\displaystyle \textbf{Hundredth Part}
\displaystyle \text{If a quantity is divided into }100\text{ equal parts, each part is called}
\displaystyle \text{one hundredth of the quantity}.
\displaystyle \text{One hundredth}=\frac{1}{100}=0.01.
\displaystyle \text{Thus, }1\%\text{ of a quantity means }\frac{1}{100}\text{ of that quantity}.
\displaystyle \\

\displaystyle \textbf{Percentage}
\displaystyle \text{Percentage means ``per hundred'', that is, a quantity expressed out of }100.
\displaystyle \text{The symbol used for percentage is }\%.
\displaystyle x\%=\frac{x}{100}.
\displaystyle \text{Thus, }25\%=\frac{25}{100}=\frac{1}{4}.
\displaystyle \text{A percentage is another way of expressing a fraction, decimal, or ratio}.
\displaystyle \\

\displaystyle \textbf{Convert a Percentage into a Fraction}
\displaystyle \text{To convert a percentage into a fraction, divide the percentage by }100
\displaystyle \text{and remove the }\%\text{ sign. Then simplify the fraction, if possible}.
\displaystyle x\%=\frac{x}{100}.
\displaystyle \text{Example: }5\%=\frac{5}{100}=\frac{1}{20}.
\displaystyle \\

\displaystyle \textbf{Convert a Fraction into a Percentage}
\displaystyle \text{To convert a fraction into a percentage, multiply the fraction by }100\%.
\displaystyle \frac{a}{b}=\frac{a}{b}\times100\%,\qquad b\ne0.
\displaystyle \text{Example: }\frac{3}{5}=\frac{3}{5}\times100\%=60\%.
\displaystyle \\

\displaystyle \textbf{Convert a Percentage into a Ratio}
\displaystyle \text{To convert a percentage into a ratio, write it as a fraction with}
\displaystyle \text{denominator }100\text{ and then reduce it to its simplest form}.
\displaystyle x\%=x:100.
\displaystyle \text{Example: }5\%=5:100=1:20.
\displaystyle \\

\displaystyle \textbf{Convert a Ratio into a Percentage}
\displaystyle \text{To convert a ratio into a percentage, first write it as a fraction and}
\displaystyle \text{then multiply the fraction by }100\%.
\displaystyle a:b=\frac{a}{b}\times100\%,\qquad b\ne0.
\displaystyle \text{Example: }1:4=\frac{1}{4}\times100\%=25\%.
\displaystyle \\

\displaystyle \textbf{Convert a Percentage into a Decimal}
\displaystyle \text{To convert a percentage into a decimal, divide the percentage by }100
\displaystyle \text{and remove the }\%\text{ sign}.
\displaystyle \text{Example: }75\%=\frac{75}{100}=0.75.
\displaystyle \\

\displaystyle \textbf{Convert a Decimal into a Percentage}
\displaystyle \text{To convert a decimal into a percentage, multiply the decimal by }100
\displaystyle \text{and add the }\%\text{ sign}.
\displaystyle \text{Example: }0.40=0.40\times100\%=40\%.
\displaystyle \\

\displaystyle \textbf{Finding a Percentage of a Quantity}
\displaystyle \text{To find }x\%\text{ of a quantity, multiply the quantity by }\frac{x}{100}.
\displaystyle x\%\text{ of }a=\frac{x}{100}\times a.
\displaystyle \text{Example: Find }25\%\text{ of }240.
\displaystyle 25\%\text{ of }240=\frac{25}{100}\times240=60.
\displaystyle \therefore 25\%\text{ of }240\text{ is }60.
\displaystyle \\

\displaystyle \textbf{Finding What Percentage One Quantity is of Another}
\displaystyle \text{To find what percentage one quantity is of another, divide the first}
\displaystyle \text{quantity by the second quantity and multiply the result by }100\%.
\displaystyle \text{Percentage}=\frac{\text{Part}}{\text{Whole}}\times100\%.
\displaystyle \text{Example: What percentage of }180\text{ is }45?
\displaystyle \text{Required percentage} =\frac{45}{180}\times100\%=25\%.
\displaystyle \therefore 45\text{ is }25\%\text{ of }180.
\displaystyle \\

\displaystyle \textbf{Finding the Whole Quantity}
\displaystyle \text{If }x\%\text{ of a quantity is }a,\text{ then the whole quantity is}
\displaystyle \frac{a\times100}{x},\qquad x\ne0.
\displaystyle \text{Example: }30\%\text{ of a number is }72.\text{ Find the number}.
\displaystyle \text{Number}=\frac{72\times100}{30}=240.
\displaystyle \therefore \text{the required number is }240.
\displaystyle \\

\displaystyle \textbf{Increasing a Quantity by a Certain Percentage}
\displaystyle \text{Suppose a quantity }a\text{ is increased by }x\%.
\displaystyle \text{Increase}=\frac{x}{100}\times a=\frac{ax}{100}.
\displaystyle \text{New quantity}=a+\frac{ax}{100}
\displaystyle =a\left(1+\frac{x}{100}\right).
\displaystyle \text{Thus, to increase a quantity by }x\%\text{, multiply it by}
\displaystyle \left(1+\frac{x}{100}\right).
\displaystyle \text{Example: Increase }500\text{ by }20\%.
\displaystyle \text{New quantity} =500\left(1+\frac{20}{100}\right)=500(1.2)=600.
\displaystyle \\

\displaystyle \textbf{Decreasing a Quantity by a Certain Percentage}
\displaystyle \text{Suppose a quantity }a\text{ is decreased by }x\%.
\displaystyle \text{Decrease}=\frac{x}{100}\times a=\frac{ax}{100}.
\displaystyle \text{New quantity}=a-\frac{ax}{100}
\displaystyle =a\left(1-\frac{x}{100}\right).
\displaystyle \text{Thus, to decrease a quantity by }x\%\text{, multiply it by}
\displaystyle \left(1-\frac{x}{100}\right).
\displaystyle \text{Example: Decrease }500\text{ by }20\%.
\displaystyle \text{New quantity} =500\left(1-\frac{20}{100}\right)=500(0.8)=400.
\displaystyle \\

\displaystyle \textbf{Finding Percentage Increase}
\displaystyle \text{When a quantity increases, first find the increase by subtracting the}
\displaystyle \text{original value from the new value}.
\displaystyle \text{Increase}=\text{New value}-\text{Original value}.
\displaystyle \text{Percentage increase} =\frac{\text{Increase}}{\text{Original value}}\times100\%.
\displaystyle \text{Example: A quantity increases from }80\text{ to }100.
\displaystyle \text{Increase}=100-80=20.
\displaystyle \text{Percentage increase} =\frac{20}{80}\times100\%=25\%.
\displaystyle \\

\displaystyle \textbf{Finding Percentage Decrease}
\displaystyle \text{When a quantity decreases, first find the decrease by subtracting the}
\displaystyle \text{new value from the original value}.
\displaystyle \text{Decrease}=\text{Original value}-\text{New value}.
\displaystyle \text{Percentage decrease} =\frac{\text{Decrease}}{\text{Original value}}\times100\%.
\displaystyle \text{Example: A quantity decreases from }200\text{ to }150.
\displaystyle \text{Decrease}=200-150=50.
\displaystyle \text{Percentage decrease} =\frac{50}{200}\times100\%=25\%.
\displaystyle \\

\displaystyle \textbf{Successive Percentage Changes}
\displaystyle \text{Successive percentage changes are applied one after another to the}
\displaystyle \text{new value obtained at each stage, not always to the original value}.
\displaystyle \text{Therefore, equal percentage increase and decrease do not cancel each other}.
\displaystyle \text{Example: Increase }100\text{ by }20\%\text{ and then decrease it by }20\%.
\displaystyle \text{After a }20\%\text{ increase, the new value} =100\times1.20=120.
\displaystyle \text{After a }20\%\text{ decrease, the new value} =120\times0.80=96.
\displaystyle \text{Overall decrease}=100-96=4.
\displaystyle \text{Overall percentage decrease} =\frac{4}{100}\times100\%=4\%.
\displaystyle \therefore \text{a }20\%\text{ increase followed by a }20\%\text{ decrease results}
\displaystyle \text{in an overall decrease of }4\%.
\displaystyle \\

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