CBSE Class 9 Mathematics Ratio and Proportion Notes Set 07

Ratio

The ratio of two quantities of the same kind is the fraction that tells how many times one is of the other. The ratio \( a:b \) has first term (antecedent) \( a \) and second term (consequent) \( b \), and equals \( \dfrac ab \) with \( b\ne0 \). A ratio does not change if both terms are multiplied or divided by the same non-zero number, so \( a:b=ka:kb \). When quantities are in the ratio \( a:b \), write them as \( ak \) and \( bk \).

Compounding and Special Ratios

Compound ratio of \( a:b \) and \( c:d \) is \( ac:bd \).
Duplicate ratio of \( a:b \) is \( a^2:b^2 \); triplicate ratio is \( a^3:b^3 \); sub-duplicate ratio is \( \sqrt a:\sqrt b \).
A ratio is of greater inequality when the first term is larger. Adding the same positive number to both terms brings the ratio closer to 1: if \( a\lt b \) then \( \dfrac{a+x}{b+x}>\dfrac ab \), and if \( a>b \) then \( \dfrac{a+x}{b+x}<\dfrac ab \).

Proportion

Four quantities are in proportion if \( a:b=c:d \), that is \( ad=bc \). Here \( a,d \) are the extremes and \( b,c \) the means. The fourth proportional to \( a,b,c \) is \( \dfrac{bc}a \); the third proportional to \( a,b \) is \( \dfrac{b^2}a \); the mean proportional between \( a \) and \( b \) is \( \sqrt{ab} \). Quantities \( a,b,c,d,\dots \) are in continued proportion if \( \dfrac ab=\dfrac bc=\dfrac cd=\dots \); they then form a geometric progression.

Properties of Equal Ratios

If \( \dfrac ab=\dfrac cd \) then
Invertendo: \( \dfrac ba=\dfrac dc \); Alternendo: \( \dfrac ac=\dfrac bd \)
Componendo: \( \dfrac{a+b}b=\dfrac{c+d}d \); Dividendo: \( \dfrac{a-b}b=\dfrac{c-d}d \)
Componendo and dividendo: \( \dfrac{a+b}{a-b}=\dfrac{c+d}{c-d} \)
If \( \dfrac ab=\dfrac cd=\dfrac ef=k \), then each ratio equals \( \dfrac{a+c+e}{b+d+f} \). In problems, put the common value equal to \( k \) and write \( a=bk,\ c=dk,\ e=fk \).

Variation

\( y \) varies directly as \( x \) (written \( y\propto x \)) if \( y=kx \); \( y \) varies inversely as \( x \) if \( y=\dfrac kx \). Joint variation: \( y\propto xz \) means \( y=kxz \). If \( y \) is the sum of two parts, one varying as \( x \) and the other as \( \dfrac1x \), then \( y=ax+\dfrac bx \). To solve, find the constant from the given pair of values, then use it for the required value.

Worked Examples

Example 1. The angles of a triangle are in the ratio 2 : 5 : 11. Find them.
The angles are \( 2k,5k,11k \) with \( 18k=180^\circ \), so \( k=10^\circ \). The angles are \( 20^\circ,50^\circ,110^\circ \).

Example 2. Find the fourth proportional to 7, 8 and 28.
Fourth proportional \( =\dfrac{8\times28}7=32 \), and indeed \( 7:8=28:32 \).

Example 3. \( x \) varies as the square of \( y \), and \( x=2 \) when \( y=5 \). Find \( x \) when \( y=15 \).
\( x=ky^2 \) with \( 2=25k \), so \( k=\dfrac2{25} \). Then \( x=\dfrac2{25}\times225=18 \).