Types of Profit and Loss Questions
In this article, we are going to have a look at the various kinds of questions formed on Profit and Loss.
Type 1: Items buyed at one price and sold at another
Type 1a
Q. Gaurav purchased an article at 4/5th of its marked price and sold it at 20% more than the marked price. What was Gaurav’s profit percent?
(a) 50% (b) 40% (c) 20% (d) 25%
Explanation:
Let marked price = 50 units
For Gaurav, cost price of the article = 4/5th of its marked price = 40 units
Now, Selling price of the article = (50 x 120)/100 = 60 units
So, the required profit percentage = [(60 - 40)/40] x 100 = (20/40) x 100 = 50%
Answer: (a)
Type 1b
Q. Some items were bought at 8 items for Rs. 5 and sold at 5 items for Rs. 8. What is the profit percentage?
(a) 108% (b) 56% (c) 156% (d) 68%
Explanation:
Let the number of items bought = L.C.M. of 8 and 5 = 40. (We will take the LCM of the number of articles)
C.P. of 40 items = (5/8) × 40 = Rs. 25
S.P. of 40 items = (8/5) × 40 = Rs. 64
So, profit percentage = (39/25) × 100 = 156%
Answer: (c)
Q. Some items were bought at 12 items for Rs. 15 and sold at 10 items for Rs. 14. What is the profit percentage?
(a) 12% (b) 18% (c) 15% (d) 9%
Explanation:
Let the number of items bought = L.C.M. of 12 and 10 = 60. (We will take the LCM of the number of articles)
C.P. of 60 items = (60/12) × 15 = Rs. 75
S.P. of 60 items = (60/10) × 14 = Rs. 84
So, profit percentage = (9/75) × 100 = 12%
Answer: (a)
Q. If cost price of 12 articles is Rs. 15 and the selling price of 8 articles is Rs. 12, then find the profit/loss percent.
Explanations :
C.P. of 12 articles = Rs. 15
So, C.P. of 1 article = Rs. 15/12 = Rs. 5/4
S.P. of 8 articles = Rs. 12
So, S.P. of 1 article = Rs. 12/8 = Rs. 3/2
Profit percent = [(S.P. – C.P.) / C.P.] × 100 = [(1/4)/(5/4)] × 100 = 20%
To apply the formula, we need to make C.P. = S.P.
L.C.M (15, 12) = 60
Cost price of 12 articles is Rs. 15
So, cost price of 48 articles is Rs. 60
Selling price of 8 articles is Rs. 12
So, cost price of 40 articles is Rs. 60
If cost price of 48 articles is equal to the selling price of 40 articles, then
Profit/loss percent = ((a−b))/b × 100 = ((48−40))/40 × 100 = 8/40 × 100 = 20%
Profit/loss percent = [(144−120)/120] × 100 = (24/120) × 100 = 20%Let there be LCM (12, 8) = 24 articles
So, CP of 24 articles = 15 × 2 = Rs. 30
SP of 24 articles = 12 × 3 = Rs. 36
So, Profit on 24 articles = 36 - 30 = Rs. 6
Profit/loss percent = (6/30) × 100 = 20%
Q. A shopkeeper buys some articles at a rate of 9 articles for Rs. 3. If he sells all of them at a rate of 4 articles for Rs. 2, and makes a profit of Rs. 18, then find the number of articles purchased.
Explanation:
Profit of 18 – 12 = Rs 6 on 36 articlesSo, profit of Rs. 18 will be on (36/6) × 18 = 108 articles
Type 2
Q. Government announced that if an electricity bill is paid before due date, one gets a concession of 5% on the amount of the bill. By paying a bill before due date, a person got a concession of Rs. 17. What was the original amount of his electricity bill?
(a) Rs. 325 (b) Rs. 350 (c) Rs. 340 (d) Rs. 334
Explanation:
Let the original amount of bill = Rs. x
According to the question,
5% of x = Rs. 17
Or 5x/100 = 17
Or x = (17 x 100)/5 = Rs. 340
Answer: (c)
Type 3: Profit/Loss in terms of SP
We know that Profit/Loss percentage is calculated on the basis of the Cost Price (CP). However, what if we are given Profit/Loss percentage as compared to the Selling Price (SP)?
Let’s see some of such questions.
Q. A person sells an article at a profit which is 25% of the selling price. If the cost price is reduced by 10% and he also reduces the selling price by 10%, then what is the profit percentage on the cost price?
(a) 30% (b) 33.33% (c) 33.67% (d) 15%
Explanation:
Let SP = 40 units
Hence, Profit = 25% of S.P. = 1/4 of S.P. = 10 units
So, C.P. = S.P. – Profit = 40 – 10 = 30 units
As Cost price is reduced by 10%, New CP = (30 x 90)/100 = 27 units
Similarly, Selling price is reduced by 10%, so new SP = (40 x 90)/100 = 36 units
New profit = SP – CP = 36 – 27 = 9 units
So, new profit percentage = (9/27) x 100 = 33.33%
Answer: (b)
Type 4: Profit = CP/SP of n items
Type 4a: Profit = CP of n items
Q. By selling 24 litres of juice at Rs. 50 per litre, a merchant earns a profit equivalent to the cost price of 6 litres. What must be his profit percentage?
(a) 15% (b) 25% (c) 20% (d) 18%
Explanation:
Let cost price per litre of juice be Rs. x
So, cost price of 24 litres of juice = 24x, and profit = 6x
Profit percentage = (Profit / Cost Price) X 100 = (6x/24x) X 100 = 25%
Answer: (b)
Type 4b: Profit = SP of n items
Q. A shopkeeper sells 100 pens, and in the process gains the selling price of 40 pens. What is the profit percentage?
(a) 40%
(b) 66.67%
(c) 33.33%
(d) Cannot be determined
Explanation:
Let the cost price of one pen = C
And the selling price of one pen = S
As given in the question,
100S – 100C = 40S
Or 60S = 100C
Or S = 5C/3
So, Gain Percent = [(SP - CP)/CP] × 100 % = [(5C/3 - C)/C] × 100% = (2/3) × 100% = 66.67%
Answer: (b)
Type 5: SP of n items is equal to CP of m items
If cost price of ‘a’ articles is equal to the selling price of ‘b’ articles, then Profit/loss percent = $\frac{a - b}{b}$ × 100
If a > b, it is profit percentage; if a < b, it is loss percentage.
Q. If cost price of 20 articles is equal to the selling price of 15 articles, then find the profit/loss percent.
Explanations :
Let C.P. of article = Rs. x
So, C.P. of 20 articles = Rs. 20x
Selling price of 15 articles = Cost price of 20 articles = 20x
So, Selling price of 1 article = Rs. 20x/15 = Rs. 4x/3
Profit percent = [(S.P. – C.P.) / C.P.] × 100 = [(x/3)/x] × 100 = 33.33%
If cost price of ‘a’ articles is equal to the selling price of ‘b’ articles, then
Profit/loss percent = ((a−b))/b × 100 = ((20−15))/15 × 100 = 5/15 × 100 = 33.33%
Let number of articles be 20.
We get the C.P. back by just selling 15 articles. So, 5 articles are left, the S.P. of which is our profit.
So, profit = S.P. of 5 articles AND C.P. = S.P. of 15 articles
Profit percent = (profit/C.P.) × 100 = (S.P. of 5 articles/S.P. of 15 articles) × 100 = (5/15) × 100 = 33.33%
Q. If the selling price of 10 articles is equal to the cost price of 12 articles, then what is the gain percent?
(a) 10% (b) 15% (c) 11% (d) 20%
Explanation:
Let the cost price of each article be Rs. 1
∴ Cost price of 10 articles = Rs. 10
As we know that the selling price of 10 articles is equal to the cost price of 12 articles.
∴ Selling price of 10 articles = Rs. 12
Now, Profit percent = [(S.P - C.P)/(C.P)] × 100 = (12 - 10)/10 × 100 = 20%.
Answer: (d)
Type 6: Mixed and Sold
Q. Setu purchased a certain number of articles at Rs. 10 each, and the same number of another articles for Rs. 16 each. He mixed them together and sold them for Rs. 14 each. What is his gain or loss percent?
(a) Loss of 7.69%
(b) Loss of 7.14%
(c) Gain of 7.69%
(d) Gain of 7.14%
Explanations :
Let us assume that he bought 10 articles of each kind.
∴ Total cost = Rs. (10 × 10 + 16 × 10) = Rs. 260
Total selling price = Rs. (14 × 20) = Rs. 280
∴ Gain percentage = [(S.P. - C.P.)/(C.P.)] × 100 = [(280 - 260)/260] × 100 = (20/260) × 100 = 7.69%
As the number of articles bought are the same in both the cases, the Average C.P. = (10 + 16)/2 = Rs. 13
S.P. = Rs. 14
∴ Gain percentage = [(S.P. - C.P.)/(C.P)] × 100 = [(14 - 13)/(13)] × 100 = (1/13) × 100 = 7.69%
Type 7: Two items sold (CP is same)
Q. Mak buys 2 books for Rs. 600 each, and makes a profit of 15% on one of them and incurs a loss of 15% on the other. What is the overall profit/loss on the entire transaction?
(a) Rs. 36 loss
(b) Rs. 38 loss
(c) Rs 36 profit
(d) None of the above
Explanation:
SP of the first book = 600 + 15% of 600 = 600 + 90 = Rs. 690
SP of the second book = 600 - 15% of 600 = 600 - 90 = Rs. 510
Total SP = 690 + 510 = Rs. 1200
Total CP = 2 x 600 = Rs. 1200
Hence, there is no profit or loss.
Answer: (d)
Type 8: Two items sold (SP is same)
Concept 1: Finding Ratio of C.P.s
When SP of two articles are same:
(a) First one is sold at a profit of x% and second one is sold at a profit of y%.
Then Ratio of $CP_1$ : $CP_2$ = (100 + y) : (100 + x)
(b) First one is sold at a profit of x% and second one is sold at a loss of y%.
Then, Ratio of $CP_1$ : $CP_2$ = (100 – y) : (100 + x)
(c) First one is sold at a loss of x% and second one is sold at a profit of y%.
Then, Ratio of $CP_1$ : $CP_2$ = (100 + y) : (100 – x)
(d) First one is sold at a loss of x% and second one is sold at a loss of y%.
Then, Ratio of $CP_1$ : $CP_2$ = (100 – y) : (100 – x)
Concept 2: Finding net profit/loss percentage
A person sells two objects at the same price, one at a profit/loss of a% and another at a profit/loss of b%.
Then, net profit/loss percentage = $\frac{100 (a + b) + 2ab}{200 + a + b}$
(a) we use - sign for a, b in case of loss and + sign in case of profit.
(b) If the formula evaluates to a negative value, it means there is a net loss and vice-versa.
Q. Two cars were sold as the same price, one at a profit of 10% and other at a loss of 20%. What must be the ratio of their cost prices?
Explanations :
Let S.P. of both cars be Rs. x
Then C.P. of first car = [100/(100 + Profit%)] × SP = [100/(100 + 10%)] × x = Rs. 10x/11
And C.P. of second car = [100/(100 - Loss%)] × SP = [100/(100 - 20%)] × x = Rs. 5x/4
Required ratio = (10x/11) : (5x/4) = 8/11, i.e. 8 : 11
When SP of two articles are same:
First one is sold at a profit of x% and second one is sold at a loss of y%.
Then, Ratio of CP1 : CP2 = (100 – y) : (100 + x) = (100 - 20) : (100 + 10) = 80 : 110 = 8:11
Q. Mak sells 2 books for Rs. 782 each, thereby making a profit of 15% on one of them and incurring a loss of 15% on the other. What is the overall profit/loss on the entire transaction?
(a) Rs. 36 loss
(b) Rs. 38 loss
(c) Rs 36 profit
(d) Rs. 38 profit
Explanation:
Let the cost price of the book on which Mak earns profit be Rs. x and that of the book on which he incurs loss be Rs. y.
Thus, 1.15x = 782 and 0.85y = 782
Solving them, we get: x = 680 and y = 920
Total cost or CP = 680 + 920 = Rs. 1600
Total SP = 2 x 782 = Rs. 1564
Hence, there is a loss of Rs. (1600 — 1564), i.e. Rs. 36
Answer: (a)
Concept 3: Special Case
If SP of two articles are same and one is sold at a profit of x% and another is sold at a loss of x%, then in that case there will always be a loss.
∴ Loss percentage = $\frac{𝑥^2}{100}$ = ($\frac{x}{10})^2%$
Note: We can drive this formula using the net profit/loss percentage formula.
Net profit percentage = $\frac{100 (a + b) + 2ab}{200 + a + b}$ = $\frac{100 (x - x) – 2x^2}{200 + x - x}$ = – $\frac{2𝑥^2}{200}$ = – ($\frac{𝑥}{10})^2%$
Q. Two shirts were sold as the same price, one at a profit of 10% and other at a loss of 10%. What is the overall gain/loss percentage?
Explanations :
Let S.P. of both shirts be Rs. x
Then C.P. of first shirt = $\frac{100}{(100 + Profit \hspace{1ex} Percent)}$ × SP = $\frac{100}{(100 + 10)}$ × x = Rs. $\frac{10x}{11}$
And C.P. of second shirt = $\frac{100}{(100 - Loss \hspace{1ex} Percent)}$ × SP = $\frac{100}{(100 - 10)}$ × x = Rs. $\frac{10x}{9}$
Total C.P. = $\frac{10x}{11}$ + $\frac{10x}{9}$ = Rs. $\frac{200x}{99}$
Total S.P. = Rs. 2x
Loss = $\frac{200x}{99}$ – 2x = Rs. $\frac{2x}{99}$
Required loss percent = $\frac{\frac{2x}{99}}{\frac{200x}{99}}$ × 100 = 1%
If SP of two articles are same and one is sold at a profit of x% and another is sold at a loss of x%, then in that case there will always be a loss.
∴ Loss percentage = $\frac{𝑥^2}{100}$% = ($\frac{𝑥}{10})^2$% = ($\frac{10}{10})^2$% = 1%
Q. Two shirts were purchased as the same price, one at a profit of 10% and other at a loss of 10%. What is the overall gain/loss percentage?
Explanations :
Let C.P. of both shirts be Rs. x
Then S.P. of first shirt = x + 10% of x = Rs. 1.1 x
And S.P. of second shirt = x – 10% of x = Rs. 0.9 x
Total C.P. = Rs, 2x
Total S.P. = 1.1 x + 0.9 x = Rs. 2x
Loss = 2x – 2x = Rs. 0
Required loss percent = $\frac{0}{2x}$ × 100 = 0%
Let C.P. of a shirt be Rs. x
In first transaction we get 10% of Rs. x as profit.
In second transaction we get 10% of Rs. x as loss.
Hence, there is neither gain nor loss, in the overall transaction.
Concept 4: Finding individual C.P., when total C.P. given
A person buys two articles for x. He sells one at a profit of a% and the other at a loss of b%. If each item was sold at the same price, then
C.P. of the article sold in profit = $\frac{x(100 - b)}{200 + a - b}$
C.P. of the article sold in loss = $\frac{x(100 + a)}{200 + a - b}$
Type 9: Two items sold (Profit/Loss is same)
Q. Two items A and B are sold at a profit of 10% and a loss of 20%, respectively. If the amount of profit and loss is the same, then what may be the cost prices of A and B, respectively?
(a) Rs. 1,500 & Rs. 1,000
(b) Rs. 2,000 & Rs. 1,000
(c) Rs. 3,000 & Rs. 2,000
(d) Rs. 3,000 & Rs. 5,000
Explanation:
Let the cost price of items A and B be A and B respectively.
According to the question,
Profit on A = Loss on B
Or 10% of A = 20% of B
Or (A x 10)/100 = (B x 20)/100
Or A/B = 2/1
Or A : B = 2 : 1
Now with the help of options, check the ratio of C.P. of A and B. Only option (b) renders the above ratio.
Option (b): 2000 : 1000 → 2 : 1
Answer: (b)
Type 10: Things sold in parts
Concept 1: Finding overall gain or loss percentage
We already saw how to find overall gain or loss percentage in case there are two items, whose S.P. is the same.
However, sometimes more items are involved and the S.P. is not necessarily the same.
Let’s see how to solve such questions.
Q. Some apples are purchased for Rs. 400. One fourth of the apples are sold at a loss of 10% and the remaining at a gain of 20%. What is the overall gain or loss percentage?
Explanations :
C.P. = Rs. 400
S.P. of one-fourth apples = (1/4) × 400 × (90/100) = Rs. 90
S.P. of the remaining three-fourth apples = (3/4) × 400 × (120/100) = Rs. 360
Total S.P. = 90 + 360 = Rs. 450
So, gain = S.P. – C.P. = 450 – 400 = Rs. 50
Gain percent = (Gain/C.P.) × 100 = (50/400) × 100 = 12.5%
Let overall gain/loss percent be x. Some apples are sold at a loss of 10% and the remaining at a gain of 20%.
(x – 20) : (-10 - x) = (1/4) : (3/4)
Or 3 (x - 20) = -10 – x
Or 3x – 60 = -10 – x
Or x = 12.5%
So, overall gain percentage = 12.5%
Concept 2: Finding total C.P.
If ‘a’ part of an article is sold at x% profit or loss, ‘b’ part at y% profit or loss and ‘c’ part at z% profit or loss and finally there is a profit or loss of p,
Then, cost price of entire articles = $\frac{p×100}{(±ax±by±cz)}$
Concept 3: Finding S.P. of a part so that net gain = 0
If pth part of an item is sold at x% loss, then
Required gain percent in selling rest of the item in order that there is neither gain nor loss in whole transaction = $\frac{px}{(1−p)}$%
Q. One-third of the pencils are sold at a loss of 20%, then at what percent of profit should the rest of them be sold so that there is no overall gain or loss percentage?
Explanations :
Let C.P. = Rs. 300
S.P. of one- third pencils = (1/3) × 300 × (80/100) = Rs. 80
As there is no overall gain or loss percentage, Total S.P. = Total C.P. = Rs. 300
So, S.P. of rest of the two- third pencils = 300 – 80 = Rs. 220
So, Gain = S.P. – C.P. = 220 – 200 = Rs. 20
Required Gain percent = (Gain/C.P.) × 100 = (20/200) × 100 = 10%
Overall gain/loss percent is 0%. Some pencils are sold at a loss of 20% and let the remaining at a gain of x%.
(0 – x) : (-20 - 0) = (1/3) : (2/3)
Or -2x = -20
Or x = 10%
So, required gain percentage = 10%
If pth part of an item is sold at x% loss, then
Required gain percent in selling rest of the item in order that there is neither gain nor loss in whole transaction = $\frac{px}{(1−p)}$%
Here, p = $\frac{1}{3}$
So, required gain percentage = $\frac{px}{(1−p)}$% = $\frac{(\frac{1}{3})20}{1−(\frac{1}{3})}$% = 10%