5. Proportions

There are many times when two ratios are equal. Then the four quantities involved in these two ratios are said to be in proportion or proportionals. Thus if \(a:b=c:d\) then \(a, b, c\) and \(d\) are proportionals or are in proportion. It is also written as

\[a:b::c:d.\]

The terms \(a\) and d are called the extremes and \(b\) and \(c\) are the means. Clearly, as you can see that if there is such a proportion then product of extremes is equal to the product of means i.e. \(ad=bc\).

5.1. Continued Proportion

Quantities are said to be in continued proportion when first is to second is equal to second is to third, third is to fourth is equal to first two; and so on. Therefore, if

\[\frac{a}{b}=\frac{b}{c}=\frac{c}{d}=.....\]

then \(a, b, c, d, ...\) are in continued proportion.

If three quantities \(a, b\) and \(c\) are in proportion then,

\[ \begin{align}\begin{aligned}a:b=b:c;\\\therefore ac=b^2\end{aligned}\end{align} \]

Here, \(b\) is called mean proportional between \(a\) and \(c\) and \(c\) is called third proportional between \(a\) and \(b\).

If three quantities are in proportionals then the first is to the third is in the duplicate ratio of the first to the second.

Let us say three quantities are \(a, b\) and \(c\). Then we have,

\[\frac{a}{b} = \frac{b}{c}\]

Now we can say following:

\[\frac{a}{c} = \frac{a}{b}*\frac{b}{c} = \frac{a}{b}*\frac{a}{b}=\frac{a^2}{b^2}\]

If \(a:b=c:d\) and \(e:f=g:h\), the we will have \(ae:bf=cg:dh\).

For \(\frac{a}{b}=\frac{c}{d}\) and \(\frac{e}{f}=\frac{g}{h}\);

\[\therefore \frac{ae}{bf}=\frac{cg}{fh},\]

or \(ae:bf=cg:dh\)

If \(a:b=c:d\) and \(b:x=d:y\) then \(a:x=c:y\).

Proportions are essentially fractions and therefore we can apply properties of fractions to them and many useful relations can be established, which, can be very useful later on. Names of some of these operations come from Geometry.

5.1.1. Invertendo

If \(a:b=c:d\), then \(b:a=d:c\).

This is too easy and therefore left as an exercise to the reader.

5.1.2. Alternando

If \(a:b=c:d\), then \(a:c=b:d\).

We have,

\[ \begin{align}\begin{aligned}\frac{a}{b}=\frac{c}{d}\\\implies ad=bc\\\text{Dividing both side by}~cd~\text{we get}\\\frac{ad}{cd}=\frac{bc}{cd};\\\implies \frac{a}{c}=\frac{b}{d};\\\therefore a:c=b:d\end{aligned}\end{align} \]

5.1.3. Componendo

If \(a:b=c:d\), then \(a+b:b=c+d:d\).

We have,

\[ \begin{align}\begin{aligned}\frac{a}{b}=\frac{c}{d}\\\text{adding 1 to both side}\\\frac{a}{b}+1=\frac{c}{d}+1\end{aligned}\end{align} \]
(1)\[\therefore \frac{a+b}{b}=\frac{c+d}{d}\]
\[\text{or}~~a+b:b=c+d:d\]

5.1.4. Dividendo

If \(a:b=c:d\), then \(a-b:b=c-d:d\).

We have,

\[ \begin{align}\begin{aligned}\frac{a}{b}=\frac{c}{d}\\\text{subtracting 1 from both side}\\\frac{a}{b}-1=\frac{c}{d}-1\end{aligned}\end{align} \]
(2)\[\therefore \frac{a-b}{b}=\frac{c-d}{d}\]
\[\text{or}~~a-b:b=c-d:d\]

Dividing (1) by (2) we get,

(3)\[\frac{a+b}{a-b}=\frac{c+d}{c-d}\]

This relation is known as Componendo & Dividendo.

5.2. Problems

  1. Find the fourth proportional to following:
    1. \(3, 4, 9\).
    2. \(34, 47, 85\).
    3. \(x, y, xy\).
  2. Find the mean proportional between following:
    1. \(a\) and \(b\)
    2. \(12a^{3.4}\) and \(14a^{5.7}\).
  3. Find the third proportional of the following:
    1. \(7\) and \(9\).
    2. \(ab\) and \(a+b\).

If \(a:b=c:d\), prove that

  1. \(a^2c+ac^2:b^2d+bd^2=(a+c)^3:(b+d)^3\).
  2. \(pa^2+qb^2:pa^2-qb^2=pc^2+qd^2:pc^2-qd^2\).
  3. \(a-c:b-d=\sqrt{a^2+c^2}:\sqrt{b^2+d^2}\).
  4. \(\sqrt{a^2+c^2}:\sqrt{b^2+d^2}=\sqrt{ac+\frac{c^3}{a}}:\sqrt{bd+\frac{d^3}{b}}\).

If \(a, b, c\) and \(d\) are in continued proportion prove that

  1. \(a:b+d=c^3:c^2d+d^3\).
  2. \(2a+3d:3a-4d=2a^3+3b^3:3a^3-4b^3\).
  3. \((a^2+b^2+c^2)(b^2+c^2+d^2)=(ab+bc+cd)^2\).
  4. If \(b\) is a mean proportional between \(a\) and \(c\), prove that
\[\frac{a^2-b^2+c^2}{a^{-2}-b^{-2}+c^{-2}}=b^4.\]
  1. If \(a:b=c:d\) and \(e:f=g:h\), prove that
\[ae+bf:ae-bf=cd+dh:cg-dh.\]
  1. If \(a:b=c:d\), prove that
\[(a+b+c+d)(a-b-c+d)=(a-b+c-d)(a+b-c-d)\]
  1. If the work done by \(x-1\) men in \(x+1\) days is to work done by \(x+2\) men in \(x-1\) days is in the ratio of \(9:10\), find \(x\).
  2. Find four proportionals such that sum of the extremes is 21, the sum of the means 19, and the sum of square of all four numbers is 442.
  3. Two casks \(a\) and \(b\) were filled with two kinds of sherry, mixed in the cask \(a\) in the ratio \(2:7\), and in the cask \(b\) in the ratio \(1:5\). What quantity must be taken from each to form a mixture which shall consist of 2 gallons of one kind and 9 gallons of the other?
  4. Nine gallons are drawn from a cask full of wine; it is then filled with water, then nine gallons of the mixture are drawn, and the cask is again filled with water. If the quantity of wine now in the cask to be quantity of water in it as 16 to 9, how much does the cask hold?
  5. In a certain country the assumption of tea is five times the consumption of coffee. If \(a\) more tea and \(b\) percent more coffee were consumed, the aggregate amount consumed would be \(7c\) percent more; but if \(b\) percent more tea and \(a\) more coffee were consumed, the aggregate amount consumed would be \(3c\) percent more. Compare \(a\) and \(b\).
  6. If four quantities are in continued proportion show that the difference between the first and last is at least three times as great as the difference between the other two.
  7. In England the population increased 15.9 per cent, between1871 and 1881 ; if the town population increased 18 per cent, and the country population 4 per cent, compare the town and country populations in 1871.
  8. Brass is an alloy of copper and zinc; bronze is an alloy containing 80 percent, of copper, 4 of zinc, and 16 of tin. A fused mass of brass and bronze is found to contain 74 percent, of copper, 16 of zinc, and 10 of tin : find the ratio of copper to zinc in the composition of brass.
  9. A crew can row a certain course up stream in 84 minutes;they can row the same course downstream in 9 minutes less tan they could row it in still water : how long would they take to row down with the stream?