16808ratio proportion

7/27/2019 16808Ratio Proportion

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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 & BusinessStatistics: Ratio and Proportion3

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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Quantitative Aptitude & Business

Statistics: Ratio and Proportion21

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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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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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

16808ratio proportion

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