# Profit and Loss - Important Formulas, Tricks and Tips

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## 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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