Percentages Part 1

How to convert fractions into percentage

To convert a fraction into percentage just multiply with 100

• Example1: Convert  $\frac{1}{2}$  into the percentage

solution:$\frac{1}{2}\times&space;100=50$%

• Example 2: Convert  $\frac{1}{6}$  into the percentage

Solution:$\frac{1}{6}\times&space;100=\frac{100}{6}=&space;16\tfrac{2}{3}$% or 16.66%

How to convert percentages into fractions

To convert a percentage into fraction just divide by 100

• Example1: Convert 20% into the fraction

Solution: $\frac{20}{100}=\frac{1}{5}$

• Example 2: Convert $14\tfrac{2}{7}$% into the fraction

Solution:$14\frac{2}{7}=&space;\frac{100}{7\times&space;100}=\frac{1}{7}$

Remember this chart shown below which is used to solve problems in a  shortcut way

$1=100$%                                        $\frac{1}{11}=9\tfrac{1}{11}$% or 9.09%

$\frac{1}{2}=50$%                                         $\frac{1}{12}=8\tfrac{1}{3}$% or 8.33%

$\frac{1}{3}=33\tfrac{1}{3}$%                                      $\frac{1}{13}=7\tfrac{9}{13}$% or7.69%

$\frac{1}{4}=25$%                                        $\frac{1}{14}=7\tfrac{1}{7}$% or 7.142%

$\frac{1}{5}=20$%                                        $\frac{1}{15}=6\tfrac{2}{3}$% or 6.66%

$\frac{1}{6}=16\tfrac{2}{3}$% or 16.66%                $\frac{1}{16}=6\tfrac{1}{4}$$%$% or 6.25%

$\frac{1}{7}=14\tfrac{2}{7}$% or 14.28%               $\frac{1}{20}=5$%

$\frac{1}{8}=12\tfrac{1}{2}$% or 12.5%                  $\frac{1}{25}=4$%

$\frac{1}{9}=11\tfrac{1}{9}$% or 11.11%                $\frac{1}{30}=3\tfrac{1}{3}$% or 3.33%

$\frac{1}{10}=10$%                                    $\frac{1}{50}=2$%

Percentages Part 2

Based on the concept in part-1, let’s try some problems which are asked in many bank exams.

Problem 1: 65% of a number is 21 less than that  $\frac{3}{4}$ of the number. What is the number?

Solution: Convert $\frac{3}{4}$  into percentage

$\frac{3}{4}$=75%

65% $\sim$ 75%   $\rightarrow$  21

10%  $\rightarrow$  21

100%  $\rightarrow$  ?

$\frac{21\times&space;100}{10}=210$

The number is 210.

Problem 2: If $66\tfrac{2}{3}$ a number is added with itself then result becomes 3900. Find the original number?

Solution: Convert $66\tfrac{2}{3}$% into fraction

Split $66\tfrac{2}{3}$% into 50%+$16\tfrac{2}{3}$%

50%=$\frac{1}{2}$    ,     $16\tfrac{2}{3}$%=$\frac{1}{6}$

$\frac{1}{2}$+$\frac{1}{6}$ = $\frac{3+1}{6}=\frac{4}{6}=\frac{2}{3}$

$66\tfrac{2}{3}$% = $\frac{2}{3}$

Here 3 is the original number

3+2  $\rightarrow$  3900

$\rightarrow$  3900

$\rightarrow$  ?

$\frac{3900\times&space;3}{5}=2340$

The original number is 2340.

Problem 3: If 96 is added in the number then number becomes $157\tfrac{1}{7}$% of itself. Find the number?

Solution : Convert $157\tfrac{1}{7}$% into fraction

Split $157\tfrac{1}{7}$% into 100% and $57\tfrac{1}{7}$%

100%=1    ,     $57\tfrac{1}{7}$%=$\frac{4}{7}$

1+$\frac{4}{7}$=$\frac{11}{7}$

Here 7 is the original number and 11 occur when we add 96 to the original number

11-7=4

$\rightarrow$  96

$\rightarrow$  ?

$\frac{96\times&space;7}{4}=168$

The number is 168.

Percentages Part 3

Problem 1: If the radius of the right circular cylinder is increased by 40% and its height is reduced by 37.5% then find the percentage change in its volume?

Solution: Radius increased to 40% = $\frac{2}{5}$

Height decreased to 37.5%=$\frac{3}{8}$

Volume of a cylinder  =  $\pi&space;r^{2}h$

Actual volume  =  $\pi&space;\times&space;5\times&space;5\times&space;8=&space;200\pi$

Height decreased  =  8-3  =5

Changed volume=$\pi&space;\times&space;7\times&space;7\times&space;5=245\pi$

% change in its volume = $\frac{Final&space;volume-initial&space;volume}{initial&space;volume}\times&space;100$

=$\frac{245-200}{200}\times&space;100$

=22.5

Problem 2: If the length of the rectangle is increased by 20% and the breadth of the rectangle is decreased by  20% find the percentage change in its area?

Solution: Length increased to 20%

Area of the rectangle  =  Length * Breadth

Actual area  =  10*10  =  100

Changed area  =  12*8  =96

% change in the area  =  $\frac{Final&space;volume-initial&space;volume}{initial&space;volume}\times&space;100$

=$\frac{96-100}{100}\times&space;100$

=-4%

Here ‘-‘ indicates area decreased to 4%

Shortcut method: Direct formula $\frac{a\times&space;a}{100}$ = $\frac{20\times&space;20}{100}$  =  4

Note:

• The above method is applicable when there is increase or decrease in the value of 10%,20%,30,%40%,50%,60%, etc.
• If such values occur follow first problem method. It is not applicable when the fractional values occur.
• The fractional values like  37.5%,$57\tfrac{1}{7}$%, etc., If such values occur follow first problem method.

Percentages Part 4

Income and expenditure concept

Problem 1: A person spends 40% of his salary on house rent, on remaining 10%  spends on travels, on remaining $16\tfrac{2}{3}$% spends on food and remaining is saved. If he saved RS 6750 what amount he spent on food?

Solution : Assume total income=100

40% spends on salary=100-40=60

on remaining 10% spends on travels=$60\times&space;\frac{10}{100}=6$ =60-6=54

On remaining $16\tfrac{2}{3}$% spends on food=$54\times&space;\frac{1}{6}=9$ =54-9=45

Saved=45

Amount he spent on food=9

45  $\rightarrow$  6750

9    $\rightarrow$  ?

$\frac{9\times&space;6750}{45}=1350$

The amount spent on food is 1350

Observe the below figure

Assume income =100

Problem 2: A man spends 75% of his income. His income is increased by 20% and he increased his expenditure by 10%. Are his savings increased by?

Solution: A man sends his income  is 75%

The fractional value of 75%=$\frac{3}{4}$

Here 4 is the actual income and 3 is the expenditure

Savings are 4-3=1

He increases his income by 20%

$4\times&space;\frac{20}{100}=0.8$

4+0.8=4.8

He increases his expenditure by 10%

$3\times&space;\frac{10}{100}=0.3$

3+0.3=3.3

Savings = 4.8-3.3=1.5

Percentage change = $\frac{Final&space;volume-initial&space;volume}{initial&space;volume}\times&space;100$

= $\frac{1.5-1}{1}\times&space;100$

=50%

Observe the below figure

Problem 3: The price of sugar is increased by $16\tfrac{2}{3}$% and the consumption of a family is decreased by 20%. Find the percentage change in expenditure?

Solution :$16\tfrac{2}{3}$%=$\frac{1}{6}$       20%=$\frac{1}{5}$

Actual price is 6 and increased price is 7

Actual consumption is 5 and decreased consumption is 4

Price * Consumption = Expenditure

6*5=30        7*4=28

Percentage change=$\frac{Final&space;volume-initial&space;volume}{initial&space;volume}\times&space;100$

=$\frac{30-28}{30}\times&space;100=6.66$%

Observe the below figure

Problem 4: The sale of a cinema ticket is increased by $57\tfrac{1}{7}$% and the price of a ticket is increased by $16\tfrac{2}{3}$%. Find the change in the return?

Solution:   $57\tfrac{1}{7}$%=$\frac{4}{7}$         $16\tfrac{2}{3}$%=$\frac{1}{6}$

Percentage change = $\frac{Final&space;volume-initial&space;volume}{initial&space;volume}\times&space;100$

= $\frac{77-42}{42}\times&space;100$

=$83\tfrac{1}{3}$%

Percentages Part 5

More or less concept

Problem 1: A reduction of 25% in the price of the sugar enables a person to purchase 4kg more for 800 RS. Find the original and current price per kg?

Solution: Let the original price = x

Reduced price = $\frac{3x}{4}$

$\frac{800}{\frac{3x}{4}}-\frac{800}{x}=4$

$\frac{800}{3x}=4$

Original price $x=66.66$

Reduced price=$\frac{3x}{4}=\frac{3\times&space;66.66}{4}=50$

Shortcut: 25% $\rightarrow$ $\frac{1}{4}$

4-3=1

1 $\rightarrow$ 4

3 $\rightarrow$ 3*4=12

4 $\rightarrow$ 4*4=16

$\frac{800}{12}=66.66$         $\frac{800}{16}=50$

Problem 2: Due to 30% increase in the price of apples, 6 apples are less available for RS 520. Find the new price and the old price of an apple?

Solution :30%  $\rightarrow$ $\frac{3}{10}$

3 $\rightarrow$ 6

1 $\rightarrow$ 2

13*2=26

10*2=20

$\frac{520}{26}=20$            $\frac{520}{20}=26$

old price=20               new price=26