# Formula - Profit & Loss

#### INTRODUCTION

Cost Price : The amount paid to purchase an article or the price at which an article is made, is known as its cost price.The cost price is abbreviated as C.P..

Selling Price : The price at which an article is sold, is known as its selling price.The selling price is abbreviated as S.P..

Profit : If the selling price (S.P.) of an article is greater than the cost price (C.P.), then the difference between the selling price and cost price is called profit.
Thus, If S.P. > C.P., then

Profit = S.P. – C.P.

=> S.P. = C.P. + Profit

=> C.P. = S.P. – Profit.

Loss : If the selling price (S.P.) of an article is less than the cost price (C.P.), then the difference between the cost price (C.P.) and the selling price (S.P.) is called loss.Thus, if S.P. < C.P., then Loss = C.P. – S.P. => C.P. = S.P. + Loss

=> S.P. = C.P. – Loss

Example 1

1. An article was bought for 2000 and sold for 2200. Find the gain or loss.

Sol. C.P. of the article = 2000

S.P. of the article = 2200

Since S.P. > C.P. So there is gain.

Gain (profit) = S.P. – C.P.

=2200 –2000 =200

Profit and Loss percentage

The profit percent is the profit that would be obtained for a C.P.of 100.

Similarly, the loss percent is the loss that would be made for a C.P. of 100.

REMEMBER

$Profit = \frac{C.P \times Profit\%}{100}$

$Loss = \frac{C.P \times Loss\%}{100}$

$S.P. = \left ( \frac{100 + Profit\%}{100} \right ) \times C.P.$

$S.P. = \left ( \frac{100 - Loss\%}{100} \right ) \times C.P.$

$C.P. = \frac{100 \times S.P.}{100 + Profit\%}$

$C.P. = \frac{100 \times S.P.}{100 - Loss\%}$

Example 2

2. A cycle was purchased for 1600 and sold for1400. Find the loss and loss %.

Sol. C.P. of the cycle = 1600

S.P. of the cycle = 1400

Since S.P < C.P, so there is a loss.

Loss = C.P. – S.P.

= 1600 – 1400 = 200.

Loss% = $\frac{Loss}{C.P.} \times 100 = \frac{200}{1600} \times 100 = 12\tfrac{1}{2}$%

Example 3

3. By selling a table for 330, a trader gains 10%. Find the cost price of the table.

Sol. S.P. = 330, Gain = 10%

∴ C.P. = $\left ( \frac{100}{100 + Gain} \right ) \times S.P.$

= $\left ( \frac{100}{100 + 10} \right ) \times 330$

= $\left ( \frac{100}{110} \right ) \times 330$ = 300.

Example 4

4. A sells a bicycle to B at a profit of 20% and B sells it to C at a profit of 25%. If C pays 225 for it, what did A pay for it.

Sol. C.P. of A = $225 \times \frac{100}{100 + 20} \times \frac{100}{100 + 25}$

= $225 \times \frac{100}{120} \times \frac{100}{125}$ = 150

Example 5

5. A mobile phone is sold for 5060 at a gain of 10%. What would have been the gain or loss percent if it had been sold for 4370 ?

Sol. S.P. = 5060, gain = 10%

∴ C.P. = $\frac{5060 \times 100}{100 + 10}$ = 4600

2nd S.P. = 4370

Since, S.P. < C.P., so there is loss.

∴ Loss% = $\frac{(4600 - 4370) \times 100}{4600}$ = 5%

Real Profit/Loss percentage :

If the profit or loss is calculated on S.P., then it is not actual profit or loss.

Real profit (loss)% is the profit (loss)% on C.P.

Real Profit% = $\mathbf{\frac{\% Profit \; on \; S.P.}{100 - \%Profit \; on \; S.P.} \times 100 }$

Dishonest dealing

$\mathbf{Gain\% = \frac{Error}{True \; Value - Error} \times 100}$
$\mathbf{\frac{True \; Scale}{False \; Scale} = \frac{100 + gain\%}{100 - loss\%}}$
Example 6

6. A cloth merchant says that due to slump in the market, he sells the cloth at 10% loss, but he uses a false metre-scale and actually gain 15%. Find the actual length of the scale.

(a) 78 cm
(b) 78.25 cm
(c) 78.5 cm
(d) 78.75 cm

Sol.(b)

$\frac{True \; Scale}{False \; Scale} = \frac{100 + gain\%}{100 - loss\%}$

$\frac{100}{False \; Scale} = \frac{100 + 15}{100 - 10}$

=> False Scale = $\frac{100 \times 90}{115}$ = 78.26 cm.

Example 7

7. A dishonest dealer professes to sell his goods at cost price, but he uses a weight of 960 g for the kg weight.Find his gain per cent.

Sol. Error = 1 kg – 960 g

= 1000 g – 960 g = 40 g.

$Gain\% = \frac{40}{1000 - 40} \times 100$

= $\frac{40}{960} \times 100 = 4\tfrac{1}{6}\%$

Goods passing through successive hands

When there are two successive profits of a% and b%, then the resultant profit per cent is given by

$\left ( a + b + \frac{ab}{100} \right )\%$

When there is a profit of a% and loss by b% in a transaction,then the resultant profit or loss percent is given by

$\left ( a - b - \frac{ab}{100} \right )\%$ according to the +ve or –ve sign respectively.

When cost price and selling price are reduced by the same amount (A) and profit increases then cost price (C.P.)

= $\frac{\left [ Initial \; Profit\% + Increase \; in \; Profit\% \right ] \times A}{Increase \; in \; Profit\%}$

Example 8

8. A table is sold at a profit of 20%. If the cost price and selling price are 200 less, the profit would be 8% more. Find the cost price.

Sol. By direct method,

C.P. = Rs.$\frac{\left ( 20 + 8 \right ) \times 200}{8}$ = 28 × 25 = 700.

If cost price of x articles is equal to the selling price of y articles, then profit/loss percentage = $\frac{x - y}{y} \times 100\%$ according to +ve or –ve sign respectively.

Example 9

9. If the C.P. of 15 tables be equal to the S.P. of 20 tables, find the loss per cent.

Sol. By direct method,

Profit/Loss% = $\frac{-5}{20} \times 100 = 25\%$ loss, since it is –ve.

Example 10

10. If the C.P. of 6 articles is equal to the S.P. of 4 articles. Find the gain per cent.

Sol. Let C.P. of an article be x; then,

C.P. of 6 articles =6x

C.P. of 4 articles =4x

But S.P. of 4 articles = C.P. of 6 articles

∴ S.P. of 4 articles = 6x

Thus, gain = S.P – C.P. = (6x – 4x) =2x

∴ Gain% = $\frac{2x}{4x} \times 100 = 50$

Thus, gain in the transaction = 50%

Example 11

11. By selling 33 metres of cloth, a man gains the sale price of 11 metres. The gain % is

(a) 50%
(b) 25%
(c) 33⅓%
(d) 20%

Sol.(a) Gain = S.P. of 33 metres – C.P. of 33 metres

= S.P. of 11 metres

=> S.P. of 22 metres = C.P. of 33 metres

∴ % gain = $\frac{gain}{C.P. \; of \; metres} \times 100$

= $\frac{S.P. \; of \; 11 \; metres}{C.P. \; of \; 33 \; metres} \times 100$

= $\frac{S.P. \; of \; 11 \; metres}{S.P. \; of \; 22 \; metres} \times 100$ = $\frac{11}{22} \times 100 = 50\%$

Shortcut method :

If on selling ‘x’ articles a man gains equal to the S.P. of y articles. Then,

% gain = $\frac{y}{x - y} \times 100 = \frac{11}{33 - 11} \times 100 = \frac{11}{22} \times 100 = 50\%$

Marked Price : The price on the lable is called the marked price or list price.

The marked price is abbreviated as M.P.

Discount : The reduction made on the ‘marked price’of an article is called the discount.

NOTE : When no discount is given, ‘selling price’ is the same as‘marked price.

• Discount = Marked price × Rate of discount.

• S.P. = M.P. – Discount.

• Discount% = $\frac{Discount}{M.P.} \times 100.$

• Buy x get y free i.e., if x + y articles are sold at cost price of x articles, then the percentage discount = $\frac{y}{x + y} \times 100.$.

Example 12

12. How much % must be added to the cost price of goods so that a profit of 20% must be made after throwing off a discount of 10% from the marked price?

(a) 20%
(b) 30%
(c) 33⅓%
(d) 25%

Sol.(c)

Let C.P.= 100, then S.P. = 120

Also, Let marked price be x. Then

90% of x = 120

=> x = $\frac{120 \times 100}{90}$ = 133⅓

∴ M.P. should be 133⅓

or M.P. = 33⅓% above C.P.

Example 13

13. At a clearance sale, all goods are on sale at 45% discount. If I buy a skirt marked 600, how much would I need to pay?

Sol. M.P. = 600, Discount = 45%

Discount = $\frac{M.P. \times Discount\%}{100} = \frac{600 \times 45}{100}$ = 270

S.P. = M.P. – Discount

= 600 – 270 =330.

Hence, the amount I need to pay is 330.

• A man purchases a certain number of articles at x a rupee and the same number at y a rupee. He mixes them together and sells them at z a rupee. Then his gain or loss %

= $\left [ \frac{2xy}{z\left ( x + y \right )} - 1 \right ] \times 100$ according as the sign is +ve or –ve.

• If two items are sold, each at. x, one at a gain of p% and the other at a loss of p%, there is an overall loss given by p2100% .

The absolute value of the loss is given by $\frac{2p^{2}x}{100^{2} - p^{2}}$.

• If CP of two items is the same and % Loss and % Gain on the two items are equal, then net loss or net profit is zero.

Example 14

14. A shopkeeper sold two radio sets for 792 each, gaining 10% on one, and losing 10% on the other. Then he

(a) neither gains nor loses
(b) gains 1%
(c) loses 1%
(d) gains 5%

Sol.(c) When selling price of two articles is same and

% gain = % loss

then there will be always loss.

and overall % loss = (10)2100% = 1%.

Example 15

15. A man bought two housing apartments for 2 lakhs each. He sold one at 20% loss and the other at 20% gain. Find his gain or loss.

(a) 4% loss
(b) 4% gain
(c) No loss, no gain
(d) 10 % loss

Sol.(c) When C.P. of two articles is same and

% gain = % loss

Then, on net, there is no loss, no gain

when two different articles sold at same S.P. and x1 and x2 are %gain (or loss) on them. Then, overall %gain or loss

= $\left [ \frac{100 \; - \; 2\left ( 100 \; \pm \; x_{1} \right )\left( 100 \; \pm \; x_{2} \right )}{\left( 100 \; \pm \; x_{1} \right ) \; + \; \left( 100 \; \pm \; x_{2} \right )} \right ]\%$

(Taking + or – according to gain or loss)

Example 16

16. A man sold two watches for 1000 each. On one he gains 25% and on the other 20% loss. Find how much% does he gain or lose in the whole transaction?

(a) 100/41% loss
(b) 100/41% gain
(c) No gain, no loss
(d) Cannot be determined

Sol.(b) When S1 = S2, then

overall % gain or % loss

$\left [ 100 - \frac{2\left ( 100 \; + \; x_{1} \right )\left( 100 \; + \; x_{2} \right )}{\left( 100 \; + \; x_{1} \right ) + \left( 100 \; + \; x_{2} \right )} \right ]\%$

= $\left [ 100 - \frac{2\left ( 125 \right )\left( 80 \right )}{\left( 125 \right ) + \left( 80 \right )} \right ]\%$

= $\left [ 100 - \frac{2 \times 125 \times 80}{205} \right ]\%$

= 10041% gain (∵ it is +ve)

Example 17

17. After allowing a discount of 12% on the marked price of an article, it is sold for 880. Find its marked price.

Sol. S.P. = 880 and Discount % = 12

Let M.P. = x

Discount = $\frac{M.P. \times Discount\%}{100} = \frac{x \times 12}{100} = \frac{3}{25}x$

Now, M.P. = S.P. + Discount

x = 880 + 325x

=> x - 325x = 880 => 22x25 = 880

=> x = $\frac{880 \times 25}{22} = 40 \times 25 = 1000$

∴ Marked price of the article is 1000.

Example 18

18. A shopkeeper offers his customers 10% discount and still makes a profit of 26%. What is the actual cost to him of an article marked 280?

Sol. M.P. = 280 and Discount % = 10

Discount = $\frac{M.P. \times Discount\%}{100} = \frac{280 \times 10}{100} = 28$

S.P. = M.P. – Discount = 280 – 28 = . 252

Now, S.P. = . 252 and profit = 26%

$C.P. = \frac{100}{100 + Gain\%} \times S.P.$

= $\frac{100}{100 + 26} \times 252 = 200$

Hence, the actual cost of the article is 200.

Successive Discounts

REMEMBER
In successive discounts, first discount is subtracted from the marked price to get net price after the first discount. Taking this price as the new marked price, the second discount is calculated and it is subtracted from it to get net price after the second discount. Continuing in this manner, we finally obtain the net selling price.

In case of successive discounts a% and b%, the effective discount is $\left ( a + b - \frac{ab}{100}\right )\%$

Example 19

19. Find the single discount equivalent to successive discounts of 15% and 20%.

Sol. By direct formula,

Single discount = $\left ( a + b - \frac{ab}{100}\right )\%$

= $\left ( 15 + 20 - \frac{15 \times 20}{100}\right )\%$ = 32%

NOTE : If the list price of an item is given and discounts d1 and d2 are given succesively on it then,

Final price = list price $\left ( 1 - \frac{d_{1}}{100} \right )\left ( 1 - \frac{d_{2}}{100} \right )$

Example 20

20. An article is listed at 65. A customer bought this article for 56.16 and got two successive discounts of which the first one is 10%. The other rate of discount of this scheme that was allowed by the shopkeeper was :

(a) 3%
(b) 4%
(c) 6%
(d) 2%

Sol.(b) Price of the article after first discount

65 – 6.5 = 58.5

Therefore, the second discount

= $\frac{58.5 - 56.16}{58.5} \times 100 = 4\%$

Example 21

21. A shopkeeper offers 5% discount on all his goods to all his customers. He offers a further discount of 2% on the reduced price to those customers who pay cash. What will you actually have to pay for an article in cash if its M.P. is4800?

Sol. M.P. = 4800

First discount = 5% of M.P.

= 5100 × 4800 = 240

Net price after discount = 4800 –240

= 4560

Second discount = 2% of 4560

= 2100 × 4560 = 91.20

Net price after discount = 4560 – 91.20

=4468.80

By Direct Method :

S.P. = 4800$\left ( 1 - \frac{5}{100} \right )\left ( 1 - \frac{2}{100} \right )$

= 4468.80

#### SALES TAX

To meet government’s expenditures like construction of roads,railway, hospitals, schools etc. the government imposes different types of taxes. Sales tax (S.T.) is one of these tax.Sales tax is calculated on selling price (S.P.)

NOTE : If discount is given, selling price is calculated first and then sales tax is calculated on the selling price of the article.
Example 22

22. Sonika bought a V.C.R. at the list price of 18,500. If the rate of sales tax was 8%, find the amount she had to pay for purchasing the V.C.R.

Sol. List price of V.C.R. =18,500

Rate of sales tax = 8%

∴ Sales tax = 8% of 18,500

= 8100 × 18500 = 1480

So, total amount which Sonika had to pay for purchasing the V.C.R. = 18,500 + 1480

= 19,980.

Example 23

23. The sale price of an article including the sales tax is616. The rate of sales tax is 10%. If the shopkeeper has made a profit of 12%, then the cost price of the article is :

(a) 500
(b) 515
(c) 550
(d) 600

Sol.(a) Let the CP of the article be x

Then, SP = x × 1.12 × 1.1

Now, x × 1.12 × 1.1 = 616

=> x = 6161.232 = 500