An Introduction to Percentages

Let us start with a daily life example where we compare different quantities, suppose the height of two students named Rahim and Karim, and we conclude Rahim is taller than Karim or Kaarim is shorter than Rahim or Rahim's height is the double of Karims or Rahim's height is half the height of Karim etc. etc. In all the cases we're basically talking about height which has to be in same unit in all the cases, which may be cm, feet, etc. so, when we compare two quantities, the results obtained is called a ratio, a rayio has no unit. A ratio is nothing but the division of one quantity by another of the same type (unit). Moving back to the first example, suppose the height of Rahim is 150 cm and the height of Karim is 100 cm. Then the ratio of their height is $\ \frac{150cm}{100cm}$ = $\ \frac{150}{100}$ = $\ \frac{3}{2}$ or 3:2. The symbol ":" stands for ratio.
Let us take another example, Aman got 45 marks out of 50. Now, what is the ratio of his obtained marks to the total marks. It is $\ \frac{obtained~ marks}{total~ marks}$ = $\ \frac{45}{50}$ = $\ \frac{9}{10}$ or 4:5. This time the obvious question in our mind is what percentage of marks he got. What we are now asking is that what marks he would have got if the total marks were 100. So, our job now is to make the total marks (denominator) 100. We'll be able to answer the question in a while once you've gone through the conversion section.

A fractional number which has 100 as its denominator is known as percentage. The word 'percent' is derived from a Latin word 'per centum' means 'out of a hundred'. The symbol '%' stands for percent.
When a fraction is having 100 in its denominator then the numerator is the percentage. When a quantity is expressed in percent, then it is called percentage.

Conversion of fractions into percentage

(i) To make the denominator 100.
Example, $\ \frac{16}{20} =\frac{16×5}{20×5}=\frac{80}{100} = 80$ %
(ii) To multiply the fraction by 100 and and put the symbol '%'.
Example, $\ \frac{16}{20}×100 = 16×5 = 80$ %

Conversion of decimals into percentage

To convert the decimal into fraction and then multiply the fraction by 100 and put the '%' symbol.
Example, (i) $\ 0.27 = \frac{27}{100}$
Now, $\ \frac{27}{100}×100 = 27$ %
(ii) $\ 0.09 = \frac{9}{100}$
Now, $\ \frac{9}{100}×100 = 9$ %

Conversion of percentage into fraction in lowest term

To make a fraction whose denominator is 100 by removing the '%' symbol and then reduce it to the lowest form.
Example, (i) 20% $\ = \frac{20}{100} = \frac{1}{5}$
(ii) 150% $\ = \frac{150}{100} = \frac{3}{2} =1\frac{1}{2}$

Conversion of percentage into decimal

To make a fraction whose denominator is 100 by removing the '%' symbol and then write the decimal value.
Example, (i) 85% $\ = \frac{85}{100} = 0.85$

Money and Metric measures as percentage

Money

we know,
100 paise = Re. 1
1 paise = Re. $\ \frac{1}{100}$ = 1% of Re. 1
Similarly,
5 paise = 5% of Re. 1
10 paise = 10% of Re. 1
100 paise = 100% of Re. 1

Length

we know,
100 cm = 1 m
1 cm = $\ \frac{1}{100}$ m = 1% of 1 m
Similarly,
5 cm = 5% of 1 m
100 cm = 100% of 1 m

Mass

we know,
1000 g = 1 kg
1 g = $\ \frac{1}{1000}$ kg $\ = \frac{1}{10}×\frac{1}{100}$ kg
$\ = 0.1×\frac{1}{100}$ kg = 0.1% of 1 kg
Similarly,
20 g = $\ \frac{20}{1000}$ kg $\ = \frac{20}{10}×\frac{1}{100}$ kg
$\ = 2×\frac{1}{100}$ kg = 2% of 1 kg
100 g = $\ \frac{100}{1000}$ kg $\ = \frac{100}{10}×\frac{1}{100}$ kg
$\ = 10×\frac{1}{100}$ kg = 10% of 1 kg

Capacity

we know,
1000 ml = 1 L
1 ml = $\ \frac{1}{1000}$ L $\ = \frac{1}{10}×\frac{1}{100}$ L
$\ = 0.1×\frac{1}{100}$ L = 0.1% of 1 L
Similarly,
5 ml = $\ \frac{5}{1000}$ L $\ = \frac{5}{10}×\frac{1}{100}$ L
$\ = 0.5×\frac{1}{100}$ L = 0.5% of 1 L
100 ml = $\ \frac{100}{1000}$ L $\ = \frac{100}{10}×\frac{1}{100}$ L
$\ = 10×\frac{1}{100}$ L = 10% of 1 L

Given percentage of a certain number

Let us take an example, '40% of 200'. firstly, to remove the '%' symbol and make 40% a fraction as follows, 40% $\ = \frac{40}{100}$ secondly, to mutiply the other quantity by the fraction obtained in previous step as follows, $\ \frac{40}{100}×200 = 80$ % That's all.
Let us take some other examples,
(i) 56% of 450
$\ \frac{56}{100}×450 = 252$

(ii) 80% of Rs. 1200
$\ \frac{80}{100}×1200 = Rs.~ 960$

(iii) 25% of 52 weeks
$\ \frac{25}{100}×52 = 13 ~weeks$

(iii) $\ 10\frac{1}{4}$ % of Rs. 800
$\ \frac{41}{4}$ % of Rs. 800
$\ \frac{\frac{41}{4}}{100}×800 = \frac{41}{4}×\frac{1}{100} = Rs.~ 82$

(iv) $\ 9\frac{1}{2}$ % of Rs. 1200
$\ \frac{19}{2}$ % of Rs. 1200
$\ \frac{\frac{19}{2}}{100}×1200 = \frac{19}{2}×\frac{1}{100}×1200 = Rs.~ 114$

(v) 24% of 50 L
$\ \frac{24}{100}×50 = 12 L$

(vi) 81.5% of 775
$\ \frac{81.5}{100}×775 = \frac{63162.5}{100} = 631.625$