.. meta::
   :author: Shiv Shankar Dayal
   :title: Preface
   :description: Algebra
   :keywords: Algebra, ratio, proportions, variations, complex numbers,
              arithmetic progressions, geometric progressions, harmonic
              progressions, series, sequence, quadratic equations,
              permutations, combinations, lograithms, binomial theorem,
              determinant, matricesNumber System

Ratio
*****
In his book *Elements of Algebra*, Euler says that anything we can increase
or decrease is *magnitude* or *quantity*.

We will study ratios of quantities in this chapter. Many of you might
already be knowing about ratios from earlier classes. A ratio is nothing
but a comparison between two quantities of same kind. For example, 4
balls vs 7 balls or 12 shoes vs 8 shoes.

A ratio between two quantities :math:`Q1` and :math:`Q2` is written as
:math:`Q1:Q2` most of the time. :math:`Q1` or the first term is called the
**antecedent** and the :math:`Q2` or the second term is called the
**consequent**.

To measure the ratio between two quantities typically we use fractions. A
reduced fraction is tells us about one quantity being multiples of another or
part of that. For example, 2 dozen bananas compared to 6 bananas can be written
as :math:`\frac{2*12}{6}` or 4.

Ratios are *abstract quantities* because they are sheer numbers. In practice,
considering our previous example we can say 2 dozen bananas are four times 6
bananas. This four is an abstract quantity because it does not have a physical
form.

Clearly, you can see that

.. math::

  \frac{Q1}{Q1} = \frac{mQ1}{mQ2}

so the ratio does not change if numerator or antecedent and denominator or
consequent are multiplied by the same quantity.

Comparing Two Ratios
====================
To compare two ratios we need to make their denominator same the we can deduce
if they are equal or unequal. For example consider two ratios :math:`r1:r2` and
:math:`r3:r4`. Then we can have

.. math::

  \frac{r1}{r2} = \frac{r1r4}{r2r4} ~~\&~~ \frac{r3}{r4} = \frac{r3r2}{r2r4}

Now since consequent is same for both the ratios we can determine which ratio
is bigger or equal or smaller by determining if :math:`r1r4` is bigger or
equal or smaller as compared to :math:`r2r3`.

Ratio of Ratios
===============
Clearly, a ratio can be represented by a fraction of two integers. Now let us
try to see what happens to ratio of two ratios:

Consider two ratios: :math:`a:b` and math:`c:d`. Thus we can write ratio of
these two ratios as fraction

.. math::
  \frac{\frac{a}{b}}{\frac{c}{d}} ~~or~~ \frac{ad}{bc}

Therefore we can conclude that it is equivalent to :math:`ad:bc`.

Surds and Ratios
================
When I said a ratio can be applied as a fraction of two integers then that
applied to finite numbers. A *surd* like :math:`\sqrt{x}` is not a finite
number because certain portion of its fractional part if recurring.
Therefore, when a surd if involved a ratio cannot be represented as a fraction
of two integers.

When we can represent a ratio *exactly* by two integers the quantities are
called **commensurable** else they are called **incommensurable**.

Compounding Ratios
==================
Ratios are compounded by multiplying ratios. The multiplication is done in
simple fashion. Antecedents with antecedents and consequents with consequents.

For example, consider some ratios: :math:`a:b`, :math:`c:d` and :math:`e:f`.
The compounded ratio is :math:`abc:def`.

Duplicating Ratios
==================
When a ratio is compounded with itself once it is called **duplicate ratio**.
If it is compounded twice then it is called **triplicate ratio**. If a square
root is taken then it is called **subduplicate ratio**.

Antecedent vs Consequent
========================
A ratio is of *greater inequality*, or *less inequality* or of *equality*
depending on whether antecedent is greater or smaller or equal to the
consequent.

Common Properties of Ratios
===========================
A ratio of greater inequality is diminished and a ratio of less inequality is
increased if we add same quantity to both its terms.

Let :math:`\frac{a}{b}` be a ratio and we add :math:`x` to both its terms so we
have :math:`\frac{a+x}{b+x}` as our new ratio.

Now,

.. math::

  \frac{a}{b}~-~\frac{a+x}{b+x}~=~\frac{ax-bx}{b(b+x)} \\
  =\frac{x(a-b)}{b(b+x)}.

and :math:`a-b` is positive or negative depending upon the fact whether
:math:`a` is greater or smaller than :math:`b`. Hence,

.. math::

  \text{if}~a> b,~~ \frac{a}{b}>\frac{a+x}{b+x}

  \text{and if}~ a< b,~~ \frac{a}{b}<\frac{a+x}{b+x}

Similarly, it can be proved that a ratio of greater inequality is increased and
a ratio of less inequality is decreased if we subtract same quantity to both
its terms.

When two or more ratios are equal we can denote that by a single symbol which
helps us in many ways.

Consider,

.. math::

  \text{If}~\frac{a}{b}=\frac{c}{d}=\frac{e}{f}= ... ,
  
  \text{each of these ratios}~=\left(\frac{pa^n+qc^n+re^n+ ...}{{pb^n+qd^n+rf^n}+ ...}\right)^\frac{1}{n}

  \text{Let}~\frac{a}{b}=\frac{c}{d}=\frac{e}{f}=...=k;

  \text{the}~a=bk, c=dk, e=fk, ...;

  \text{Hence},~pa^n=pb^nk^n,~qc^n=qd^nk^n,~re^n=rf^nk^n,...;

  \implies \left(\frac{pa^n+qb^n+re^n+ ...}{{pb^n+qd^n+rf^n}+ ...}\right) = \left(\frac{pb^nk^n+qd^nk^n+rf^nk^n+ ...}{{pb^n+qd^n+rf^n}+ ...}\right) = k^n

  \implies \left(\frac{pa^n+qb^n+re^n+ ...}{{pb^n+qd^n+rf^n}+  ...}\right)^\frac{1}{n} = k\\

Similarly, we can say that when a a number of fractions are equal, each of them
is equal to the sum of all the numerators divided by the sum of denominators,

.. math::

  \text{If}~\frac{a}{b}=\frac{c}{d}=\frac{e}{f}= ... ,

  \implies \frac{a+c+e}{b+d+f} = \frac{bk+dk+fk}{b+d+f} = k.

if :math:`\frac{a_1}{b_1}, \frac{a_2}{b_2}, ..., \frac{a_n}{b_n}` be unequal
fractions, of which denominators are all having the same sign, then the
fraction

.. math::
  \frac{a_1 + a_2 + ... + a_n}{b_1 + b_2+ .. +b_n}

lies in magnitude between the greatest and least of them.

Let all the denominators are positive. Let :math:`\frac{a_r}{b_r}` be the least fraction,
and denote it by :math:`k`; then

.. math::

  \frac{a_r}{b_r} = k; \therefore a_r = kb_r;

  \frac{a_1}{b_1} = k; \therefore a_1 = kb_1;

  \frac{a_2}{b_2} = k; \therefore a_2 = kb_2;

  \text{and so on;}

  \therefore \text{by addition,}

  a_1 + a_2 + ... + a_n > (b_1 + b_2 + ... + b_n)k;
  
  \therefore\frac{a_1 + a_2 + ... + a_n}{b_1 + b_2 + ... + b_n}> k; \text{that is,}>\frac{a_r}{b_r}

Similarly, we may prove that

.. math::
  \frac{a_1 + a_2 + ... + a_n}{b_1 + b_2 + ... + b_n}> k; \text{that is,}<\frac{a_s}{b_s}

where :math:`\frac{a_s}{b_s}` is greatest of the given functions.

If we have two equations containing three unknown variables in the first degree
such as

.. math::
  :label: Equation 1

  a_1x + b_1y + c_1z = 0

.. math::
  :label: Equation 2

  a_2x + b_2y + c_2z = 0

Rewriting these we can have following;

.. math::
  a_1\left(\frac{x}{z}\right) + b_1\left(\frac{y}{z}\right) + c_1 =0,

  a_2\left(\frac{x}{z}\right) + b_2\left(\frac{y}{z}\right) + c_2 =0,

Solving these we obtain,

.. math::
  \frac{x}{z}=\frac{b_1c_2-b_2c_1}{a_1b_2-a_2b_1},~~~\frac{y}{z}=\frac{c_1a_2-c_2a_1}{a_1b_2-a_2b_1}

or,

.. math::
  :label: Equation 3

  \frac{x}{b_1c_2-b_2c_1} = \frac{y}{c_1a_2-c_2a_1}=\frac{z}{a_1b_2-a_2b_1}

So we can see that for :math:`x` coefficients of :math:`y` and :math:`z` are
used and for :math:`y`, that of :math:`x` and :math:`z` are used and so on in
a cyclic manner as given in image below:

.. tikz:: Cross multiplication of co-efficients

   \draw (0,0) node {$b2$};
   \draw (1.5,0) node {$c2$};
   \draw (3,0) node {$a2$};
   \draw (4.5,0) node {$b2$};
   \draw (0,1.5) node {$b1$};
   \draw (1.5,1.5) node {$c1$};
   \draw (3,1.5) node {$a1$};
   \draw (4.5,1.5) node {$b1$};
   \draw[->, >=stealth, thick] (0, 0.2) -- (1.5, 1.3);
   \draw[->, >=stealth, thick] (1.5, 0.2) -- (3, 1.3);
   \draw[->, >=stealth, thick] (3, 0.2) -- (4.5, 1.3);
   \draw[->, >=stealth, thick] (0.1, 1.3) -- (1.4, 0.2);
   \draw[->, >=stealth, thick] (1.6, 1.3) -- (2.9, 0.2);
   \draw[->, >=stealth, thick] (3.1, 1.3) -- (4.4, 0.2);
   

Multiply the coefficients as indicated by the arrows. The ascending arrows are
positive and descending ones are negative. This is called the **Rules of Cross
Multiplication**.

Now expanding on previous discussion consider following set of three equations:

.. math::
  :label: Equation 4

  a_1x + b_1y + c_1z=0

.. math::
  :label: Equation 5

  a_2x + b_2y + c_2z=0

.. math::
  :label: Equation 6

  a_3x + b_3y + c_3z=0

So by our previous rule of cross-multiplication between eq. (4) and (5) we have

.. math::

  \frac{x}{z}=\frac{b_1c_2-b_2c_1}{a_1b_2-a_2b_1},~~~\frac{y}{z}=\frac{c_1a_2-c_2a_1}{a_1b_2-a_2b_1}

Now if we take these ratios to be equal to :math:`k` then multiplying and
substituting in equation :eq:`Equation 6` we have

.. math::
  a_3(b_1c_2-b_1c_2)+b_3(c_1a_2-c_2a_1)+c_3(a_1b_2-a_2b_1) = 0

This particular relation is known as the **eliminant** of the given equations.

Problems
========
1.  Find the ratio of ratios :math:`4:7` and :math:`11:5`.
2.  Find the ratio of ratios :math:`13:24` and :math:`34:23`.
3.  Find out that among the following pair of ratios which one is greater in
    the pair:

    (a) :math:`4:5` and :math:`5:6`,
    (b) :math:`34:33` and :math:`68:67` and
    (c) :math:`123:63` and :math:`233:120`.

4.  Find the duplicate, triplicate and subduplicate of following ratios:

    (a) :math:`2(x+3):y-5`
    (b) :math:`a^2+ab+b^2:a^3-b^3`

5.  If :math:`x-2:3(x+7)` has a ratio of :math:`7:9` then find :math:`x`.
6.  Find two numbers in the ratio of :math:`11:17` where one exceeds other by
    126.
7.  What number must be added to each of the terms of :math:`5:13` to make it
    equal to :math:`5:7`?
8.  What number must be subtracted from each of the terms of :math:`33:37` to
    make it equal to :math:`7:9`?
9.  If :math:`x-2:y-3=3:5`, find the ratio of :math:`9x-2y:4x+2y`.
10. If 

.. math::
  \frac{a}{b}=\frac{c}{d}=\frac{e}{f},

prove that

.. math::
  \frac{7a^4b^2+8a^2c^2-5e^4f}{7b^6+8b^2f^2-5e^5} = \frac{a^4}{b^4}.

11. If

.. math::
  \frac{a}{b}=\frac{b}{c}=\frac{c}{d},

prove that :math:`\frac{a}{d}` is equal to

.. math::
  \sqrt{\frac{a^5+b^2c^2+a^3c^2}{b^4c+d^4+b^2cd^2}}..

12. If

.. math::
  \frac{x}{q+r-p}=\frac{y}{r+p-q}=\frac{z}{p=q-r}

prove that

.. math::
  (q-r)x + (r-p)y + (p-q)z = 0.

13. If

.. math::
  \frac{y+z}{pb+qc}=\frac{z+x}{pc+qa}=\frac{x+y}{pa+qb},

prove that

.. math::
  \frac{2(x+y+z)}{a+b+c} = \frac{(b+c)x+(c+a)y+(a+b)z}{ab+bc+ca}.

14. If

.. math::
  \frac{x}{x}=\frac{y}{b}=\frac{z}{c},

prove that

.. math::

  \frac{x^3+a^3}{x^2+a^2}+\frac{y^3+b^3}{y^2+b^2}+\frac{z^3+c^3}{z^2+c^2}=\frac{(x+y+z)^3+(a+b+c)3}{(x+y+z)^2+(a+b+c)^2}.

15. If

.. math::
  \frac{2y+2z-x}{a}=\frac{2z+2x-y}{b}=\frac{2x+2y-z},

prove that

.. math::
  \frac{x}{2b+2c-a}=\frac{y}{2c+2a-b}=\frac{z}{qa+2b-c}.

16. If :math:`(a^2+b^2+c^2)(x^2+y^2+z^2)=(ax+by+cz)^2` show that
:math:`x:a=y:b=z:c`.

17. If

.. math::
  x=\frac{a-b}{a+b},~y=\frac{b-c}{b+c},~z=\frac{c-a}{c+a}

prove that

.. math::
  (1+x)(1+y)(1+z) = (1-x)(1-y)(1-z).

18. If :math:`l(my+nz-lx)=m(nz+lx-my)=n(lx+my-nz),` prove that

.. math::
  \frac{y+z-x}{l}=\frac{z+x-y}{m}=\frac{x+y-z}{n}

19. Show that the eliminant of :math:`ax+by+cz=0, cx+by+az=0, bx+ay+cz=0`, is
:math:`a^3+b^3+c^3-3abc`.

20. Eliminate :math:`x, y, z` from the equations :math:`ax+hy+gz=0,
hx+by+fz=0, gx+fy+cz=0`.

21. If :math:`x=cy+bz, y=az+cx, z=bx+ay`, prove that

.. math::
  \frac{x^2}{1-a^2}=\frac{y^2}{1-b^2}=\frac{z^2}{1-c^2}

22. If :math:`a(y+z)=x, b(z+x)=y, c(x+y)=z` prove that :math:`bc+ca+ab+2abc=1`.

23. Solve the following equations:

    (a) :math:`3x-4y+7z=0, 2x-y-2z=0, 3x^3-y^3+z^3=18`.
    (b) :math:`x+y-z=0, 3x-2y+17z=0, x^3+3y^3+2z^3=167`.
    (c) :math:`4xy-7yz-3zx=0, 4xy-21yz+3zx=0, x+2y+3z=19`.
    (d) :math:`3x^2-2y^2+5z^2=0, 7x^2-3y^2-15z^2=0, 5x-4y+7z=0`.

24. If

.. math::
  \frac{l}{\sqrt{a}-\sqrt{b}}+\frac{m}{\sqrt{b}-\sqrt{c}}+\frac{n}{\sqrt{c}-\sqrt{a}}=0,

  \frac{l}{\sqrt{a}+\sqrt{b}}+\frac{m}{\sqrt{b}+\sqrt{c}}+\frac{n}{\sqrt{c}+\sqrt{a}}=0,

show that

.. math::
  \frac{l}{(a-b)(c-\sqrt{ab})}=\frac{m}{(b-c)(a-\sqrt{bc})}=\frac{n}{(c-a)(b-\sqrt{ac})}

25. Solve the following equations:

    (a) :math:`ax+by+cz=0, bcx+cay+abz=0,xyz+abc(a^3x+b^3y+c^3z)=0`.
    (b) :math:`ax+by+cz=a^2x+b^2y+c^2z=0,x+y+z+(a-b)(b-c)(c-a)=0`.

26. If :math:`a(y+z)=x, b(z+x)=y, c(x+y)=z` prove that

.. math::
  \frac{x^2}{a(1-bc)}=\frac{y^2}{b(1-ca)}=\frac{z^2}{c(1-ab)}.

27. If :math:`ax+hy+gz=0, hx+by+fz=0, gx+fy+cz=0`, prove that

.. math::
  \frac{x^2}{bc-f^2}=\frac{y^2}{ca-g^2}=\frac{z^2}{ab-h^2}~~\text{and}

.. math::
  (bc-f^2)(ca-g^2)(ab-h^2)=(gh-af)(fg-ch)(hf-bg)