# CAT Quant Percentages – Important Formulas and Concepts

## Meaning and Definition

- Percent literally means “for every 100”.
- It is one of the simplest tool for comparison of data.
- Any percentage can be expressed as a decimal fraction by dividing the percentage figure by 100 and conversely, any decimal fraction can be converted to percentage by multiplying it by 100.

## Importance of Base / Denominator

- The most crucial aspect of percentage is the denominator which in other words is also called the base value of the percentage.

## Percentage Change

- Also known as Percentage Increase / Decrease of a quantity, it is the ratio expressed in percentage of the actual increase or decrease or decrease of the quantity to the original amount of the quantity. $$Percentage \ Change = \frac{Absolute \ Value \ Change}{Original \ Quantity}*100$$ $$Percentage \ Increase \ / \ Decrease = \frac{Absolute \ Value \ Increase \ / \ Decrease}{Original \ Quantity} * 100$$
**Note –**The base used for the sake of Percentage Change calculations is always the original quantity unless otherwise stated.- In general, if the percentage increase is p%, then the new value is – $$New \ Value = {\LARGE [} \frac{P}{100} + 1 {\LARGE ]}$$
- In general, if the new value is “k” times the old value, then the percentage increase is – $$Percentage \ Increase = [k – 1]*100$$
**Three different cases of Percentage Increase / Decrease –**- If the value of an item goes up/down by x%, the percentage reduction/increment to be now made to bring it back to the original level is – $${\LARGE [} \frac{100x}{100 \pm x}{\LARGE ]} {\%}$$
- If A is x% more/less then B, then B is what percent less/more than A is calculated as – $${\LARGE [} \frac{100x}{100 \pm x}{\LARGE ]} {\%}$$
- IF the price of an item goes up/down by x%, then the quantity consumed should be reduced/increased by what percentage so that the total expenditure remains the same – $${\LARGE [} \frac{100x}{100 \pm x}{\LARGE ]} {\%}$$

## Successive Change

- If there are successive increases / decrease of p%, q% and r% in three stages, the effective percentage increase is – $${\LARGE [} {\LARGE (} \frac{100+p}{100} {\LARGE )} {\LARGE (} \frac{100+q}{100} {\LARGE )} {\LARGE (} \frac{100+r}{100}{\LARGE )} – 1 {\LARGE ]}$$ If one or more of p, q, and r are decrease percentage figures, then it will be taken as a negative figure and not as a positive figure.
- Another method , If 1st change = a% and 2nd Change = b% then, $$Overall \ \% \ Change = a + b + \frac{ab}{100}$$

## Difference between Percentage Point Change and Percentage Change

- The difference is illustrated through an example –
**Percentage Point Change**– 30% – 25% = 5% points**Percentage Change**– [(30 – 25) / 25 ]*100 = 20%

## Multiplying Factor

- We use multiplying Factor whenever there is a % percent increase / decrease – $$MF \ [Multiplying \ Factor] = \frac{100 \pm r}{100}$$ where, r = % change [% increase(+) % decrease(+)]
- $$\% Change \ (r) = \frac{Final \ Value \ (FV) – Initial \ Value \ (IV)}{Initial \ Value \ (IV)} * 100$$
- $$FV = IV * MF \\ Final \ Value = Initial \ Value * Multiplying \ Factor$$

## Base

- $$A \ is \ what \ \% \ of \ B(base) = {\LARGE[} \frac{A}{B} * 100{\LARGE]}$$
- $$A \ is \ what \ \% \ more \ than \ B(base) = {\LARGE[} \frac{A-B}{B} * 100{\LARGE]}$$
- $$B \ is \ what \ \% \ less \ than \ A(base) = {\LARGE[} \frac{A-B}{A} * 100{\LARGE]}$$

## Product Consistency Table

Product XY is Constant | X increases (%) | Y decreases (%) |

X is inversely proportional to Y | X increases (%) | Y decreases (%) |

Ratio Change effect of Denominator Change | Denominator Increases (%) | Ratio Decreases (%) |

Denominator Change effect of Ratio Change | Ratio Increases (%) | Denominator decreases (%) |

$$Standard \ Value \ 1$$ | $$5$$ | $$4.76$$ |

$$Standard \ Value \ 2$$ | $$9.09$$ | $$8.33$$ |

$$Standard \ Value \ 3$$ | $$10$$ | $$9.09$$ |

$$Standard \ Value \ 4$$ | $$11.11$$ | $$10$$ |

$$Standard \ Value \ 5$$ | $$12.5$$ | $$11.11$$ |

$$Standard \ Value \ 6$$ | $$14.28$$ | $$12.5$$ |

$$Standard \ Value \ 7$$ | $$16.66$$ | $$14.28$$ |

$$Standard \ Value \ 8$$ | $$20$$ | $$16.66$$ |

$$Standard \ Value \ 9$$ | $$25$$ | $$20$$ |

$$Standard \ Value \ 10$$ | $$33.33$$ | $$25$$ |

$$Standard \ Value \ 11$$ | $$40$$ | $$28.57$$ |

$$Standard \ Value \ 12$$ | $$50$$ | $$33.33 |

$$Standard \ Value \ 13$$ | $$60$$ | $$37.5$$ |

$$Standard \ Value \ 14$$ | $$66.66$$ | $$40$$ |

$$Standard \ Value \ 15$$ | $$75$$ | $$42.85$$ |

$$Standard \ Value \ 16$$ | $$100$$ | $$50$$ |

**Examples –**- The selling price of a biscuit is decreased by 20%. The current price is 100. By what percent should the new price be increased to bring it back to the original price. $$ 25 \% \ – [using \ standard \ value \ 9]$$
- If the price of milk goes up by 10%, then what should be the percentage decrease in the quantity consumed so that the total expenditure on tea remains the same. $$9.09 \% \ – [using \ standard \ value \ 3]$$

## Percentage to Fraction Conversion Table

6.25 % = \( \frac{1}{16} \) | 11.11 % = \( \frac{1}{9} \) | 9.09 % = \( \frac{1}{11} \) | 7.14 % = \( \frac{1}{14} \) | 8.33 % = \( \frac{1}{12} \) |

12.50 % = \( \frac{2}{16} \) | 22.22 % = \( \frac{2}{9} \) | 18.18 % = \( \frac{2}{11} \) | 14.28 % = \( \frac{2}{14} \) | 16.66 % = \( \frac{2}{12} \) |

18.75 % = \( \frac{3}{16} \) | 33.33 % = \( \frac{3}{9} \) | 27.27 % = \( \frac{3}{11} \) | 28.56 % = \( \frac{3}{14} \) | 25 % = \( \frac{3}{12} \) |

and so on | and so on | and so on | and so on | and so on |

20 % = \( \frac{1}{5} \) | 25 % = \( \frac{1}{4} \) |

40 % = \( \frac{2}{5} \) | 50 % = \( \frac{2}{4} \) |

60 % = \( \frac{3}{5} \) | 75 % = \( \frac{3}{4} \) |

80 % = \( \frac{4}{5} \) | 100 % = 1 |