Profit and Loss - Important Formulas, Tricks and Tips

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"Without at least one profit and loss question an entrance exam is incomplete". This statement may emphasis the relevancy of this topic. Except CAT, all the other placements test and competitive entrance exams such as IBPS PO, SNAP, FMS, IIFT, NMAT, CMAT are including multiple questions from this area. Especially in MAT exam this is one of the lucrative areas by means of scoring. Read on to learn important formulas, tricks and tips to solve aptitude questions on profit and loss with the help of examples and detailed explanations.

Most attractive and very favorable fact related to this topic is, the base of profit, loss and discount calculation is nothing but the application of percentage and ratio, in which you already made a strong foundation!!!

The main objectives of this module:
  • Pr-requisites (Percentage applications)
  • Percentage table
  • Successive variations
  • Constant Product rule
  • Basic Terminologies
  • Basic Problems on Profit, Loss and Discount
  • Application of RPV (Ratio, Proportion and Variation) in Profit & Loss
  • Profit or loss percentage with respect to CP or SP

Prerequisites for this topic

The requirement for a better performance in this topic is an in-depth understanding of the concepts and applications of 'Percentages', which was already discussed in the mentioned topic. Here we are going to consider some very important numerical applications of 'Percentages'.

Percentage Table

The table of basic percentage values is a very relevant tool for your better numerical performance of the application of percentages. Hence, you should by heart the below table and understand properly the application of this table.

 123456789101112
1 100 200 300 400 500 600 700 800 900 1000 1100 1200
2 50 100 150 200 250 300 350 400 450 500 550 600
3 33.33 66.66 100 133.33 166.66 200 233.33 266.66 300 333.33 366.66 400
4 25 50 75 100 125 150 175 200 225 250 275 300
5 20 40 60 80 100 120 140 160 180 200 220 240
6 16.66 33.33 50 66.66 83.33 100 116.66 133.33 150 166.66 183.33 200
7 14.28 28.57 42.85 57.14 71.42 85.71 100 114.28 128.57 142.85 157.14 171.42
8 12.5 25 37.5 50 62.5 75 87.5 100 112.5 125 137.5 150
9 11.11 22.22 33.33 44.44 55.55 66.66 77.77 88.88 100 111.11 122.22 133.33
10 10 20 30 40 50 60 70 80 90 100 110 120
11 9.09 18.18 27.27 36.36 45.45 54.54 63.63 72.72 81.81 90.9 100 109.09
12 8.33 16.66 25 33.33 41.66 50 58.33 66.66 75 83.33 91.66 100

Way to remember the relations between values.

Root value (base) is 1/2 = 0.5 = 50%

2/2 = 2 * 50% = 100%

3/2 = 3 * 50% = 150% and so on

Root Value is 1/3 = 0.333.... ≅ 33.33%

2/3 = 2 * 33.33% = 66.66 %

3/3 = 99.99% ( in calculation 99.99% = 100%)

And so on.

Root Value is 1/4 = 0.25 = 25%

2/4 = 2 * 25% = 50%

3/4 = 3 * 25% = 75% and so on.

Root value is 1/5 = 0.2 = 20%

2/5 = 2 * 20% = 40%

3/5 = 3 * 20% = 60% and so on.

Root value is 1/6 = 0.1666.... ≅ 16.66%

2/6 = 1/3 = 33.33%

3/6 = 1/2 = 50%

4/6 = 2/3 = 66.66%

5/6 = 5 * 16.66% = 83.33% and so on.

Root value is 1/7 = 0.142857 142857... ≅ 14.28%

1/7 is an interesting recurring decimal. A group of 6 digits '142857' is repeating infinitely and the repetition start immediately after the decimal point. For catching the further multiples of this root value, its require to by-heart this order of decimals such as 14-28-57 [twice of 7 , 4 times of 7 , (8 times of 7 + 1)]

 

In the above circular arrangement of the decimal order of 1/7, we are approaching it is in a clock-wise direction, and the values in ascending order is 1, 2, 4, 5, 7 and 8.

Pattern of these decimal values are in a cyclic order and starting values are in ascending order can be written in the following manner;

0.142857 142857 .... = 1/7 ≅ 14.28%

0.285714 285714 .... = 2/7 ≅ 28.57%

0.428571 428571 ..... = 3/7 ≅ 42.85%

0.571428 571428 ..... = 4/7 ≅ 57.14%

0.714285 714285 .... = 5/7 ≅ 71.42%

0.857142 857142 .... = 6/7 ≅ 85.71%

Root value is 1/8 = 0.125 = 12.5%

2/8 = 1/4 = 25%

3/8 = 3 * 12.5% = 37.5%

4/8 = 1/2 = 50%

5/8 = 5 * 12.5% = 62.5%

6/8 = 3/4 = 75%

7/8 = 7 * 12.5 = 87.5%

Root value is 1/9 = 0.1111..... 11.11%

2/9 = 2 * 11.11% = 22.22%

3/9 = 3 * 11.11% = 33.33%

4/9 = 44.44% and so on.

Root value is 1/10 = 0.1 = 10%

2/10 = 20%

3/10 = 30% and so on.

Root value is 1/11 = 0.09 09.... 9.09%

2/11 = 2 * 9.09% = 18.18% (18 is the 9th multiple of 2)

3/11 = 3 * 9.09% = 27.27% (27 is the 9th multiple of 3)

4/11 = 4 * 9.09% = 36.36% (36 is the 9th multiple of 4)

5/11 = 5 * 9.09% = 45.45% (45 is the 9th multiple of 5)

And so on.

Root value is 1/12 = 0.083333.... ≅ 8.33%

2/12 = 1/6 = 16.66%

3/12 = 1/4 = 25%

4/12 = 1/3 = 33.33%

5/12 = 5 * 8.33% = 41.66%

6/12 = 1/2 = 50%

7/12 = 7 * 8.33% = 58.33%

8/12 = 2/3 = 66.66%

9/12 = 3/4 = 75%

10/12 = 5/6 = 83.33%

11/12 = 11 * 8.33% = 91.66%

As your mathematical interest, you can expand this table to higher quantities. For the normal level of percentage calculations this table values are enough.

Successive percentage variation.

Suppose a quantity first increased by 20% and successively increased by another 10%, then we can find the net percentage change happened to the quantity after the two successive increments.

Let the initial quantity be 100, then;

100 → 20% ↑ → 120 → 10% ↑ → 132

100 became 132 after two successive increments of 20% and 10%. Here the order of percentage increments doesn't matter; means if the quantity first increased by 10% then 20% also give the same result.

The above concept we can illustrate algebraically in the following manner;

100 → x% ↑ → 100 [1+ x / 100] → y% ↓ → 100 [1 + x / 100][1 + y / 100]=100 [1 + x + y + xy / 100]

Therefore the resultant is [x + y + xy/100] greater than the initial quantity.

Result 1:
If a quantity first increased by x% and successively increased by y%, then the effective percentage of increase on the initial quantity is [x + y + xy/100]
Similarly;
Result 2:
First an increment of x% then a successive decrease of y% then the percentage of the net variation (increase/decrease) is [x - y - xy/100]
If the resultant of the above operation is positive, then the net effect is increment.
If the result is negative, then the effect is decrease.
If the result is '0', then there is no change.
Result 3:
First decreased by x% then a successive decrease of y%, then the net variation is [ - x - y + xy/100].

Successive increase and decrease by same percentage value.

If a quantity first increased/decreased by x% and successively decreased/increased by again x%, then the net variation is always x2 / 100 decrease.

Successive percentage variation - three times.

If a quantity successively increased by x%, y% and z%, then the net variation happened the quantity can be expressed in the following manner;

x + y + z + (xy + yz + xz)/100 + xyz/100

Example:

Price of a product is successively increased by 10%, 20% and 30% in three consecutive years. Find the total percentage variation happen the price of the product after three successive variations.

Solution:

Method1: Applying formula.

As per the formula, the net variation is

10 + 20 + 30 + ((10 * 20) + (20 * 30) + (30 * 10))/100 + (10 * 20 * 30)/10000

= (60 + 11 + 0.6)%

= 71.6%.

Hence the price is increased by a total of 71.6% from its initial price.

Method 2: Split the formula as two - two variations.

Percentage change after the first two successive variations = [10 + 20 + (10 * 20)/100] = 32

Include the third successive variation with this percentage of variation, then the resultant variation is

[32 + 30 + (32 * 30)/100 = (62 + 9.6) = 71.6]

Method 3: Base 100 approach.

Let us assume the initial price of the product is Rs.100.

Then; 100 → 10% ↑ → 110 → 20% ↑ → 132 → 30% ↑ → 171.6

After the three successive variations initial 100 amounted to 171.6, therefore the percentage of increment is 71.6%.

General approach towards three successive variation:
When applying the operator for the three successive variations, enter the percentage of increment as a positive value and the percentage of decrease as a negative value. If the output of the operator is a positive quantity, then the net effect is an increment, if the output is a negative quantity then the net effect is a decrease and if the output is zero, it means there was no change.
Example:
If a quantity first increase by 20%, follow with a 10% decrease and then 15% increase successively. Then the net change in percentage is
20+(-10) + 15 + [20 * (-10)] + [(-10) * 15] + [15 * 20]100+ 20 * (-10) * 1510000 = 25 - 50100 - 30001000 = 24.2%.
Here the output is a positive quantity, then the net effect is 24.2% increment.

Constant Product Rule

First of all consider the situation when or where the constant product rule is applicable.

If the distance between two cities P and Q is 120 km and a car X is travel from P to Q at a constant speed of 30 kmph, then it will take 4 hours to cover the mentioned distance. If the car will increase its speed to 40kmph, then it can cover the distance in 3 hours.

Here speed and the travelling time are inversely proportional- means travelling time will decrease as per the corresponding increment in speed. In this aspect the distance is a constant product of the factors speed and time. In such a situation the percentage increment/decrease in speed will make a corresponding percentage decrease/increase in its other part, time. This percentage relation we can catch with the help of Constant Product Rule.

This rule is one of key strategic application in most of the percentage related problems. So the aspirant should be skillful in the effective application of the constant product rule.

Here the rule explained below;

Method I: Percentage approach.

If the product of two inversely proportional quantities X and Y is a constant, then;

Variation in XResultant effect in Y
r % increase (100 * r / 100 + r) decrease
r % decrease (100 * r / 100 - r) decrease

Example 1:

If the length of a rectangle is increased by 20% then how much percentage decrease should be in its breadth to keep the same area?

Solution:

Area = length * breadth.

Area is a constant product of the factor values length and breadth.

20% increment in length → (20 * 100) / (100 + 20) % decrease in breadth.

Hence the breadth should decrease by 16.66% to keep the same area.

Example 2:

If the speed of a train is decreased by 25%, then how much percentage more time is required to cover a constant distance?

Solution:

25% decrease in speed → (25 * 100) / (100 - 25) % increase in time.

Hence the train will take 33.33% more time to cover the same constant distance.

Method II: Fractional approach.

In this fractional approach of the constant product rule, we are expressing the percentage values in its corresponding fractional form. The calculation process with the percentage-equivalent-fractions will support the student to do the mental calculations so easily and quickly.

If the product of two inversely proportional quantities X and Y is a constant, then;

Variation in XResultant variation in Y
a/b increase a/(b+a) decrease
a/b decrease a/(b-a) increase

Example 1:

Travelling from city A to B a car keep a constant speed of 30 kmph and while the return journey from B to A, it increased the speed to 40 kmph. So the time taken for the return journey is how much percentage more or less than that for the first half of the trip?

Solution:

Speed for A to B = 30 kmph.

Speed for B to A = 40 kmph.

Increase in speed = 10 kmph.

Fraction of increment in speed = 10/30 = 1/3

As per the rule; 1/3 increment in speed → 1/(3+1) decrease in time.

Therefore the time consumption is decreased by 1/4 or 25%.

The above discussed three main pre-requisites are frequently applicable in most of the Profit and Loss questions. Hence, you should prepare well on the above three Percentage applications.

Basic Terminologies.

Cost Price (CP) : This is the real worth of a product. From customer's end CP is the amount he spent for

purchasing a product.

Selling Price (SP): This is the revenue generated from a product while selling it. From the merchant's end

SP is the amount he received from the customer while selling a product to customer.

Profit (P)/ Gain: While selling an article if the selling price is greater than cost price then the seller

receives a profit. In this sense Profit = SP - CP.

Loss (L): if the selling price is lesser than the Cost Price then the seller incurred a loss in the transaction.

Hence Loss = CP- SP.

Profit Percentage: This is an expression of the profit amount in terms of Cost Price.

Profit % = Profit/Cost Price * 100 or (SP - CP)/Cost Price * 100 .

Loss Percentage: Here loss is expressed in terms of Cost Price.

Loss % = Loss % = Loss/Cost Price * 100 or (CP - SP)/Cost Price * 100.

Marked Price/ Listed Price: Marked price is the price printed on a product. A customer can see this as the value/price for the product.

Discount: If the merchant is selling the product for a price which is lesser than the Marked Price, then he offered a discount to the customer on that particular product.

Discount = Marked Price - Selling Price.

Discount percentage: Here discount is expressed as the percentage of Marked Price.

Discount Percentage = (Marked Price - Selling Price)/Marked Price * 100 = Discount/Marked Price * 100.

Illustration:

If a merchant purchased a product by Rs.100, and he mark up it to Rs.150 finally sold it by Rs.120, then:

CPSPMP
Rs.100 Rs.120 Rs.150
  • Profit = SP - CP = Rs.120 - Rs.100 = Rs. 20.
  • Profit Percentage = Profit/Cost Price * 100 = 20/100 * 100 = 20%
  • Mark up by Rs.50 and Mark up to Rs.150.
  • Discount = MP - SP = Rs.150 - Rs. 120 = Rs.30.
  • Discount Percentage = Discount/Marked Price * 100 = 30/150 * 100 = 20%

Application of RPV (Ratio, Proportion and Variation) in Profit & Loss

When we are considering the questions from this topic, we can see a plethora of questions have a ratio wise approach. Here we can categorize such types of questions and can form some tips to solve those kinds of problems. Understanding the below mentioned approaches towards such problems, will lead a student to do the effective mental calculations (without a rough work) in those problems and make them able to reach a quick conclusion easily.

 

Type I: Basic CP-SP ratio.
Example: 1.
The cost price of 10 articles is equal to the Selling price of 8 articles. Find the profit or loss percentage.
Explanation:
Here the given data is the rate at which the product purchased and sold. It doesn't mean the purchasing quantity is exactly 10 and the selling quantity is exactly 8 and it means the merchant purchased a certain number of mentioned products in the given rate and sold them all at the rate mentioned in the question.
First of all just frame a relation, i.e. 10 CP = 8 SP
(In this type of a relation, if the coefficient of CP is greater than the coefficient of SP, then there is a profit. If the coefficient of CP is equal to the coefficient of SP, then there is neither profit nor gain. If the coefficient of Cp is less than the coefficient of the SP, then there is a loss.)
Frame a ratio between CP and SP; CP/SP= 8/10 = 45
Assume; CP = Rs.4
SP = Rs.5

Profit percentage = (5 - 4)/4 * 100 = 25%

Example: 2

A trader is selling rice at the cost price but weighing 100 grams in every 1 kilogram. Find the percentage of profit from the sale.

  • A)10%  B) 8.8%  C) 11.11%  D)25%

Solution

For selling 900 grams he is generating the revenue which is equal to the CP of 1000 grams.

900 SP = 1000 CP

9 SP = 10 CP

SP/ CP = 10/ 9

Let SP = Rs.10

CP = Rs.9

Profit = 10 - 9 =Rs.1

Profit percentage = 1/9 * 100 = 11.11%.

Ans: C.

Alternate Method (Plug in)

Assume the cost price of 1 kg of rice = Rs.100.

Therefore the CP for 900 grams = Rs.90

For making the transaction of 1 kg of rice, he needs to spend Rs.90 only but the revenue from the transaction is Rs.100.

Profit = Rs.10.

Profit percentage = 10/90 * 100 = 11.11%.

Example: 3

While selling 1 kg of sugar, a shop keeper weighing only 0.9 kg and he professes to sell sugar at a loss of 8%. Find his actual loss or gain percentage from the sale of sugar.

  • A)3.33% profit  B) 3.33% loss  C) 2.22% profit  D) 2.22% loss.

Solution

Let the cost price of 1 kg of sugar = Rs.100.

Loss = 8%

Selling price of 1 kg of sugar = 100-8 = Rs.92.

But for selling 1 kg of sugar, the shop keeper is weighing only 900 grams. His cost per 1 kg is Rs.90 only.

CP = Rs.90

SP = Rs.92

Profit = 92 - 90 = Rs.2

Profit percentage = 2/90 * 100 = 2.22% profit

Ans: C.

Type II: Profit/loss in terms of CP.

Illustrated example:

While selling 10 apples;

  • Merchant gain CP of 2 apples. Find profit percentage.
  • Merchant lost CP of 2 apples. Find loss percentage.

Consider both situations together.

Profit = 2 CPLoss = 2 CP
SP = CP + Profit
10 SP = 10 CP + 2 CP
10 SP = 12 CP
SP : CP = 12 : 10
SP = 12
CP = 10
Profit = 12 - 10 = 2
Profit Percentage = 2/10 * 100 = 20%
CP = SP + Loss
10 CP = 10 SP + 2 CP
8 CP = 10 SP
SP : CP = 8 : 10
SP = 8
CP = 10
Loss = 10- 8 = 2
Loss percentage = 2/10 * 100 = 20%

Generalization of the above concepts.

While selling 'x' number of articles;

Profit = CP of 'y' number of articlesLoss = CP of 'y' number of articles
Profit percentage = y/x * 100 % Loss percentage = y/x * 100 %

Example: 1

Shop keeper gained the cost price of 3 kg of rice in a sale of 11 kg of rice. Find his profit percentage.

  • A)20%  B)30%  C)27.27%  D)33.33%

Solution:

Profit percentage = 3/11 * 100 = 27.27%

Ans : C

Example: 2

In a sale of 12 machines the sales person received revenue which is equal to the cost price of 10 machines. Find his loss percentage.

  • A)16.66%  B)22.22%  C)8.33  D)18.18%

Solution:

Loss = 2 CP

Loss percentage = 2/12 * 100 = 16.66%

Ans : A.

Type III: Profit/loss in terms of SP.

Illustrated example:

While selling 10 books;

  • Shop keeper gained SP of 2 books. Find his profit percentage.
  • Shop keeper lost SP of 2 books. Find his loss percentage.
Profit = 2 SPLoss = 2 SP
SP = CP + profit
10 SP = 10 CP + 2 SP
8 SP = 10 CP
SP : CP = 10 : 8
Sp = 10
Cp = 8
Profit = 10 - 8 = 2
Profit percentage = 2/8 * 100% = 25%
CP = SP + Loss
10 CP = 10 SP + 2 SP
10 CP = 12 SP
SP : CP = 10 : 12
SP = 10
CP = 12
Loss = 12- 10 = 2
Loss percentage = 2/12 * 100 = 16.66%

Generalization of the above concepts.

While selling 'x' number of articles;

Profit = SP of 'y' number of articlesLoss = SP of 'y' number of articles
Profit percentage = y/(x-y) * 100 % Loss percentage = y/(x+y) * 100 %

Example: 1

In a sale of 12 dozen pencils the shop keeper gained an amount which is equal to the Selling price of 3 dozen pencils. Find his profit percentage.

  • A)25%  B)33.33%  C)8.33%  D)10%

Solution:

Sold quantity = 12 dozen..... ( x= 12)

Profit = SP of 3 dozen.... (y = 3)

Profit percentage = y/(x-y) * 100% = 3/(12-3) * 100 = 33.33%.

Ans : B.

Example: 2

In a sale of refrigerators an appliance shop reported a loss of certain amount on a particular day due to transportation costs and maintenance cost for some of the damaged items. On that day if they sold a total of 25 refrigerators and the loss amount is equal to the selling price of 5 refrigerators. Find the loss percentage reported on the sales of refrigerators on that particular day.

  • A)12.5%  B)14.4%  C)15.6%  D)16.6%

Solution:

Sold quantity = 25

Loss = 5 SP

Loss percentage = 5/(25 + 5) * 100 = 16.66%

Ans: D.


Profit / Loss percentage on the base of SP/CP.

Type I: Percentage wise approach.

Rule 1: If profit is r% of Cost price then the profit is r/(100 + r) * 100 % of Selling price.
Rule 2: If profit is r % of Selling price then the profit is r/(100 - r) * 100 % of Cost price.
Rule 3: If loss is r% of Cost price then the loss is r/(100- r) * 100 % of Selling price.

Rule 4: if loss is r % of Selling price then the loss is r/(100 + r) * 100 % of Cost price.

Tips for remembering the rules:
Profit percentage on Selling price is lesser than the profit percentage on cost price, because if there is a profit then the selling price is greater than the cost price. From this basic concept you can easily reach a conclusion that whether the denominator of the calculation part is 100 + r or 100 - r. for getting the largest percentage value keep the denominator as comparatively less, i.e. 100 - r and for least percentage value keep the denominator as 100 + r.

 

Example: 1

If the profit is 25% of cost price then finds the profit is how much percentage of the selling price.

Solution:

As per the rule 1, 25/(100+25) * 100 = 20%.

Example:2

If the profit is 25% of the selling price then finds the profit is what percentage of cost price.

Solution:

As per rule 2, 25/(100 -25) * 100 = 33.33%.

Example: 3

If the loss is 20% of cost price then find the loss is what percentage of selling price.

Solution:

As per rule 3, 20/(100 - 20) * 100 = 25%

Example: 4

If the loss is 20% of the selling price then find the loss is what percentage of the cost price.

Solution:

As per rule 4, 20/(100 + 20) * 100 = 16.66%

Type II: Fractional approach

Rule 1: if the profit is of cost price than the profit is x/(y+x) of selling price.

Rule 2: if the profit is of selling price then the profit is x/(y- x) of the cost price.

Rule 3: if the loss is of cost price then the loss is x/(y- x) of selling price.

Rule 4: if the loss is of selling price then the loss is x/(y+x) of cost price.

Example: 1

If the profit is 37.5% of the cost price, find the profit is what percentage of selling price.

Solution:

37.5% = 3/8

As per rule 1, 3/(8+3) = 3/11 = 27.27%

Example: 2

If the profit is 28.57% of the selling price, find the profit is what percentage of the cost price.

Solution:

28.57% = 2/7

As per Rule 2, 2/(7-2) = 2/5 = 40%

Example: 3

If the loss is 22.22% of the cost price, then find the loss is what percentage of the selling price.

Solution:

22.22% = 2/9

As per rule 3, 2/(9-2) = 2/7 = 28.57%

Example: 4

If the loss is 44.44% of the selling price, find the loss is what percentage of the cost price.

Solution:

44.44% = 4/9

As per rule 4, 4/(9-4) = 4/5 = 80%

In the first module of Profit, Loss and Discount, we mainly discussed the basic types of problems. So, here in this second module, we are going to deal with some important extended applications of the concept. In this module, we will discuss the frequently asked typical formats of the questions from this topic.

Main objectives of this module:

  • Sale of multiple articles of equal cost prices
  • Sale of multiple articles at equal selling prices
  • Percentage of profit or loss for neutralizing the net effect

Profit / loss percentage of a transaction of two articles of same CP.

It is a situation of selling two products of equal cost prices at different selling prices. The sale of each product will make a profit or loss. The main requirement is to find out the percentage/ amount of the profit or loss from the entire transaction. The same logic applied in the transactions of such two products can be extended in a large scale such as the transactions of multiple products.

The following examples will give a clear idea about the nature of questions from this area.

Examples:

Athul bought two mobile phones of same cost prices and sold one among them at a profit of 10% and the other at a loss of 10%. Find his profit or loss in the entire transaction.

  • 5% profit
  • 5% loss
  • No profit, no loss
  • Can't be determined.

Solution:

Let the cost price of each item is Rs.100

Total cost price for two items = Rs.200

SP of first item at a profit of 10% = Rs.110.

SP of the second item at a loss of 10% = Rs.90.

Total SP = 110 = 90 = Rs.200.

Total SP = Total CP

Therefore, there is neither profit nor loss.

Ans: C

From the above question; if Athul sold the first mobile at a profit of 20% and the second at a loss of 10%. Find the net percentage of profit or loss from the entire transaction.

  • 10% profit
  • 5%profit
  • 2.5%loss
  • 8%loss

Solution:

CP for one item = Rs.100

CP for two items = Rs. 200

SP of first item at a profit of 20% = Rs.120.

SP of second item at a loss of 10% = Rs.90.

Total SP = 120 + 90 = Rs.210.

Profit = 210 - 200 = Rs.10

Profit percentage =

Ans : B.

From question 1, if he sold the first mobile phone at a profit of 10% and the second at a loss of 15%, find his profit or loss from the entire transaction.

  • 5% loss
  • 5% profit
  • 2.5% profit
  • 2.5% loss.

Solution:

CP for one item = Rs.100

Total CP for two items = Rs.200.

SP of the first item at a profit of 10% = Rs.110.

Sp of the second item at a loss of 15% = Rs.85.

Total SP for two items together = 110 + 85 = Rs.195

Loss = Rs.5

Loss percentage = 5/200 * 100 = 2.5%

Ans : D.

Result:
Cost prices of two articles are same and one among them sold at a profit of 'x%' and the other sold at a loss of 'y%'.
  • If x > y, then there is a profit and the profit percentage is
  • If x < y, then there is a loss and the loss percentage is

Cost prices of two articles are same and one among them sold at a profit of 'x%' and the other sold at a profit of 'y%' then the net effect of the entire transaction is profit.

Cost prices of two articles are same and one among them sold at a loss of 'x%' and the other sold at a loss of 'y%', then the net effect of the entire transaction is loss.

Transaction of three articles of equal CPs

The method applied for the transaction of two articles is applicable in the case of the transaction of three articles. You will get a clear understanding through the following example.

Example: Three articles of equal cost prices sold respectively at 10% profit, 8% loss and 13% profit. Find the net percentage of profit/ loss from the entire transaction?

A. Neither profit nor loss

B. 5 % profit

C. 2.5 % loss

D. 10.5 % profit

Solution:

Let the CP of each product = Rs. 100

Total CP of three products = Rs. 300

SP of the product which made 10% profit = Rs. 110

SP of the product which made 8% loss = Rs. 92

SP of the product which made 13% profit = Rs. 113

Total Sp of three products = 110 + 92 + 113 = Rs. 315

Net profit = Rs. 15

Profit percentage = 15/300 * 100 = 5%

Ans: B

Alternate method:

Net effect of the entire transaction is always the average of the individual variations.

i.e. Net effect of 10% profit, 8% loss and 13% profit = (10-8+13)/3=5

Answer is positive 5, hence the net effect is 5% profit.

Trick
While selling multiple articles of equal cost prices at different selling prices, the net percentage of variation (profit/loss)the entire transaction is always the average of the individual variations.
Remember...!!!!
While entering the individual variations for finding the net variation, individual profits should enter as positive quantities and individual losses should enter as negative quantities.
If the resultant is positive, then the net variation is profit.
If the resultant is zero, then there is no variation, is neither profit nor loss.
If the resultant is negative, then the net variation is loss.

Example:

Ram bought three identical toys at Rs.160 each and sold them all at a net profit of 2.5%. If he sold the first and second toys at 6.25% profit and 3.75% loss, what is the selling price of the third toy?

A. Rs. 168

B. Rs. 170

C. Rs. 152

D. Rs. 160

Solution:

Net effect of entire transaction = 2.5% profit

i.e. (6.25 - 3.75 + x)/3 = 2.5

6.25 - 3.75 + x = 7.5

2.5 + x = 7.5

x = 5

i.e. The third toy sold at a profit of 5% profit.

Hence the selling price of the third toy = 160 + 5% of 160 = Rs. 168

Ans: A

Findings.
While selling two articles of equal cost prices, one at x% of profit and the other at y% of loss.

If x = y, then there is no profit or loss.

If x > y, then there is a profit.

If x < y, then there is a loss.

Example: Data Sufficiency Question

Sona bought two mobile phones at Rs. 17,000 each, then sold one among them at a profit of p% and the other sold at a loss of q%. Is the net of the entire transaction made a profit to Sona?

Statement I: p ≥ q

Statement II: p < q

Solution:

As per the first statement, the profit percentage is more or equal to the loss percentage. Hence net effect of the entire transaction may be a profit or there is no profit or no loss.

Therefore the statement I alone is not sufficient.

As per the second statement, the profit percentage is less than the loss percentage; hence there should be a loss in the entire transaction. i.e. The transaction won't make a profit to Sona.

Therefore statement II alone is sufficient.


Profit / loss percentage of a transaction of multiple articles of same SP

This is about the transaction of multiple products at the same selling prices. Some among the products made profit and some of them made loss, ultimately we need to find out the net percentage/amount of profit or loss from the entire transaction. This is a frequently asked type of question. Irrespective of any formulae wise approach, it is better to learn the method of effective assumption to tackle such questions.

 

From the following illustrated examples, you can easily catch the logical method assumption.
Example:
Rahul sold two bags, both at the same price.
  • On one bag he gained 10% and another at 10% loss. Find his profit or loss in the entire transaction.
  • 1% loss
  • 2.5 % profit
  • 3% loss
  • No profit, no loss.
Solution:
For assuming the suitable quantities as the respective cost prices for making the same selling prices in both the situations, you can follow the following method.
On the first bag he made a profit of 10%, so just assume a quantity which is 10 more than 100 as the CP for the second bag. i.e. 110.
On the second bag he faced a loss of 10%, so assume a quantity which is 10 less than 100 as the CP for the first bag. i.e. 90.
We can arrange the assumed values in the following manner;
ItemsCPVariationSP
Bag 1 90 10% profit 99
Bag 2 110 10% loss 99
Total 200   198
Total CP = Rs.200
Total SP = Rs.198.
Loss = CP - SP = 200 - 198 = Rs.2
Loss percentage = 2/200 * 100 = 1%

Ans: A.

Result:
While selling two articles at the same selling prices, if one of the items gained 'r' % and the other incurred a loss of 'r' %, then the net effect of the entire transaction is loss.
  • On one bag he gained 20% and another at 10% loss. Find his profit or loss in the entire transaction.

Solution:

First bag → 20% profit → assume the CP of the second bag as 20 more than 100. i.e. 120.

Second bag → 10% loss → assume the CP of the first bag as 10 less than 100. i.e. 90.

ItemsCPVariationSP
Bag 1 90 20 % profit 108
Bag 2 120 10 % loss 108
Total 210   216

Total CP = 210

Total SP = 216

Profit = 6

Profit percentage = 6/210 * 100 % ≅ 2.857

Net effect is 2.86 % profit.

Note: If the percentage of profit and the percentage of loss are expressed in two different values, the above mentioned assumption method is the easiest and the effective method to find the answer.

Result:
While selling two articles at equal selling prices, if one among them sold at a profit of 'P%' and the other sold at a loss of 'L%', then the net percentage of profit or loss from the entire transaction can be found by using the following formula.
((P-L) * 100 - 2 PL)/(200 + (P - L)) %

If the output of the above formula is a positive value, then the net effect of the entire transaction is a profit.

If the output is negative, then the net effect is a loss.

If the output is zero, then the net effect is 'neither profit nor loss.

Findings:
While selling two articles at equal Selling prices, one at a profit of p% and the other at a loss of q%;
If p ≤ q, then there should be loss in the entire transaction.
If p > q, then there is no assurance about the net effect of the entire transaction, that may be a profit, a loss or a neutral result (neither profit nor loss).

Example: Data Sufficiency question

Abhijith sold two radios at Rs. 2500 each then he made a profit of m% from one among the radio and incurred a loss of n% from the sale of the other radio. Whether the sale of radios made a profit to him?

Statement I: m ≤ n

Statement II: m > n

Solution:

As per the statement I, the profit percentage is less than or equal to the loss percentage. Hence there should be a loss.

Therefore the statement I alone is sufficient.

As per the second statement, the profit percentage is more than the loss percentage. Then there are three possibilities for the net effects of the entire transactions. i.e. The net effect of the entire transaction may be profit, a loss or neither profit nor loss.

Therefore the statement II alone is not sufficient.

Example:

While selling two different articles at equal Selling Prices, Raji made a loss of 22.22% from the sale of the first article. At what percentage of profit she should sell the second article to never make a profit or loss to her from the entire transaction?

A. 22.22%

B. 28.57%

C. 40%

D. 33.33%

Solution:

Let the CP of the first article = Rs. 9 {this article sold at a loss of 22.22% = 2/9, therefore it is better to assume the CP of this product as 9, then we can easily find out the corresponding selling prices}

Therefore SP of the first article = 9 - 2/9 (9) = Rs. 7

i.e. Selling prices of each articles = Rs. 7 {SP are same}

Hence the total selling price = Rs. 14

As the transaction didn't make a profit or loss, the total CP should be equal to the total SP.

So, the CP of the second article = Rs. 5 {because, 9 + 5 = 14}

And the SP of the second article = Rs. 7

Hence the required profit percentage = (7-5)/5 * 100 = 40%

Ans: C

Result:
While selling two articles of equal selling prices, one among them made a loss of r % (or ), then the required percentage of profit in the sale of the second article to avoid any profit or loss from the entire transaction is r/(100-2r) % or x/(y-2x)
While selling two articles of equal selling prices, one among them made a profit of r % (or ), then the required percentage of loss in the sale of the second article to avoid any profit or loss from the entire transaction is r/(100 + 2r) % or x/(y + 2x)

Examples for the applications of the above results:

1. In the transaction of two articles at the same selling prices, if the first article made a profit of 28.57% then what should be percentage of loss incurred from the second article to avoid any profit or loss from the entire transaction?

Solution:

Profit made from first article = 2/7

Hence the required loss from the second article = 2/(7 + 2(2)) = 2/11 18.18%

2. Two articles of equal selling prices made 12.5% loss and 'r%' profit respectively, hence there is neither profit nor loss from the entire transaction. What is the approximate value of 'r'?

Solution:

Loss incurred from the first article = 1/8

Hence the required profit from the second article = 1/(8- 2(1)) = 1/6 = 16.66%

Therefore the value of r = 16.66

Selling three articles at the same selling prices

Example: (Application of constant product rule)

Anand sold three articles at equal selling prices. The first article made a profit of 20%, second article made a loss of 20% and the third article made a profit of 14.28%. Find his net percentage of profit or loss from the entire transaction.

A. 1.4% profit

B. 25.72 % profit

C. 2.73% loss

D. Neither loss nor profit

Solution:

For solving this question, you should learn the method of effective assumption through the application of constant product rule.

Here selling prices are equal; hence you should find a computationally convenient value aas the selling price of each product.

Go through the following method to get a suitable value as the SP of each product.

Product 1 → SP is 1/5 more than CP → CP is 1/6 less than SP (Hence Sp should be divisible by 6)

Product 2 → SP is 1/5 less than CP → CP is 1/4 more than SP (Hence SP should be divisible by 4)

Product 3 → SP is 1/7 more than CP → CP is 1/8 less than SP (Hence SP should be divisible by 8)

From the above findings, we can fix the SP of each product as the LCM of 6, 4 and 8.

LCM of 6, 4 and 8 = 24

i.e. Let us assume the SP of each product = Rs. 24

Therefore the total SP of three products = Rs. 72

CP of product 1 = 24 - 1/6 (24) = Rs. 20

CP of product 2 = 24 + 1/4 (24) = Rs. 30

CP of product 3 = 24 - 1/8 (24) = Rs. 21

Total CP of three products = Rs. 71

Profit percentage = (72-71)/71 * 100 ≅ 1.408 %

Ans: A

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