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Arithmetic I

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Percentages
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CAT 2025 Lesson : Percentages - Concepts & Cheatsheet

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Note: The video for this module contains a summary of all the concepts covered in this 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

1) %\% = 1100\dfrac{1}{100} and 100%=1100\% = 1

2) xx% of yy = x100×y\dfrac{x}{100} \times y

3) If a certain value is increased by x%x\% or decreased by y%y\%, then (1+x100)\left( 1 + \dfrac{x}{100} \right) or (1−y100)\left( 1 - \dfrac{y}{100} \right) are the respective multiplication factors.

4) Where xx and yy are two numbers,
- if xx has increased or decreased to yy, then xx is now equal to yy.
- if xx has increased by yy, then xx is now equal to x+yx + y.
- if xx has decreased by yy, then xx is now equal to x−yx - y.

5) If the quantity pp has now changed to qq, percentage change = q−pp×100%\dfrac{q - p}{p} \times 100\%

6) If there are successive changes of p%p\%, q%q\% and r%r\% in three stages, the effective percentage change is

((1+p100)(1+q100)(1+r100)−1)×100\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 x%\bm{x}\% grew by y\bm{y} percentage points, then the new rate is (x+y)\bm{(x + y)}%.

8) Where the product of two variables - x\bm{x} and y\bm{y}, is constant,
- if x\bm{x} increases by ab\dfrac{a}{b} of itself, then y\bm{y} decreases by ab+a\dfrac{a}{b + a} of itself

- if x\bm{x} decreases by ab\dfrac{a}{b} of itself, then y\bm{y} increases by ab−a\dfrac{a}{b - a} of itself

9) Where the base year's value is 100, index value in year x\bm{x} = Price of goods/stock in year xPrice of goods/stock in base year×100\dfrac{\text{Price of goods/stock in year} \space x}{\text{Price of goods/stock in base year}} \times 100

10) Where xx and yy are two variables,
- if xx is ab\dfrac{a}{b} more than y\bm{y}, then y\bm{y} is ab+a\dfrac{a}{b + a} less than x\bm{x}.

- if xx is ab\dfrac{a}{b} less than y\bm{y}, then y\bm{y} is ab−a\dfrac{a}{b - a} more than x\bm{x}.

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