goods at cost price but gives 900 gms for a kg. Find his profit percentage?

Solution:
Let the cost price = selling price = Rs. 1/gm

For a kg (1000 gms) he actually gives only 900 gm.

His cost price for 900 gms = 900 × 1 = Rs. 900

His selling price for 1000 gms = 1000 × 1 = Rs. 1000

∴ he earns a profit of = 1000 - 900 = Rs. 100

⇒ profit % = 100/900 × 100 = 11.11%

Example: A shopkeeper sells goods at 20% more than his cost price but gives only 800 gms for a kg. Find his profit percentage?

Solution:
Let the cost price = 10/gm
selling price = Rs. 12/gm

For a kg (1000 gms) he actually gives only 800 gm.

His cost price for 800 gms = 800 × 10 = Rs. 8000

His selling price for 1000 gms = 1000 × 12 = Rs. 12000

∴ he earns a profit of = 120000 - 8000 = Rs. 4000

⇒ profit % = 4000/8000 × 100 = 50%

Alternately:
Same question can also be solved using the concept of successive percentage change.

We know, total price = (price/gm) × (quantity)

multiplifation factor (total price) = multiplication factor (price) × multiplication factor (quantity)

[Using the concpet of successive % change]

Multiplication factor for price (20% increase) = (1 + 20/100) = 6/5

An initial quantity of 800 gms is quoted to be as 1000 gms, hence multiplication factor for quantity = 1000/800 = 5/4

∴ multiplication factor for total price = 6/5 × 5/4 = 3.2 = 1.5

⇒ % change in price = (1.5 - 1) × 100 = 50%

∴ percentage profit = 50%

Example: A shopkeeper sells goods at 10% less than his cost price but gives only 800 gms for a kg. Find his profit percentage?

Solution:
We know, total price = (price/gm) × (quantity)

multiplifation factor (total price) = multiplication factor (price) × multiplication factor (quantity)

Multiplication factor for price (10% decrease) = 9/10

Multiplication factor for quantity = 1000/800 = 5/4

∴ multiplication factor for total price = 9/10 × 5/4 = 9/8 = 1.125

⇒ % change in price = (1.125 - 1) × 100 = 12.5%

∴ percentage profit = 12.5%

Example: A shopkeeper sells goods at 20% less than his cost price but to the customer he gives quantity less than the amount he charges for. Find how much quantity he actually gives for a kg if he earns 60% profit?

Solution:
Let the shopkeeper gives x gms for a kg.

We know, total price = (price/gm) × (quantity)

multiplifation factor (total price) = multiplication factor (price) × multiplication factor (quantity)

Multiplication factor for price (20% decrease) = 8/10 = 4/5

Multiplication factor for quantity = 1000/x

∴ multiplication factor for total price = 1 + 60/100 = 8/5

⇒ $\frac{8}{5}$ = $\frac{4}{5}$ × $\frac{1000}{\mathrm{x}}$

⇒ x = $\frac{\mathrm{1000\; \times \; 4}}{8}$ = 500

∴ For every kg, the shopkeeper actually gives only 500 gms.

Example: A shopkeeper sells goods at his cost but cheats by 20% while buying as well as while selling. Find his profit %?

Solution:
We know, total price = (price/gm) × (quantity)

multiplifation factor (total price) = multiplication factor (price) × multiplication factor (quantity)

Multiplication factor for price (no change) = 1

while buying the shopkeer gets 1200 gms for every 1000 gms he pays for, hence multiplication factor for quantity while buying = 1200/1000 = 6/5

while selling for every 800 gms he actually gives the shopkeer charges for 1000 gms, hence multiplication factor for quantity while selling = 1000/800 = 5/4

∴ Multiplication factor for quantity = 6/5 × 5/4 = 3/2

∴ multiplication factor for total price = 1 × 3/2 = 3/2 = 1.5

⇒ % change in total price = (1.5 - 1) × 100 = 50%

∴ percentage profit for the shopkeeper = 50%