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$$
These are short and easy approach for quick percentage change calculations.
Examples –
The selling price of a biscuit is decreased by 20%. The current price is 100. By what percent should the new price be increased to bring it back to the original price. $$ 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]$$
