"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!!!

- 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.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|

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.**1**42857 142857 .... = 1/7 ≅ 14.28%

0.**2**85714 285714 .... = 2/7 ≅ **28.57%**

0.**4**28571 428571 ..... = 3/7 ≅ **42.85%**

0.**5**71428 571428 ..... = 4/7 ≅ ** 57.14%**

0.**7**14285 714285 .... = 5/7 ≅ ** 71.42%**

0.**8**57142 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 9^{th} multiple of 2)

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

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

5/11 = 5 * 9.09% = 45.45% (45 is the 9^{th} 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 x^{2} / 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. |