What is a Percentage? Some Basic concepts of Percentage and Real Life Situations involving Percentage With Verified Solutions

Percentage:

Percent is defined as a ratio with a second term of 100 or a fraction with 100 as a denominator. It help us to understand and compare information about quantity quickly and easily.

For Example, If Ali earns 100 rupees and gives 3 rupees out of them to the government as a tax. We can say that he pays 3 percent as a tax which can be represented as 3% as shown below.

Ali paid tax = Rs. 3 out of Rs. 100

Ali paid tax = `\frac{3}{100}`

Ali paid tax = 3% or 3 percent of the whole.

Similarly, if Ali earns 350 rupees and he has to pay the tax at the same rate (3%) then how much will he pay?

We can calculate the amount of the tax by multiplying 3% by 350 rupees. Here the result is called the percentage.

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Some Basic concepts of Percentages:

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(i) The symbol ( % ) is used instead of writing the word percent. So, we can write 65 percent as 65%.

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(ii) Any fraction with a denominator of 100 is called a percentage.

For example; `\frac{65}{100}` can be written as 65 %. Or we can write 65 % as `\frac{65}{100}`.

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(iii) A fraction `\frac{65}{100}` means 65 parts out of 100 parts. Therefore, percent means per hundred or out of a hundred.

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(iv) To change a fraction into a percentage:

To change a fraction into a percentage we will just multiply the fraction by 100% Or multiply and divide the fraction by 100. Then simplifying except the 100 in the denominator, we will get the required percentage (%)

For example: Changing the fraction `\frac{6}{25}` to percentage

1st Method: (just simply multiplying by 100%)

`\frac{6}{25}` = `\frac{6}{25}` ｘ 100% (multiplying the fraction by 100%)

so, by simplifying the numbers only and leaving the % symbol, we get 24%

2nd Method: (by multiplying and dividing 100)

`\frac{6}{25}` = `\frac{6}{25}` ｘ `\frac{100}{100}`

`\frac{6}{25}` = `\frac{6 ｘ 100}{25}`ｘ `\frac{1}{100}`

`\frac{6}{25}` = 24 ｘ `\frac{1}{100}`

`\frac{6}{25}` = `\frac{24}{100}`

`\frac{6}{25}` = 24%

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(v) To change a percentage to a decimal:

Using basic concept (ii) we will convert it into a fraction, then change it to a decimal.

For example, 65% = `\frac{65}{100}` = 0.65

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(vi) Expressing One Quantity as a Percentage of Another:

Explanation:

Example 1:

The height of Amina is 4 feet while the height of his brother Asad is 5 feet. Let us express the height of Amina as a percentage of the height of Asad.

Step 1: Divide 4 by 5.

Step 2: Multiply the result by 100% and simplify to obtain quantity in percentage. {as 100% = `\frac {100}{100}` = 1}

Step 3: Place the sign of percentage after the result.

The correct answer is `\frac {4}{5}`ｘ100% = 80%

Therefore, Amina's height is 80% of the height of Asad.

Example 2:

In a quiz program, Urooba answered 63 questions out of 75. What percent of questions were answered? Let us see:

Correct answers = 63

Total questions = 75

Percentage of questions that were answered `\frac {63}{75}`ｘ100% = 84%

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(vii) Comparison of Two Quantities by Percentage:

In a test, Hassan got 44 out of 50 marks in mathematics and 72 out of 80 marks in general science. We are not sure in which subject he is better

For better understanding, we convert quantities into a percentage.

Pass percentage in mathematics `\frac {44}{50}`ｘ100% = 88%

Pass percentage in general science `\frac {72}{80}`ｘ100% = 90%

So, Hassan did better in general science

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(viii) Increase or Decrease in a Quantity by a Given Percentage

Example:

Usman's elder brother works in an office. Last month his salary increased from Rs. 50000 to Rs. 55000. Let us find the percent increase in salary.

Increase in salary = Rs 55000 - Rs 50000 = Rs. 5000

Increase percentage in salary = `\frac {Rs. 5000}{Rs. 50000}`ｘ100% = 10%

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