ratio and proportion · 2016-06-24 · 39 statistics: ratio and proportion . logarithms the...
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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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Quantitative Aptitude & Business Statistics: Ratio and Proportion 3
Ratios can be written, or expressed, three (3) different ways.
1. a to b 2. a:b
3.
ba
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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.
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 5
Inverse Ratio
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
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 6
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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Quantitative Aptitude & Business Statistics: Ratio and Proportion 7
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 : ba3:2
33 27:8
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 8
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
cannot be expressed as the ratio of two integers.
2:3
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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
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 10
For example Continued ratio of Rs.200,Rs.400 and Rs.600 is Rs200:Rs400:Rs.600.=
1:2:3
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 11
Example-1
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.
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 12
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
97
505504
=−−
xx
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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)
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 14
Example -2
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
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 15
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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Quantitative Aptitude & Business Statistics: Ratio and Proportion 16
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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Quantitative Aptitude & Business Statistics: Ratio and Proportion 17
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.
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 18
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.
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 19
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 .
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 20
Thus, b is the mean proportional between a and c ,then b2 =ac i.e
b=
ac
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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.
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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)
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 23
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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Quantitative Aptitude & Business Statistics: Ratio and Proportion 24
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-….)
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 25
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
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 26
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
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 27
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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Quantitative Aptitude & Business Statistics: Ratio and Proportion 28
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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Quantitative Aptitude & Business Statistics: Ratio and Proportion 29
Indices
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:
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 30
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’.
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 31
Law’s of Indices
Law.1: am X an =a m+n, where m and n are positive integers
Law.2: =a m-n where m and
n are positive integers
n
m
aa
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 32
( ) mnnm 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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Quantitative Aptitude & Business Statistics: Ratio and Proportion 33
Find x ,if
Solution
XXXXX )(=
XXXX )()( 23
21
=
xXXX
.23
23
21
1 )()( ==+
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 34
(If bases are equal ,then power is also equal) ie 3/2=3/2* x X =1
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 35
Example
=1
ac
a
ccb
c
bba
b
a
xx
xx
xx
+++
..
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 36
Example
=1
222222
..lnln
l
nnmnm
n
mmlml
m
l
xx
xx
xx
++++++
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 37
If
Then 3X3-9x=10
31
31
33−
+=X
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 38
Solution
)33(3.3.3)3()3()33(
)(3)(
31
31
31
31
331
331
331
31
333
−−−−+++=+
+++=+ baabbaba
109910
3313
3
3
3
=−
+=
++=
xXxX
xX
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 39
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 are related as follows:
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 40
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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Quantitative Aptitude & Business Statistics: Ratio and Proportion 41
Definition of Logarithms
Suppose b>0 and b≠1, there is a number ‘p’ such that:
logb n = p if and only if bp = n
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 42
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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Quantitative Aptitude & Business Statistics: Ratio and Proportion 43
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 nlogmlog aa −
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 44
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 a
na =
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 45
Why Logarithms
Logarithms were originally developed to simplify complex arithmetic
calculations. They were designed to transform multiplicative processes into additive ones.
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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.
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 47
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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Quantitative Aptitude & Business Statistics: Ratio and Proportion 48
Characteristic
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.
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 49
Examples for Characteristic Number 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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Quantitative Aptitude & Business Statistics: Ratio and Proportion 50
Examples for Characteristic
Number Characteristic
0.00507
0.000670
-3
-4
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 51
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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Quantitative Aptitude & Business Statistics: Ratio and Proportion 52
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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Quantitative Aptitude & Business Statistics: Ratio and Proportion 53
For Example If log 61720=4.7904 Then 61720=anti log 4.7904
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 54
Example-1
Solution: log2 8 = 3
3Write 2 8 in logarithmic form.=
We read this as: ”the log base 2 of 8 is equal to 3”.
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 55
Example-2 Write 42 = 16 in logarithmic form.
Solution:
log4 16 = 2
Read as: “the log base 4 of 16 is equal to 2”.
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 56
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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Quantitative Aptitude & Business Statistics: Ratio and Proportion 57
Solve: log3 (4x +10) = log3 (x +1)
Since the bases are both ‘3’ we simply set the arguments equal.
4x +10 = x +13x +10 = 13x = − 9x = − 3
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 58
Example
Solve: log8 (x2 −14) = log8 (5x)Solution: Since the bases are both ‘8’ we
simply set the arguments equal. x2 −14 = 5xx2 − 5x −14 = 0(x − 7)(x + 2) = 0
Factor
(x − 7) = 0 or (x + 2) = 0x = 7 or x = −2 continued on the
next page
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 59
Example continued
Solve: log8 (x2 −14) = log8 (5x)Solution:
x = 7 or x = −2
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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.
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 61
Example If log a bc=X, log bca=y, log cab=z prove that
11z
11y
11x
1=
++
++
+
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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
11
11
11
++
++
+ zyx
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 63
log abc a+ log abc b + log abc c
=log abc abc =1
abcabcabc cba log1
log1
log1
++
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 64
Multiple Choice Questions
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Quantitative Aptitude & Business Statistics: Ratio and Proportion 65
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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Quantitative Aptitude & Business Statistics: Ratio and Proportion 66
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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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
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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
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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
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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
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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
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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
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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.
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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.
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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THE END
Ratio and Proportion