Percentages Part 3

Increase and decrease concept

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 r^{2}h

Actual volume  =  \pi \times 5\times 5\times 8= 200\pi

Radius increased=5+2=7

Height decreased  =  8-3  =5

Changed volume=\pi \times 7\times 7\times 5=245\pi

% change in its volume = \frac{Final volume-initial volume}{initial volume}\times 100

=\frac{245-200}{200}\times 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%

Breadth decreased to 20%

Area of the rectangle  =  Length * Breadth

Actual area  =  10*10  =  100

Changed area  =  12*8  =96

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

=\frac{96-100}{100}\times 100

=-4%

Here ‘-‘ indicates area decreased to 4%

Shortcut method: Direct formula \frac{a\times a}{100} = \frac{20\times 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 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 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 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 7}{4}=168

The number is 168.

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 100=50%

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

Solution:\frac{1}{6}\times 100=\frac{100}{6}= 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}= \frac{100}{7\times 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%