**1. 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)

8.A:B & B:C & C:D Then A:B:C:D = ?

# SOLVED PROBLEMS

**Ex. 1. If a : b = 5 : 9 and b : c = 4: 7, find a : b : c.**

**Sol.**a:b=5:9 and b:c=4:7= (4X9/4): (7x9/4) = 9:63/4

a:b:c = 5:9:63/4 =20:36:63.

**Ex. 2. Find:**

**(i) the fourth proportional to 4, 9, 12;**

**(ii) the third proportional to 16 and 36;**

**(iii) the**

**mean proportional between 0.08 and 0.18.**

**Sol.**

i) Let
the fourth proportional to 4, 9, 12 be x.

Then, 4 : 9 : : 12 : x ó 4
x x=9x12 ó
X=(9 x 12)/14=27;

Fourth proportional to 4, 9, 12 is 27.

(ii) Let
the third proportional to 16 and 36 be x.

Then, 16 : 36 : : 36 : x ó16 x x = 36 x 36 ó x=(36 x 36)/16 =81

Third proportional to 16 and 36 is 81.

(iii) Mean
proportional between 0.08 and 0.18

Ö0.08
x 0.18 =Ö8/100
x 18/100= Ö144/(100
x 100)=12/100=0.12

**Ex. 3. If x : y = 3 : 4, find (4x + 5y) : (5x - 2y).**

**Sol.**X/Y=3/4 ó (4x+5y)/(5x+2y)= (4( x/y)+5)/(5 (x/y)-2) =(4(3/4)+5)/(5(3/4)-2)

=(3+5)/(7/4)=32/7

**Ex. 4. Divide Rs. 672 in the ratio 5 : 3**.

**Sol.**Sum of ratio terms = (5 + 3) = 8.

First part = Rs. (672 x (5/8)) = Rs. 420; Second part = Rs. (672 x
(3/8)) = Rs. 252.

**Ex. 5. Divide Rs. 1162 among A, B, C in the ratio 35 : 28 : 20.**

**Sol.**Sum of ratio terms = (35 + 28 + 20) = 83.

A's share = Rs. (1162 x (35/83))= Rs. 490; B's share = Rs. (1162
x(28/83))= Rs. 392;

C's share = Rs. (1162 x (20/83))= Rs. 280.

**Ex. 6. A bag contains 50 p, 25 P and 10 p coins in the ratio 5: 9: 4, amounting to**

**Rs. 206. Find the**

**number of coins of each type.**

**Sol.**Let the number of 50 p, 25 P and 10 p coins be 5x, 9x and 4x respectively.

(5x/2)+( 9x/ 4)+(4x/10)=206ó 50x
+ 45x + 8x = 4120ó1O3x = 4120 óx=40.

Number of 50 p coins = (5 x 40) = 200; Number of 25 p coins = (9 x 40) =
360;

Number of 10 p coins = (4 x 40) = 160.

**Ex. 7. A mixture contains alcohol and water in the ratio 4 : 3. If 5 litres of water is added to the mixture, the ratio becomes 4: 5. Find the quantity of alcohol in the given mixture**

**Sol.**Let the quantity of alcohol and water be 4x litres and 3x litres respectively

4x/(3x+5)=4/5 ó20x=4(3x+5)ó8x=20 óx=2.5

Quantity of alcohol = (4 x 2.5) litres = 10 litres.