## 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.**

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 CP | Loss = 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 articles | Loss = 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 SP | Loss = 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 articles | Loss = 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.