\displaystyle \textbf{RATIO AND PROPORTION}

\displaystyle \textbf{Ratio}
\displaystyle \text{The ratio of two quantities }a\text{ and }b\text{ of the same kind and expressed}
\displaystyle \text{in the same units is a comparison of the two quantities by division}.
\displaystyle \text{It is written as }a:b=\frac{a}{b},\text{ where }b\neq0,\text{ and is read as}
\displaystyle \text{``}a\text{ is to }b\text{''.}
\displaystyle \text{A ratio has no unit when the two quantities being compared have the same units}.
\displaystyle \text{Example: The ratio of }12\text{ cm to }18\text{ cm is}
\displaystyle 12:18=\frac{12}{18}=\frac{2}{3}=2:3.
\displaystyle \\

\displaystyle \textbf{Terms of a Ratio}
\displaystyle \text{In the ratio }a:b,\text{ the first term }a\text{ is called the antecedent and the}
\displaystyle \text{second term }b\text{ is called the consequent}.
\displaystyle \text{Thus, in the ratio }2:5,\text{ the antecedent is }2\text{ and the consequent is }5.
\displaystyle \\

\displaystyle \textbf{Important Property of a Ratio}
\displaystyle \text{If both terms of a ratio are multiplied or divided by the same non-zero}
\displaystyle \text{quantity, the value of the ratio remains unchanged}.
\displaystyle \text{Thus, if }m\neq0,\text{ then}
\displaystyle \frac{a}{b}=\frac{ma}{mb},\qquad\text{or equivalently,}\qquad a:b=ma:mb.
\displaystyle \text{Also,}
\displaystyle \frac{a}{b}=\frac{\frac{a}{m}}{\frac{b}{m}},\qquad \text{or equivalently,}\qquad a:b=\frac{a}{m}:\frac{b}{m}.
\displaystyle \text{Example: }3:5=(3\times4):(5\times4)=12:20.
\displaystyle \text{Also, }12:20=(12\div4):(20\div4)=3:5.
\displaystyle \\

\displaystyle \textbf{Ratio in Simplest Form}
\displaystyle \text{A ratio }a:b,\text{ where }a\text{ and }b\text{ are whole numbers, is said to be in}
\displaystyle \text{its simplest form if the H.C.F. of }a\text{ and }b\text{ is }1.
\displaystyle \text{To reduce a ratio to its simplest form, divide both terms by their H.C.F.}
\displaystyle \text{Example: The H.C.F. of }12\text{ and }18\text{ is }6.
\displaystyle \therefore 12:18=(12\div6):(18\div6)=2:3.
\displaystyle \text{Since the H.C.F. of }2\text{ and }3\text{ is }1,\text{ the ratio }2:3\text{ is in}
\displaystyle \text{its simplest form}.
\displaystyle \\

\displaystyle \textbf{Proportion}
\displaystyle \text{Four quantities }a,b,c\text{ and }d\text{ are said to be in proportion if the ratio}
\displaystyle \text{of the first two quantities is equal to the ratio of the last two quantities}.
\displaystyle \text{Thus, if }a:b=c:d,\text{ then }a,b,c\text{ and }d\text{ are in proportion}.
\displaystyle \text{A proportion may also be written as }a:b::c:d.
\displaystyle \text{It is read as ``}a\text{ is to }b\text{ as }c\text{ is to }d\text{''.}
\displaystyle \text{Example: }2:3=8:12\text{ because }\frac{2}{3}=\frac{8}{12}.
\displaystyle \therefore 2,3,8\text{ and }12\text{ are in proportion}.
\displaystyle \\

\displaystyle \textbf{Means and Extremes}
\displaystyle \text{In the proportion }a:b=c:d,\text{ the first and fourth terms }a\text{ and }d
\displaystyle \text{are called the extremes, while the second and third terms }b\text{ and }c
\displaystyle \text{are called the means}.
\displaystyle \text{Thus, in }2:3=8:12,\text{ the extremes are }2\text{ and }12,\text{ while the means}
\displaystyle \text{are }3\text{ and }8.
\displaystyle \\

\displaystyle \textbf{Fundamental Property of Proportion}
\displaystyle \text{If four quantities }a,b,c\text{ and }d\text{ are in proportion, then the product}
\displaystyle \text{of the extremes is equal to the product of the means}.
\displaystyle a:b=c:d\quad\Longrightarrow\quad\frac{a}{b}=\frac{c}{d}.
\displaystyle \therefore ad=bc.
\displaystyle \text{Conversely, if }ad=bc,\text{ where }b\neq0\text{ and }d\neq0,\text{ then}
\displaystyle a:b=c:d.
\displaystyle \text{Hence, }a:b=c:d\quad\Longleftrightarrow\quad ad=bc.
\displaystyle \text{In other words,}
\displaystyle \text{Product of extremes}=\text{Product of means}.
\displaystyle \text{Example: In }2:3=8:12,\quad2\times12=3\times8=24.
\displaystyle \therefore 2:3=8:12\text{ is a proportion}.
\displaystyle \\

\displaystyle \textbf{Finding a Missing Term in a Proportion}
\displaystyle \text{The fundamental property of proportion can be used to find an unknown term}.
\displaystyle \text{Example: If }4:7=x:21,\text{ find }x.
\displaystyle \frac{4}{7}=\frac{x}{21}.
\displaystyle 4\times21=7x.
\displaystyle 84=7x.
\displaystyle \therefore x=12.
\displaystyle \\

\displaystyle \textbf{Continued Proportion}
\displaystyle \text{Three quantities }a,b\text{ and }c\text{ are said to be in continued proportion if}
\displaystyle a:b=b:c.
\displaystyle \text{Using the fundamental property of proportion,}
\displaystyle a:b=b:c\quad\Longrightarrow\quad b^2=ac.
\displaystyle \text{Example: }4,6\text{ and }9\text{ are in continued proportion because}
\displaystyle 4:6=6:9,\qquad\text{and}\qquad6^2=4\times9=36.
\displaystyle \\

\displaystyle \textbf{Mean Proportional}
\displaystyle \text{If }a,b\text{ and }c\text{ are in continued proportion, then }b\text{ is called the}
\displaystyle \text{mean proportional between }a\text{ and }c.
\displaystyle a:b=b:c\quad\Longrightarrow\quad b^2=ac.
\displaystyle \therefore b=\sqrt{ac},\text{ for positive quantities }a\text{ and }c.
\displaystyle \text{Example: The mean proportional between }4\text{ and }9\text{ is}
\displaystyle \sqrt{4\times9}=\sqrt{36}=6.
\displaystyle \\

\displaystyle \textbf{Third Proportional}
\displaystyle \text{If }a:b=b:c,\text{ then }c\text{ is called the third proportional to }a\text{ and }b.
\displaystyle a:b=b:c\quad\Longrightarrow\quad ac=b^2.
\displaystyle \therefore c=\frac{b^2}{a},\qquad a\neq0.
\displaystyle \text{Example: To find the third proportional to }4\text{ and }6,\text{ let it be }x.
\displaystyle 4:6=6:x.
\displaystyle 4x=36.
\displaystyle \therefore x=9.
\displaystyle \text{Hence, the third proportional to }4\text{ and }6\text{ is }9.
\displaystyle \\

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