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Ratio and Proportion
Quantitative Aptitude & Business Statistics
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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.
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Ratios can be written, or expressed, three (3)different ways.
1. a to b
2. a:b
3.b
a
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a is called the first term or antecedent
and b is called the second term orconsequent.
Because a ratio is a quotient (fraction), its
denominator cannot be zero.
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Inverse Ratio
One ratio is the inverse of another if their
product is 1.Thus a:b is the inverse of b:aand vice versa.
1. A ratio a:b is said to be greater inequality if
a>b and less inequality if a
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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
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4.The sub duplicate ratio of a:b is
5.The sub-triplicate ratio of a:b is
For example ,duplicate ratio of 2:3 is
Triplicate ratio of 8:27 is , 2:3
b:a
33: ba
3:2
33 27:8
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5.If the ratio of two similar quantit ies can be
expressed as a ratio of two integers ,theQuantities are said to be commensurable,
otherwise, they are said to be
incommensurable
cannot be expressed as the ratio of twointegers.
2:3
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6.Continued ratio is the relation (or
comparison) between the two magnitudesof 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
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For example Continued ratio of
Rs.200,Rs.400 and Rs.600 isRs200:Rs400:Rs.600.=
1:2:3
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Example-1
The monthly incomes of two persons are in
the ratio of 4:5 their monthly expenditure arein the ratio 7:9.If each saves Rs.50per month
,Find their monthly incomes.
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Solution
Let the monthly incomes are 4X and 5X
If each saves Rs.50.Per monthThen expenditures are Rs.(4x-50)and (5x-50)
Then X=100 9
7
505
504
=
x
x
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Hence monthly incomes of the two
persons are Rs.4X100(Rs.400)andRs.5x100(Rs.500)
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Example -2
Find in what ratio will the total wages of
the workers of a factory be increased ordecreased if there be a reduction in the
number of workers in the ratio 15:11and
increment in theirwages in the ratio
22:25
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Solution
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
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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
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Proportion An equality of two ratios is called Proportion .
Four quantities a,b,c,d are said to be inproportion 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 Thequantities are a,b,c,d are terms of theproportion ;a,b,c and d are called its first,second ,third and fourth terms respectively.
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First and fourth terms called are called extremes.
The second and third terms are called means (ormiddle 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 meansThis is called cross product rule.
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Three quantit ies a,b,c are same kind (in sameunits) are said to be continuous proportion) ifa:b=b:c i.e b2 =ac If a,b ,c are continuousproportion ,then middle term b is called thenthe middle term b is called mean proportionalbetween a and c ,a is called the firstproportional and c is third proportional .
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Thus, b is the mean proportional between a
and c ,then b
2
=ac i.eb= ac
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In a ratio a:b ,both quantities must be of
the same kind while in a proportiona:b=c:d ,all the quantities need not be
same type. The first two quantities of
same kind and last two quantit ies should
be same kind.
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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)
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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)
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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 theseratios (Subtrahendo) is equal to
(a- c e-.):(b d- f-.)
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Example -1
Find the value of x if 10/3:x:: 5/2:5/4
Using the cross product ruleX*5/2=(10/3)5/4
Or X=(10/3)*5/4=5/3
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Example2
Find the fourth proportional to
2/3 ,3/7,4Solution: Let the fourth proportional be X
then 2/3,3/7,4 and x are in proport ion.
Using the cross product rule,
(2/3)*x=(3*4)/7
Or X=(3*4*3)/7=18/7
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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
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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
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Indices
If n is a positive integer, and a is a real
number ,i.e nN 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 ofn factors each equal
to a as shown as bellow:
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an=a X a X a.to n factors
Here an
is a power of awhose base is a andindex or power is n.
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Laws of Indices
Law.1: am X an =a m+n, where m and n are
positive integers
Law.2: =am-n
where m and
n are positive integers
n
m
a
a
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( )mn
n
m aa =
Law.3:
where m and n are positive integers
Law.4:
where n takes all positive values.
( ) nnn b.aab =
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Find x ,if
Solution
XXXXX )(=
XXXX )()( 2
321
=
x
XXX.
2
3
2
3
2
11 )()( ==+
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(If bases are equal ,then power is also equal)
ie 3/2=3/2* xX =1
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Example
=1
ac
a
ccb
c
bba
b
a
xx
xx
xx
+++
..
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Example
=1
222222
..
lnln
l
nnmnm
n
mmlml
m
l
x
x
x
x
x
x++++++
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If
Then 3X3-9x=10
3
1
3
1
33
+=X
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Solution
)33(3.3.3)3()3()33(
)(3)(
3
1
3
1
3
1
3
1
33
1
33
1
33
1
3
1
333
+++=+
+++=+ baabbaba
109
910
33
13
3
3
3
=
+=
++=
xX
xX
xX
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Logarithms
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 arerelated as follows:
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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
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Definition of Logarithms
Suppose b>0 and b1,
there is a number psuch that:
logb n = p if and only if bp = n
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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
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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
=
n
mlog
a
nlogmlogaa
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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
mlognmlog an
a =
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Why Logarithms
Logarithms were originally
developed to simplify complex arithmeticcalculations.
They were designed to transform
multiplicative processes into additive ones.
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Logarithm Tables The Logarithms of a number consists of twoparts ,the whole part or integral part is called thecharacteristic and the decimal part is called the
mantissa. Where the former can be known by
mere inspectiom,the later has to be obtainedfrom logarithms tables.
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Characteristic
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.
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Characteristic
The Characteristic of the logarithm of anynumber less than one (1)is negative and
numerically one more than the number ofZeros to the right of decimal point .If thereis no Zero then obviously it wil l -1.
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Examples for CharacteristicNumber Characteristic
37
4623
6.21
0.07
1(2-1)
3(4-1)
0(1-1)
-2(number of Zeros on)
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Examples for Characteristic
Number Characteristic
0.00507
0.000670
-3
-4
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Mantissa
The mantissa is the fractional part of the
logarithm of a given number
Number Mantissa Logarithm
Log 4597 =6625(6618+7(Mean
Difference)
=3.6625
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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
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For Example
If log 61720=4.7904Then 61720=anti log 4.7904
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Example-1
Solution: log2 8 = 3
3Write 2 8 in logarithmic form.=
We read this as: the logbase 2 of 8 is equal to 3 .
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Example-2Write 4
2= 16 in logarithmic form.
Solution:
log4 16 = 2
Read as: the log base 4 of 16
is equal to 2 .
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Write 2 3
=18
in logarithmic form.
log2 18
= 3
Solution:
1Read as: "the log base 2 of is equal to -3".8
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Solve: log3(4x+10)= log3(x+1)
Since the bases are both 3 we simply set the
arguments equal.
4x +10 = x +13x + 10 = 1
3x = 9
x = 3
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Example
Solve: log8 (x2 14) = log8 (5x)
Solution: Since the bases are both 8 wesimply set the arguments equal.
x2 14 = 5x
x2
5x
14 = 0(x 7)(x + 2) = 0Factor
(x 7) = 0 or (x + 2) = 0x = 7 or x = 2 continued on the
next page
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Example
continued
Solve: log8 (x2
14) = log8 (5x)Solution:
x = 7 or x = 2
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It appears that we have 2 solutions here.
If we take a closer look at the definit ion ofa logarithm however, we will see that not
only must we use posit ive bases, but also
we see that the arguments must be
positive as well. Therefore -2 is not asolution.
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Example If log a bc=X, log bca=y, log cab=z prove that
11z
1
1y
1
1x
1=
++
++
+
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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
11
11
11
+
+
+
+
+ zyx
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log abc a+ log abc b + log abc c
=log abc abc =1
abcabcabc cba log
1
log
1
log
1++
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Multiple Choice Questions
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1________ is the mean proportional
between 12x2 and 27y2.
A) 18xy
B) 81 xy
C) 8 xy
D) 19.5 xy
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1________ is the mean proportional
between 12x2 and 27y2.
A) 18xy
B) 81 xy
C) 8 xy
D) 19.5 xy
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2.log 32/4 is equal to
A) log 32/log4
B) log 32 log4
C)23
D) None of these
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2.log 32/4 is equal to
A) log 32/log4
B) log 32 log4
C)23
D) None of these
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3.The logarithm of a number consists oftwo parts, the whole part or the integral
part is called the ______ and the decimalpart is called the _______.
A) Characteristic, Number
B) Characteristic, Mantissa
C) Mantissa, Characteristic
D) Number, Mantissa
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3.The logarithm of a number consists oftwo parts, the whole part or the integral
part is called the ______ and the decimalpart is called the _______.
A) Characteristic, Number
B) Characteristic, Mantissa
C) Mantissa, Characteristic
D) Number, Mantissa
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4.The value of (8/27)1/3 is
A) 2/3
B) 3/2
C) 2/9
D) None of these
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4.The value of (8/27)1/3 is
A) 2/3
B) 3/2
C) 2/9
D) None of these
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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.
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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.
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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
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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
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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
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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
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8.If p: q = r: s, implies q: p = s: r, then the
process is called
A) Componendo
B) Invertendo
C) Alternendo.
D) Dividendo
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8.If p: q = r: s, implies q: p = s: r, then the
process is called
A) Componendo
B) Invertendo
C) Alternendo.
D) Dividendo
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9. log (3 5 7)2 is equal to __________
A) 2(log 3 + log 5 + log7)
B) log (2357)
C) 2(log 3 log 5 log 7)
D) None of these
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9. log (3 5 7)2 is equal to __________
A) 2(log 3 + log 5 + log7)
B) log (2357)
C) 2(log 3 log 5 log 7)
D) None of these
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10. The triplicate ratio of 4: 5 is ________.
A) 125: 64
B)16:25
C)64:125
D) None of these
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10. The triplicate ratio of 4: 5 is ________.
A) 125: 64
B)16:25
C)64:125
D) None of these
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THE END
Ratio andProportion
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