Quantitative Aptitude - SPLessons

Profit – Loss Problems

Chapter 12

SPLessons 5 Steps, 3 Clicks
5 Steps - 3 Clicks

Profit – Loss Problems

shape Introduction

Profit and Loss Problems is related to profit, loss, selling price and cost price.


shape Methods

Profit (P): When selling price is greater than the cost price then the seller is said to have profit. Profit is also known as gain.


Loss (L): When selling price is less than the cost price then the seller is said to have loss.


Selling price (S.P.): The price at which a store sells the goods, is called its selling price.


Cost price (C.P.): The price at which a store owner purchases goods, is called its cost price.


Take a scenario to have detailed explanation about profit, loss, selling price, and cost price:

Consider a shopkeeper. To sell something, the person need to have something in the shop. So, for example the person bought a soap with some cost price of 5 rupees from a whole sale dealer. Now, a customer comes to buy a soap and the person had to sell it for atleast 6 rupees. One rupee extra is profit. Suppose if the person sells it for 4 rupees then loss of one rupee incurs. Therefore, if selling price is more then it is profit and if cost price is more then it is loss.

Example 1:
An article is purchased for Rs. 450 and sold for Rs. 500. Find the gain percent.

Solution:

    Profit = Selling price (S.P.) – Cost price (C.P.) = 500 – 450 = 50.

    Gain% = (\(\frac{50}{450} \times 100\))% = \(\frac{100}{9}\)%


Example 2:
A man buys an article for Rs. 27.50 and sells it for Rs. 28.60. Find his gain percent.

Solution:

    C.P.= Rs. 27.50, S.P.= Rs. 28.60.

    So, Gain = Rs. (28.60 – 27.50) = Rs. 1.10.

    Therfore, Gain% = (\(\frac{1.10}{27.50}\) x 100)% = 4%.


Example 1:
If a radio is purchased for Rs. 490 and sold for Rs. 465.50, find the loss percent.

Solution:

    C.P.= Rs. 490, S.P.= Rs. 465.50.

    Loss = Rs. (490 – 465.50) = Rs. 24.50.

    Therefore, Loss% = (\(\frac{24.50}{490}\) x 100)% = 5%.


Example 2:
If the loss incurred in a transaction is \(\frac{3}{5}^{th}\) of the selling price, find the loss percent.

Solution:

    Let the selling price be \(x\). loss is \(\frac{3x}{5}\).

    Loss = Cost price (C.P.) – Selling price (S.P.) (Using the Formula)

    Cost price (C.P.) = Selling price (S.P.) + Loss = \(x + \frac{3x}{5}\) = \(\frac{8x}{5}\)

    Loss% = \(\frac{3x}{5}\)/\(\frac{8x}{5} \times 100\) = 37.5%


Example 1:
The C.P. of 21 articles is equal to S.P. of 18 articles. Find the gain or loss percent.

Solution:

    C.P. of 21 articles = \(x\)

    C.P. of 1 article = \(\frac{x}{21}\)

    S.P. of 18 articles = \(x\)

    S.P. of 1 article = \(\frac{x}{18}\)

    since \(\frac{x}{21}\) < \(\frac{x}{18}\), there is a gain

    Gain = S.P. – C.P. = \(\frac{x}{18}\) – \(\frac{x}{21}\)

    = \(\frac{(7x-6x)}{126}\)

    = \(\frac{x}{126}\)

    Gain % = (\(\frac{Gain \times 100}{C.P.}\))

    = (\(\frac{x}{126}\) ÷ \(\frac{x}{21}\)) x 100

    = (\(\frac{x}{126}\) x \(\frac{21}{x}\)) x 100

    = \(\frac{100}{6}\) = \(\frac{50}{3}\) = 16\(\frac{2}{3}\)%


Example 2:
By selling 33 meters of cloth, one gains the selling price of 11 meters. Find the gain percent.

Solution:

    (S.P. of 33 m) – (C.P. of 33 m) = Gain = S.P. pf 11m.

    Therefore, S.p. of 22 m = C.P. of 33 m.

    Let C.P. of each meter be Re 1. Then, C.P. of 22 m = Rs. 22, S.P. of 22 m = Rs. 33.

    Therefore, Gain% = (\(\frac{11}{22}\) x 100)% = 50%.


Example 1:
A vendor bought bananas at 6 for Rs. 10 and sold them at 4 for Rs. 6. Find his gain or loss percent.

Solution:

    Suppose, number of bananas bought = L.C.M. of 6 and 4 = 12.

    ∴ C.P. = Rs. (\(\frac{10}{6}\) x 12) = Rs. 20;

    S.P. = Rs. (\(\frac{6}{4}\) x 12) = Rs. 18.

    ∴ Loss% = (\(\frac{2}{20}\) x 100)% = 10%.


Example 2:
10% loss on selling price is what percent loss on the cost price?

Solution:

    Let S.P. = Rs. 100. Then, Loss = Rs. 10, C.p. = Rs. (100 + 10) = Rs. 110.

    ∴ Loss% = (\(\frac{10}{110}\) x 100)% = 9\(\frac{1}{11}\)%.


Example 1:
Find S.P. when C.P. = Rs. 56.25, Gain = 20%

Solution:

    S.P. = 120% of Rs. 56.25 = Rs. (\(\frac{120}{100}\) x 56.25) = Rs. 67.50.


Example 2:
Find S.P. when C.P. = 80.40, Loss = 5 %

Solution:

    S.P.= 85% of Rs. 80.40 = Rs. (\(\frac{85}{100}\) x 80.40) = Rs. 68.34.


Example 1:
Find C.P. when S.P. = Rs. 40.60, Gain = 16%

Solution:

    C.P. = Rs. (\(\frac{100}{116}\) x 40.60) = Rs. 35.


Example 2:
Find C.P. when S.P. = Rs. 51.70, Loss = 12%

Solution:

    C.P. = Rs. (\(\frac{100}{88}\) x 51.70) = Rs. 58.75.



Example:
A man sold two flats for Rs. 6, 75,958. On one he gains 16% while on the other he loses 16%. How much does he gain or lose in the whole transaction?

Solution:

    Remember: In such a case, there is always a loss. The selling price is immaterial.

    ∴ Loss% = \((\frac{Common \ Loss \ and \ Gain\%}{10})^{2}\) = \((\frac{16}{10})^{2}\)% = \(\frac{64}{25}\)% = 2.56%.



Example:
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.

Solution:

    Gain% = \([\frac{Error}{(True \ Value) – (Error)} \times 100]\)% = (\(\frac{40}{960}\) x 100)% = 4\(\frac{1}{6}\)%.

shape Formulae

1. Profit = Selling price (S.P.) – Cost price (C.P.)


2. Loss = Cost price (C.P.) – Selling price (S.P.)


3. \(Gain\%\) = \(\frac{(Gain * 100)}{Cost price}\)


4. \(Loss\%\) = \(\frac{(Loss * 100)}{Cost price}\)


5. Selling price = \(\frac{(100 + Gain\%)}{100}\) x \(Cost price\)


6. Selling price = \(\frac{(100 – Loss\%)}{100}\) x \(Cost price\)


7. Cost price = \(\frac{100}{(100 + Gain\%)}\) x \(Selling price\)


8. Cost price = \(\frac{100}{(100 – Loss\%)}\) x \(Selling price\)


9. If an object is sold at a profit of say,  \(25\%\) then
Selling price = \(125\%\) of Cost price.


10. If an object is sold at a loss of say, \(25\%\) then
Selling price = \(75\%\) of Cost price.


11. When a person sells two similar items, one at a gain of
say, \(a\%\), and the other at a loss of \(a\%\), then the seller
always incurs a loss given by
\(Loss\%\) = \((\frac{common loss and gain\%}{10})^2\) = \((\frac{x}{10})^2\)


12. If a trader professes to sell his goods at cost price, but uses false weights, then
\(Gain\%\) = [\(\frac{Error}{(True value) – (Error)} * 100]\%\)

shape Samples

1. The ratio of the cost price to the selling price is 4 : 5. Find the profit Percentage?

Solution:

    Given that ratio of cost price to selling price = 4 : 5

    Cost price = 4

    Selling price = 5

    Here selling price is more than cost price. Then,

    Profit = Selling price – Cost price

    P = 5 – 4

    P = 1

    Therefore, as profit % is calculated based on cost price

    Profit% = \(\frac{Profit}{Cost price}\) x 100

    ⇒Profit% = \(\frac{1}{4}\) x 100

    ⇒Profit% = 0.25 x 100

    ⇒Profit% = 25%

    Therefore, Profit percentage = 25%.


2. If a girl incurs 10% loss by selling her watch for Rs.1160. At what price should the watch be sold to earn 5% profit?

Solution:

    Given that

    Selling price = 1160

    Loss % = 10%

    As it is given loss, here selling price is less than cost price

    Selling price = 90% of Cost price

    If 1160 -> 0.90,

    then \(x\) -> 1.00

    By cross multiplying,

    \(x\) x 0.90 = 1160 x 1

    ⇒\(x\) = \(\frac{1160}{0.90}\)

    ⇒\(x\) = \(\frac{1160 * 100}{90}\)

    ⇒\(x\) = 1288.88 ≅ 1289

    Now Selling price = 105% of Cost price

    5% of 1289 = 64.45 ≅ 65

    so add 65 to 1289 = 1289 + 65 = 1354

    Hence, at Rs. 1354 should the watch sold to earn 5% profit.


3. A man sold two flats for Rs. 7,25,660 each. On one he gains 8% while on the other he loss 8%. How much does he gain or lose in the whole transaction?

Solution:

    Given that

    Selling price = Rs. 7,25,660

    Common loss and gain percentage = 8%

    Consider the formula,

    \(Loss\%\) = \((\frac{common loss and gain\%}{10})^2\)

    ⇒\(Loss\%\) = \((\frac{8}{10})^2\%\)

    ⇒\(Loss\%\) = \((\frac{64}{100})\%\)

    ⇒\(Loss\%\) = \(0.64\%\)


4. An article was sold for Rs. 38.50 with a profit of 10%. If it were sold for Rs. 30.75then what would have been the percentage of profit or loss?

Solution:

    Given that

    for Selling price = Rs.38.50, Profit % = 10%

    for Selling price = Rs.30.75, Profit % or loss% = ?

    Now, \(Cost price \) = \(\frac{100}{(100 + Gain\%)}\) x \(Selling price\)

    ⇒\(Cost price \) = \(\frac{100}{(100 + 10)}\) x \(38.50\)

    ⇒\(Cost price \) = 35Rs.

    Consider, Selling price = Rs.30.75, Cost price = Rs.35 then

    As selling price < cost price

    ⇒Loss = Cost price – Selling price

    ⇒Loss = 35 – 30.75 = 4.25 Rs.

    Therefore, \(Loss\%\) = \(\frac{(Loss * 100)}{Cost price}\)

    ⇒\(Loss\%\) = \(\frac{(4.25 * 100)}{35}\)

    ⇒\(Loss\%\) = 12.14%

    Hence, Loss % = 12.14 %


5. If a book is purchased for Rs. 500 and sold for Rs. 460, then find the loss percent?

Solution:

    Given that

    Cost price = Rs. 500

    Selling price = Rs. 460

    Consider, loss = cost price – selling price = 500 – 460 = 40 Rs.

    Therefore, \(Loss\%\) = \(\frac{(Loss * 100)}{Cost price}\)

    ⇒\(Loss\%\) = \(\frac{(40 * 100)}{500}\)

    ⇒\(Loss\%\) = 8%

    So, Loss percentage = 8%