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Percentages
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CAT 2025 Lesson : Percentages - Basics of Percentages

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1. Introduction

Percentages is applied in several chapters in math (including Profit & Loss, Interest & Growth, Ratio & Proportion). Quite a few Data Interpretation questions include percentage in some form.

%\% is the symbol used to denote percentage. For instance, 4040 percent is written as 40%40\%.

The value of this symbol %=1100\% = \dfrac{1}{100}.

Note that 100%=100×1100=1100\% = 100 \times \dfrac{1}{100} = 1

Percentage is useful in comparing values, which can be observed in the following example.

Example 1

Ram scored 3333, 5252 and 5454 marks in Maths, Science and English tests. If the total marks for these tests were 7575, 130130 and 120120 respectively, in which subject did he score the lowest percentage?

Solution

% in Maths =3375×100%=44%= \dfrac{33}{75} \times 100\% = 44\%

% in Science =52130×100%=40%= \dfrac{52}{130} \times 100\% = 40\%

% in English =54120×100%=45%= \dfrac{54}{120} \times 100\% = 45\%

Answer: Science

2. Conversions

2.1 Conversion of Decimal to Percent

To convert a decimal or fraction, we simply multiply it by 100100 and retain the '%\%' symbol.

Example 2

Express 0.00230.0023, 0.230.23 and 230230 as a percentage.

Solution

0.0023=0.0023×1=0.0023×100%=0.23%0.0023 = 0.0023 \times 1 = 0.0023 \times 100\% = 0.23\%

0.23=0.23×100%=23%0.23 = 0.23 \times 100\% = 23\%

230=230×100%=23000%230 = 230 \times 100\% = 23000\%

In short, the number written with the %\% symbol is 100100 times the number when written without %\%.

2.2 Conversion of Fraction to Percent

In case of fraction, the output can be written as a improper fraction, mixed fraction or as a near decimal.

Example 3

Express 211\dfrac{2}{11} as a percentage.

Solution

To convert any number to percentage we multiply by 100%.
211=211×100%\dfrac{2}{11} = \dfrac{2}{11} \times 100\%

=20011%= \dfrac{200}{11} \% (Improper Fraction)

=18211%= 18\dfrac{2}{11} \% (Mixed Fraction)

∼18.18%\sim 18.18\% (Near Decimal)

2.3 Conversion of Ratio to Percent

Ratio is a relation between two quantities.

In these questions, we are typically asked to find the percentage of an item in the total.

Example 4

The ratio of copper to iron in an alloy is 2:32 : 3. What percent of the alloy is copper?

Solution

For every 2 parts of copper in the alloy, there are 3 parts of iron. In other words, every 5 parts of the total alloy contains 2 parts of copper.

Copper %\% in the alloy =25×100%=40%= \dfrac{2}{5} \times 100\% = 40\%

Now, it is correct to state that 40% of the alloy is copper.

Answer: 40%40\%

2.4 Reconversion from Percent

Conversion from percent is easy. Simply replace %\% with 1100\dfrac{1}{100}.

Example 5

Express 45%45\% as a decimal.

Solution

45%=45×1100=920=0.4545\% = 45 \times \dfrac{1}{100} = \dfrac{9}{20} = 0.45

Answer: 0.450.45

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