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.

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

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 – $${\LARGE [} \frac{100x}{100 \pm x}{\LARGE ]} {\%}$$

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, $$Overall \ \% \ Change = a + b + \frac{ab}{100}$$

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

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

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

Product 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$$

These are short and easy approach for quick percentage change calculations.

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. $$9.09 \% \ – [using \ standard \ value \ 3]$$