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Percentages

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CAT 2025 Lesson : Percentages - Concepts & Cheatsheet

Note: The video for this module contains a summary of all the concepts covered in the Percentages lesson. The video would serve as a good revision. Please watch this video in intervals of a few weeks so that you do not forget the concepts. Below is a cheatsheet that includes all the formulae but not necessarily the concepts covered in the video.

9. Cheatsheet

\%

\dfrac{1}{100}

and

100\% = 1

x

% of

y

\dfrac{x}{100} \times y

3) If a certain value is increased by

x\%

or decreased by

y\%

, then

\left( 1 + \dfrac{x}{100} \right)

\left( 1 - \dfrac{y}{100} \right)

are the respective multiplication factors.

4) Where

x

and

y

are two numbers,
- if

x

has increased or decreased to

y

, then

x

is now equal to

y

.
- if

x

has increased by

y

, then

x

is now equal to

x + y

.
- if

x

has decreased by

y

, then

x

is now equal to

x - y

.

5) If the quantity

p

has now changed to

q

, percentage change =

\dfrac{q - p}{p} \times 100\%

6) If there are successive changes of

p\%

q\%

and

r\%

in three stages, the effective percentage change is

\left( \left( 1 + \dfrac{p}{100} \right) \left( 1 + \dfrac{q}{100} \right) \left( 1 + \dfrac{r}{100} \right) - 1 \right) \times 100

(p, q and r are integers, wherein a negative value suggests a decrease)

7) Percentage point or percent point is the arithmetic difference of two percentages. If a certain rate of

\bm{x}\%

grew by

\bm{y}

percentage points, then the new rate is

\bm{(x + y)}

%.

8) Where the product of two variables -

\bm{x}

and

\bm{y}

, is constant,
- if

\bm{x}

increases by

\dfrac{a}{b}

of itself, then

\bm{y}

decreases by

\dfrac{a}{b + a}

of itself

- if

\bm{x}

decreases by

\dfrac{a}{b}

of itself, then

\bm{y}

increases by

\dfrac{a}{b - a}

of itself

9) Where the base year's value is 100, index value in year

\bm{x}

\dfrac{\text{Price of goods/stock in year} \space x}{\text{Price of goods/stock in base year}} \times 100

10) Where

x

and

y

are two variables,
- if

x

\dfrac{a}{b}

more than

\bm{y}

, then

\bm{y}

\dfrac{a}{b + a}

less than

\bm{x}

.

- if

x

\dfrac{a}{b}

less than

\bm{y}

, then

\bm{y}

\dfrac{a}{b - a}

more than

\bm{x}

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