CHAPTER
1
Ratio and Proportion, Indices, Logarithms
Concept No. 1. Ratio A ratio is a comparison of the sizes of two or more quantities of the same kind by division. Remarks 1. Both terms of a ratio can be multiplied or divided by the same (non-zero) (non -zero) number. Usually a ratio is expressed in lowest terms (or simplest form). 2. Ratio exists only between quantities of the same kind. 3. Quantities to be compared (by division) division ) must be in the same units. 4. To compare two ratios, convert them into equivalent like fractions. The fraction by which the original quantity is multiplied to get a new quantity is called the multiplying ratio (or factor). Concept No. 2. Inverse Ratio Ra tio One ratio is the inverse of another if their product is 1. Thus a : b is the inverse of b : a and vice –versa. 1. A ratio a : b is said to be of grater inequality if a > b and of less inequality if a > b. 2. The ratio compound of the two ratios a : b and c : d is ac : bd. 3. A ratio compounded of itself is called its duplicate ratio. Thus as : b2 is the duplicate ratio of a : b. Similarly, the triplicate ratio of a : b is a3 : b3. 4. The sub –duplicate ratio of a : b is 3
a
:
3
a
:
b and the sub triplicate ratio of a : b is
b
5. Continued Ratio is the relation (or compassion) between the magnitudes of three or more quantities of the same kind. The continued ratio of three similar quantities a, b, c, is written as a : b : c. Concept No. 3. Proportion An equality of two ratios is called a proportion. Four quantities a, b, c, d are said to be in proportion if a : b = c : d (also written as a : b : : c : d) i.e. if a/b = c/d i.e. if ad = bc. If a : b = c : d then d is called fourth proportional If a : b = c : d are in proportion then a/b = c/d i.e. ad = bc i.e. product of extremes = product of means This is called cross product rule 1
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If a, b, c are in continuous proportion, then the middle term b is called the mean proportional between a and c, a is the first proportional and c is the third proportional. Concept No. 4. Properties of Proportion 1. If a : b = c : d, then ad = bc Pr oof .
a b
c d
; ad ac ( By cross multiplication )
2. If a : b = c : d, then b : a = d : c (Invertendo) a c 1 1 b d Proof. or ,or b d a/b c/d a c 3. If a : b = c : d, then a : c = b : d (Alternendo) a
Pr oof .
b
c d
or , ad bc
Dividing both sides by cd, we get ad cd
bc
a b , or , i. e. a : c b : d cd c d
4. If a : b = c : d, then a + b : b = c + d : d (Componendo) Proof. or ,
a
b a b
b
c
, or ,
d c d
d
a b
1
c d
1
, i. e. a b : b c d : d
5. If a : b = c : d, then a – b : b = c – d : d (Dividendo) a
c
a
, or ,
1
c
1 b d b d a b c d , i . e. a b : b c d : d b d
Proof.
6. If a : b = c : d, then a + b : a – b = c + d : c – d (Componendo and Dividendo) a c a c a b c d Pr oof . , or , 1 1, or .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. ..1 b
Again
a b
d
1 ,
b
c d
1, or
d
a b b
b
c d d
d
.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .2
Dividing (1) and (2) we get a b c d , i. e. a b : a b c d : c d a b c d 7. If a : b = c : d = e : f =……………………………., then each of these ratios (Addendo) is equal (a + c + e +……….) : (b + d + f +……..)
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proof
c
=
=
e
= ...........(say) k b d f a = bk, c = dk, e = fk,……………… Now a c e.......... k ( b d f )........... or
a c e..... b d f .....
Hence , ( a c e .. .. .. .. .. ) : ( b d f .. .. .. .. .. )
Concept No.5. Laws of Indices 1.
2.
am
Ex .
2
am
Ex . 3.
an 3
2
an
2
5
( a m )n
am
2
2
3
a0
n
2
am
( base must be same )
3 2
2
5
n
2
5 3
2
2
a mn
Ex . ( 2 5 ) 2 4.
2
5 2
2
10
1
Concept No.6. Properties of Logarithm 1.
log a mn log a m log a n
2.
loga (m / n) log a m log a n
3.
loga mn n log a m
4.
loga a 1
5.
loga 1 0
6.
log b a loga 1
7.
log b a logc b logc a
8.
log b a log a / logb Multiple Choice Questions
1.
61 72 The value of 2 4 6 7
(a) 0
7 / 2
62 73 3 5 6 7
5 / 2
is
(b) 252
k
3
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(c) 250 2.
(d) 248
The value of
x 2/
7
z 1/
2
x 2/
5
z 2/
3
x 9/ z 1/
7 3
z
x 3/
(a) 1 (c) 0 On simplication
2
32 x y 5x y 3 6 y1 reduces to x 1 10 y 3 15x 6
(a) – 1 (c) 1 y
4.
If
(b) 0 (d) 10 y 1
2
9 .3 ( 3
)
3 x
3
2
27 y
3
1 27
then x – y is given by
(a) – 1 (c) 0 5.
is
(b) – 1 (d) None x 3
3.
5
(b) 1 (d) None
9 x1 / 4 3.3 x Show that 33 .3 3 x
is given by
3 x 6
(a)
2
3
(c) 3
(b) –1
3x
(d) 0 x
6.
Show that
16(32) 2
3x 2
x 1 .4
x 1 x 15 (2) (16)
x 1 5(5)
5
(a) 1 (c) 4 7.
x a Show that b x
Show that
is given by
(b) –1 (d) 0 a b
x b c x
b c
x c a x
(a) 0 (c) 3 8.
2 x
c a
is given by
(b) –1 (d) 1 ( a b ) x
x
a2 b2
( b c ) x
x
b2 c2
( c a ) x
x
c2 a2
reduces to
(a) 1
(b) 0
(c) –1
(d) None
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9.
Show that
1 b c a b x c a
1 c a b c x a b
1 a b c a x bc
(b) 3
(c) –1
(d) None b
5
reduces to
(a) 1 a
c
x b x c x a 10. Show that c a b reduces to x x x
(a) 1
(b) 3
(c) 0
(d) 2
x a 11. Show that b x
( a 2 ab b2 )
x b c x
( b2 bc c2 )
x c a x
(a) 1
(b) – 1
(c) 0
(d) 3
x a 12. Show that b x
a 2 ab b 2
x b c x
b 2 bc c 2
reduces to
x c a x
(b) x 2 ( a
(a) 1 (c) x 2 ( a
3
b3 c3 )
1 2 ( a b c )
z
(d) x 2 ( a
13. On simplification (a)
( c2 ca a 2 )
1 1 z
a b
z a c
1 1 z
b c
z b a
(b)
(c) 1
c 2 ca a2
reduces to
2
3
b 2 c 2 )
b3 c3 )
1 1 z
1 ( a b c )
z
(d) 0
14. If a p b, bq c, cr a the value of pqr is given by (a) 0 (c) –1 b a a b b x x a 15. On simplication a b x ab x b a
(b) 1 (d) None a b redues to
c a
z c b
would reduces to
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(a) 1 (c) 0
(b) –1 (d) None
x ab 16. On simplication 2 2 x a b
a b
x bc 2 2 x b c
3
(a) x 2a (c) x 2 ( a
3
bc
x ca 2 2 x c a
(b) x 2a b 3 c 3 )
17. If 2 a 3b 6c then
1
a
1
b
1
c
ca
reduces to
3
(d) x 2 ( a
3
b 3 c 3 )
reduce to
(a) 0 (c) 3
(b) 2 (d) 1
18. If 2 a 4 b 8 c and abc 288 then the value 12a 1 4b 18c is given by (a) (c)
1 8
11 96
19. If a (a) 3 (c) 2
(b)
3
(d)
2 1
3
1
8 11 96
3 2 1 the the value of a 3a 2 is
(b) 0 (d) 1
20. If a 31 / 4 31/ 4 and b 31/ 4 31/ 4 then the value of 3 (a 2 b 2 )2 is (a) 67 (c) 64
(b) 65 (d) 62 Answer Sheet
1 7 13 19
(b) (d) (c) (b)
2 8 14 20
(a) (a) (b) (c)
3 9 15
(c) (a) (a)
4 10 16
(b) (a) (c)
5 11 17
(a) (a) (a)
6 12 18
(a) (c) (c)
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