Download Ratio and Proportion

Ratio and Proportion
Quantitative Aptitude & Business Statistics
Ratio and Proportion
Ratio: A ratio is a comparison of the sizes of two
or more quantities of the same kind of division.
If a and b are two quantities of the same kind by
division.
3

Ratios can be written, or expressed, three (3)
different ways.


1. a to b
2. a:b

3.
a
b
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
4

a’ is called the first term or antecedent
and b’ is called the second term or
consequent.

Because a ratio is a quotient (fraction), its
denominator cannot be zero.
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Inverse Ratio



5
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 greater inequality if
a>b and less inequality if a<b.
2.The ratio compound of the two ratios a:b
and c:d is ac:bd
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
3.A ratio is said to be compounded itself is
called duplicate ratio.
Thus a2:b2 is the duplicate ratio of a:b
Similarly ,the triplicate ratio of a:b is a3:b3
For example
Duplicate ratio of 2:3 is 4:9
Triplicate ratio of 2:3 is 8:27

6
Quantitative Aptitude & Business
Statistics: Ratio and Proportion

4.The sub duplicate ratio of a:b is
a: b

5.The sub-triplicate ratio of a:b is
3
3
a : b,duplicate ratio of 2:3 is
For example
Triplicate ratio of 8:27 is
, 2:3
3
7
2: 3
8 : 3 27
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
5.If the ratio of two similar quantities can be
expressed as a ratio of two integers ,the
Quantities are said to be commensurable,
otherwise, they are said to be
incommensurable 3 : 2
cannot be expressed as the ratio of two
integers.
8
Quantitative Aptitude & Business
Statistics: Ratio and Proportion

9
6.Continued ratio is the relation (or
comparison) between the two magnitudes
of three magnitudes of three or more
quantities of the same kind. the continued
ratio of three similar Quantities a,b and c
is a:b:c
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
 For
example Continued ratio of
Rs.200,Rs.400 and Rs.600 is
Rs200:Rs400:Rs.600.=
1:2:3
10
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Example-1

11
The monthly incomes of two persons are in
the ratio of 4:5 their monthly expenditure are
in the ratio 7:9.If each saves Rs.50per month
,Find their monthly incomes.
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Solution
Let the monthly incomes are 4X and 5X
If each saves Rs.50.Per month
Then expenditures are Rs.(4x-50)and (5x-50)

Then X=100
12
4 x − 50 7
=
5 x − 50 9
Quantitative Aptitude & Business
Statistics: Ratio and Proportion

13
Hence monthly incomes of the two
persons are Rs.4X100(Rs.400)and
Rs.5x100(Rs.500)
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Example -2

14
Find in what ratio will the total wages of
the workers of a factory be increased or
decreased if there be a reduction in the
number of workers in the ratio 15:11and
increment in their wages in the ratio
22:25
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Solution



15
Let x be the original number of workers
and Rs.Y the average wages per workers
Then the total wages before
changes=Rs.xy
After increment ,the wages per
workers=Rs.(25y)/22
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
The total wages after changes
=(11/15 X) Rs.(25y)/22= Rs.5xy/6.
 Hence the required ratio in which the total
wages decrease is xy:5xy/6=6:5

16
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Proportion
An equality of two ratios is called Proportion .
Four quantities a,b,c,d are said to be in
proportion a:b=c:d (also written as a:b :: c:d
a:b is as to c:d) if a/b =c/d i.e if ad=bc The
quantities are a,b,c,d are terms of the
proportion ;a,b,c and d are called its first
,second ,third and fourth terms respectively.

17
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
First and fourth terms called are called extremes.
The second and third terms are called means (or
middle terms)
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.
18
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Three quantities a,b,c are same kind (in same
units) are said to be continuous proportion) if
a:b=b:c i.e b2 =ac If a,b ,c are continuous
proportion ,then middle term b’ is called then
the middle term b is called mean proportional
between a and c ,a is called the first
proportional and c is third proportional .
19
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Thus, b is the mean proportional between a
and c ,then b2 =ac i.e
b= ac
20
Quantitative Aptitude & Business
Statistics: Ratio and Proportion

21
In a ratio a:b ,both quantities must be of
the same kind while in a proportion
a:b=c:d ,all the quantities need not be
same type. The first two quantities of
same kind and last two quantities should
be same kind.
Quantitative Aptitude & Business
Statistics: Ratio and Proportion




22
Properties of Proportion
if a:b =c:d ,then ad=bc
If a:b=c:d then b :a=d :c (invertendo)
if a:b=c:d then a :c=b :d (Alternendo)
if a:b =c:d ,then a + b: b=c+d :d (componendo)
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
if a:b =c:d
then a - b: b=c - d :d (Dividendo)
 if a:b =c:d then
a + b: a - b =c+d :c-d
(componendo and Dividendo)

23
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
if a:b=c:d=e:f=………….,then each of these
ratios (Addendo) is equal to (a + c +e+….):(b
+d+ f+….)
 if a:b=c:d=e :f=………….,then each of these
ratios (Subtrahendo) is equal to
(a- c –e-….):(b –d- f-….)

24
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Example -1
Find the value of x if 10/3:x:: 5/2:5/4
Using the cross product rule
X*5/2=(10/3)5/4
Or X=(10/3)*5/4=5/3

25
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Example2
Find the fourth proportional to
2/3 ,3/7,4
Solution: Let the fourth proportional be X
then 2/3,3/7,4 and x are in proportion.
Using the cross product rule,
(2/3)*x=(3*4)/7
Or X=(3*4*3)/7=18/7

26
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Example3
If a:b=c:d =2.5:1.5,what are the values of
ad: bc and a +c : b+d
Solution:
we have a/b=c /d =2.5/1.5……..(1)
From (1) ad=bc or ad/ bc=1:1
Again from (1) a/b=c /d=a + c/ b+d
a+c/b+d=2.5/1.5=5/3 =5:3

27
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Example:4
If a/3 =b/4 =c/7 ,then prove that
a+b+c/c =2
 Solution :
We have a/3=b/4=c/7=a+b+c/3+4+7
a+b+c/14=c/7 or
a+ b +c /c=14/7=2

28
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Indices

29
If n’ is a positive integer, and ‘a’ is a real
number ,i.e n€N and a € R (where ‘n’ is
the set of all positive numbers and R is
the set of all real numbers), a’ is used to
continue product of ‘n ‘factors each equal
to ‘a’ as shown as bellow:
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
an=a X a X a…….to n factors
Here an is a power of a’ whose base is ‘a and
index or power is ‘n’.
30
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Law’s of Indices

Law.1: am X an =a m+n, where m and n are
positive integers
m
a
n
 Law.2:
a
=a m-n where m and
n are positive integers
31
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
(a )
m n

Law.3:
=a
mn
where m and n are positive integers
Law.4:
where n takes all positive values.

(ab ) = a .b
n
32
n
n
Quantitative Aptitude & Business
Statistics: Ratio and Proportion

Find x ,if
X X = (X X )
1
2

33
3
2 X
X (X ) = (X )
Solution
X
X
1
1+ 2
3
2
= (X ) = (X )
3
.x
2
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
(If bases are equal ,then power is also equal)
ie 3/2=3/2* x
X =1

34
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Example
x
 b
x
a

35



a +b
x 
. c 
x 
b
b+c
x 
. a 
x 
c
c+a
=1
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Example
x 
 m 
x 
l

36
l 2 + lm + m 2
2
2
m
+
mn
+
n
m
x 
. n 
x 
x
. l
x
n



n 2 + nl + l 2
=1
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
1
3

If
X =3 +3
1
−
3
Then 3X3-9x=10
37
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Solution
(a + b) = a + b + 3ab(a + b)
3
1
3
3
1
−
3 3
3
1
3 3
1
−
3 3
1
3
1
−
3
1
3
1
−
3
(3 + 3 ) = (3 ) + (3 ) + 3.3 .3 (3 + 3 )
38
X
3
X
3
X
3
1
= 3+
+ 3x
3
= 10 + 9 x
− 9 x = 10
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Logarithms

39
The logarithm of a number to a given base
is the index or the power to which the
base must be raised to produce the
number ,i.e to make it equal to the given
number. If there are three quantities
indicated by say a, X and n, they are
related as follows:
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
If ax=n, then X is said to be the logarithm of
the numbers to the base ‘a', symbolically
it can be expressed as follows
log an=X
40
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Definition of Logarithms
Suppose b>0 and b≠1,
there is a number ‘p’
such that:

logb n = p if and only if b = n
p
41
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Fundamental Laws of Logarithm

1. Logarithm of the product of two numbers is
equal to the sum of the logarithms of the
numbers to the same base ,i.e
loga mn=loga m +loga n
42
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Fundamental Laws of Logarithm

2.Logarithm of the Quotient of two numbers
is equal to the difference of the logarithms of
the numbers to the same base ,i.e
m=
log a
n
43
log a m − log a n
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Fundamental Laws of Logarithm

3. Logarithm of the number is raised to the
power equal to the index of the power raised
by the logarithms of the number to the same
base ,i.e
log a m = n log a m
n
44
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Why Logarithms
Logarithms were originally
developed to simplify complex arithmetic
calculations.
 They were designed to transform
multiplicative processes into additive ones.

45
Quantitative Aptitude & Business
Statistics: Ratio and Proportion

46
Logarithm
Tables
The
Logarithms
of a number consists of two
parts ,the whole part or integral part is called the
characteristic and the decimal part is called the
mantissa. Where the former can be known by
mere inspectiom,the later has to be obtained
from logarithms tables.
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Characteristic

47
The Characteristic of the logarithmic of
any number greater than 1 with positive
and is one less than the number of digits
to the left the decimal point in the given
number.
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Characteristic

48
The Characteristic of the logarithm of any
number less than one (1)is negative and
numerically one more than the number of
Zeros to the right of decimal point .If there
is no Zero then obviously it will -1.
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Examples for Characteristic
Number
49
Characteristic
37
1(2-1)
4623
3(4-1)
6.21
0(1-1)
0.07
-2(number of Zeros on)
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Examples for Characteristic
50
Number
Characteristic
0.00507
-3
0.000670
-4
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Mantissa

51
The mantissa is the fractional part of the
logarithm of a given number
Number
Mantissa
Logarithm
Log 4597
=6625(6618+7 =3.6625
(Mean
Difference)
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Anti logarithms
If X is the logarithms of a given number n
with a given base then n is called the
antilogarithm (anti log) of X to that base .
 This can be expressed as follows
If log a n =X
Then n = anti log X

52
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
For Example
If log 61720=4.7904
Then 61720=anti log 4.7904

53
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Example-1
Write 2 = 8 in logarithmic form.
3
Solution:
log2 8 = 3
We read this as: ”the log
base 2 of 8 is equal to 3”.
54
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Example-2
Write 4 = 16 in logarithmic form.
2
Solution:
log4 16 = 2
Read as: “the log base 4 of 16
is equal to 2”.
55
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
1
Write 2 = in logarithmic form.
8
−3
Solution:
1
log2 = − 3
8
1
Read as: "the log base 2 of is equal to -3".
8
56
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Solve:
log3 (4 x + 10) = log3 (x +1)
Since the bases are both ‘3’ we simply set the
arguments equal.
4x +10 = x +1
3x +10 = 1
3x = − 9
x=−3
57
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Example
Solve:
Solution:
log8 (x −14) = log8 (5x)
2
Since the bases are both ‘8’ we
simply set the arguments equal.
x −14 = 5x
x 2 − 5x −14 = 0
(x − 7)(x + 2) = 0
2
Factor
(x − 7) = 0 or (x + 2) = 0
x = 7 or x = −2
58
continued on the
Quantitative
Aptitude & Business
next
page
Statistics: Ratio and Proportion
Example
continued
2
8
Solve: log (x − 14) = log8 (5x)
Solution:
x = 7 or x = −2
59
Quantitative Aptitude & Business
Statistics: Ratio and Proportion


60
It appears that we have 2 solutions here.
If we take a closer look at the definition of
a logarithm however, we will see that not
only must we use positive bases, but also
we see that the arguments must be
positive as well. Therefore -2 is not a
solution.
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Example

If log a bc=X, log bca=y, log cab=z prove that
1
1
1
+
+
=1
x +1 y +1 z+1
61
Quantitative Aptitude & Business
Statistics: Ratio and Proportion





62
X+1= loga bc+ logaa=log a abc
Y+1= logb cac+ log bb=log a abc
Z+1= log cab+log cc=log a abc
Hence
1
1
1
+
+
x +1 y +1 z +1
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
1
1
1
+
+
log a abc log b abc log c abc


63
log abc a+ log abc b + log abc c
=log abc abc =1
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
Multiple Choice Questions
64
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
1________ is the mean proportional
between 12x2 and 27y2.
A) 18xy
B) 81 xy
C) 8 xy
D) 19.5 xy
65
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
1________ is the mean proportional
between 12x2 and 27y2.
A) 18xy
B) 81 xy
C) 8 xy
D) 19.5 xy
66
Quantitative Aptitude & Business
Statistics: Ratio and Proportion





67
2.log 32/4 is equal to
A) log 32/log4
B) log 32 – log4
C)23
D) None of these
Quantitative Aptitude & Business
Statistics: Ratio and Proportion





68
2.log 32/4 is equal to
A) log 32/log4
B) log 32 – log4
C)23
D) None of these
Quantitative Aptitude & Business
Statistics: Ratio and Proportion





69
3.The logarithm of a number consists of
two parts, the whole part or the integral
part is called the ______ and the decimal
part is called the _______.
A) Characteristic, Number
B) Characteristic, Mantissa
C) Mantissa, Characteristic
D) Number, Mantissa
Quantitative Aptitude & Business
Statistics: Ratio and Proportion





70
3.The logarithm of a number consists of
two parts, the whole part or the integral
part is called the ______ and the decimal
part is called the _______.
A) Characteristic, Number
B) Characteristic, Mantissa
C) Mantissa, Characteristic
D) Number, Mantissa
Quantitative Aptitude & Business
Statistics: Ratio and Proportion





71
4.The value of (8/27)1/3 is
A) 2/3
B) 3/2
C) 2/9
D) None of these
Quantitative Aptitude & Business
Statistics: Ratio and Proportion





72
4.The value of (8/27)1/3 is
A) 2/3
B) 3/2
C) 2/9
D) None of these
Quantitative Aptitude & Business
Statistics: Ratio and Proportion





73
5.The mean proportional between 1.4 gms
and 5.6 gms is
A) 28 gms.
B) 2.8 gms
C) 3.2 gms.
D) None of these.
Quantitative Aptitude & Business
Statistics: Ratio and Proportion





74
5.The mean proportional between 1.4 gms
and 5.6 gms is
A) 28 gms.
B) 2.8 gms
C) 3.2 gms.
D) None of these.
Quantitative Aptitude & Business
Statistics: Ratio and Proportion





75
6.The ratio compound of two ratios 4: 3
and 7: 3 is
A) 12:21
B) 28:9
C) 9:28
D) None of these
Quantitative Aptitude & Business
Statistics: Ratio and Proportion





76
6.The ratio compound of two ratios 4: 3
and 7: 3 is
A) 12:21
B) 28:9
C) 9:28
D) None of these
Quantitative Aptitude & Business
Statistics: Ratio and Proportion





77
7.The ratio of two quantities is 5: 9. If the
antecedent is 25, the consequent is
A) 9
B) 45
c) 40
D)None of these
Quantitative Aptitude & Business
Statistics: Ratio and Proportion





78
7.The ratio of two quantities is 5: 9. If the
antecedent is 25, the consequent is
A) 9
B) 45
c) 40
D) None of these
Quantitative Aptitude & Business
Statistics: Ratio and Proportion





79
8.If p: q = r: s, implies q: p = s: r, then the
process is called
A) Componendo
B) Invertendo
C) Alternendo.
D) Dividendo
Quantitative Aptitude & Business
Statistics: Ratio and Proportion





80
8.If p: q = r: s, implies q: p = s: r, then the
process is called
A) Componendo
B) Invertendo
C) Alternendo.
D) Dividendo
Quantitative Aptitude & Business
Statistics: Ratio and Proportion





81
9. log (3 × 5 ×7)2 is equal to __________
A) 2(log 3 + log 5 + log7)
B) log (2×3×5×7)
C) 2(log 3 – log 5 – log 7)
D) None of these
Quantitative Aptitude & Business
Statistics: Ratio and Proportion





82
9. log (3 × 5 ×7)2 is equal to __________
A) 2(log 3 + log 5 + log7)
B) log (2×3×5×7)
C) 2(log 3 – log 5 – log 7)
D) None of these
Quantitative Aptitude & Business
Statistics: Ratio and Proportion





83
10. The triplicate ratio of 4: 5 is ________.
A) 125: 64
B)16:25
C)64:125
D) None of these
Quantitative Aptitude & Business
Statistics: Ratio and Proportion





84
10. The triplicate ratio of 4: 5 is ________.
A) 125: 64
B)16:25
C)64:125
D) None of these
Quantitative Aptitude & Business
Statistics: Ratio and Proportion
THE END
Ratio and Proportion