**I. RATIO:**The ratio of two quantities a and b in the same units, is the fraction a/b and we write it as a:b.

In the ratio a:b, we call a as the

**first term or antecedent**and b, the**second term or consequent.****Ex.**The ratio 5: 9 represents 5/9 with antecedent = 5, consequent = 9.

**Rule:**The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio.

**Ex.**4: 5 = 8: 10 = 12: 15 etc. Also, 4: 6 = 2: 3.

2.

**PROPORTION:***The equality of two ratios is called proportion.*
If

*a:*b = c: d, we write,*a:*b:: c : d and we say that*a,*b, c, d are in proportion . Here*a*and d are called extremes, while b and c are called mean terms.
Product of means = Product of
extremes.

Thus, a: b:: c : d <=> (b x c) = (a x d).

3. (i) Fourth Proportional: If

*a*: b = c: d, then d is called the fourth proportional
to

*a,*b, c.*(ii)*Third Proportional: If

*a*: b = b: c, then c is called the third proportional to

a
and b.

(iii) Mean
Proportional: Mean proportional between

*a*and*b*is*square root of ab*
4. (i) COMPARISON OF RATIOS:

We say that

*(a*:*b)*> (c:*d)*<=> (a/b)>(c /d).
(ii)
COMPOUNDED RATIO:

The
compounded ratio of the ratios (a:

*b),*(c:*d),*(e : f) is*(ace*:*bdf)*
5. (i)

*Duplicate ratio*of*(a*: b) is*(a*: b^{2}^{2}).*(ii) Sub-duplicate ratio*of (a :

*b)*is

*(*√

*a*: √

*b).*

*(iii)Triplicate ratio*of (a :

*b)*is

*(*:

^{a3}^{b3}).

*(iv)*Sub-triplicate ratio of (a :

*b)*is

*(a*⅓

*:*

*b*⅓

*).*

*(v)*If (a/b)=(c/d), then ((a+b)/(a-b))=((c+d)/(c-d))

**(Componendo and dividendo)**

6. VARIATION:

(i) We say that x is directly proportional to

*y,*if x =*ky*for some constant k and
we write,
x µ

*y.**(ii)*We say that x is inversely proportional to

*y,*if xy =

*k*for some constant k and

we write, x∞(1/y)

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