## 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.

TrickWhile 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.