CBSE Class 10 Mathematics Ratio and Proportion Notes Set 04

Ratio

A ratio compares two quantities of the same kind by division. In \(a:b\), \(a\) is the antecedent and \(b\) the consequent. Multiplying or dividing both terms by the same non-zero number does not change a ratio. Two ratios \(a:b\) and \(c:d\) compound to \(ac:bd\). The duplicate ratio of \(a:b\) is \(a^2:b^2\), the triplicate is \(a^3:b^3\) and the sub-duplicate is \(\sqrt a:\sqrt b\).

Examples. The angles of a triangle are in the ratio \(2:5:11\): each part is \(\dfrac{180}{18}=10^\circ\), so the angles are \(20^\circ,50^\circ,110^\circ\). The quantity to be subtracted from each term of \(9:16\) to make it \(1:2\) satisfies \(2(9-x)=16-x\), so \(x=2\). To express a ratio as a percentage, multiply the fraction by 100: \(3:40\) is \(7.5\%\).

Ratios from homogeneous equations: if \(3x^2-10xy+3y^2=0\), divide by \(y^2\) to get \(3t^2-10t+3=0\) with \(t=x/y\), so \(t=3\) or \(\dfrac13\). Adding the same positive number to both terms of a ratio of greater inequality (\(a>b\)) makes it smaller, and of lesser inequality makes it larger; in both cases it moves towards 1.

Equal Ratios

If \(\dfrac ab=\dfrac cd=\dfrac ef=k\), put \(a=bk,\ c=dk,\ e=fk\). The following hold: invertendo \(\dfrac ba=\dfrac dc\); alternendo \(\dfrac ac=\dfrac bd\); componendo \(\dfrac{a+b}{b}=\dfrac{c+d}{d}\); dividendo \(\dfrac{a-b}{b}=\dfrac{c-d}{d}\); and componendo-dividendo \(\dfrac{a+b}{a-b}=\dfrac{c+d}{c-d}\). Also each ratio equals \(\dfrac{a+c+e}{b+d+f}\) (when the denominator sum is not zero). Example: if \(\dfrac{a}{b+c}=\dfrac{b}{c+a}=\dfrac{c}{a+b}\), each ratio is \(\dfrac12\), or \(-1\) when \(a+b+c=0\).

Proportion

Four quantities \(a,b,c,d\) are in proportion when \(a:b=c:d\), that is \(ad=bc\): the product of the extremes equals the product of the means. The fourth proportional to \(a,b,c\) is \(\dfrac{bc}{a}\), the third proportional to \(a,b\) is \(\dfrac{b^2}{a}\) and the mean proportional between \(a\) and \(b\) is \(\sqrt{ab}\). Examples: the fourth proportional to 7, 8, 28 is \(\dfrac{8\times28}{7}=32\); the mean proportional between 2 and 18 is \(\sqrt{36}=6\). Quantities \(a,b,c,\dots\) are in continued proportion if \(\dfrac ab=\dfrac bc=\dots\); for three terms \(b^2=ac\).

Variation

Find the constant \(k\) from the given values and then use it. Examples. If \(y\) varies inversely as \(x\) and \(y=5\) when \(x=3\), then \(k=15\) and \(y=\dfrac{15}{x}\). If \(x\) varies as the square of \(y\) and \(x=2\) when \(y=5\), then \(x=\dfrac{2}{25}y^2\). The surface area of a sphere varies as the square of its radius. A quantity that is the sum of two parts, one varying as \(x\) and the other inversely as \(x\), has the form \(y=ax+\dfrac bx\); two data points fix \(a\) and \(b\).