Ratio and Proportion - icai.org · 8 Statistics: Ratio and Proportion . 5.If the ratio of two...

Ratio and Proportion

Quantitative Aptitude & Business Statistics


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.

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


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.


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


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


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


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


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


For example Continued ratio of Rs.200,Rs.400 and Rs.600 is Rs200:Rs400:Rs.600.=

1:2:3


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.


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


Hence monthly incomes of the two persons are Rs.4X100(Rs.400)and

Rs.5x100(Rs.500)


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


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


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


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.


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.


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 .


Thus, b is the mean proportional between a and c ,then b2 =ac i.e

b=

ac


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.


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)


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)


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-….)


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


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


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


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


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:


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’.


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


( ) mnnm aa =

Law.3: where m and n are positive integers Law.4: where n takes all positive values.

( ) nnn b.aab =


Find x ,if

Solution

XXXXX )(=

XXXX )()( 23

21

=

xXXX

.23

23

21

1 )()( ==+


(If bases are equal ,then power is also equal) ie 3/2=3/2* x X =1


Example

=1

ac

a

ccb

c

bba

b

a

xx

xx

xx

+++

..


Example

=1

222222

..lnln

l

nnmnm

n

mmlml

m

l

xx

xx

xx

++++++


If

Then 3X3-9x=10

31

31

33−

+=X


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


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:


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


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


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



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 −



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 =


Why Logarithms

Logarithms were originally developed to simplify complex arithmetic

calculations. They were designed to transform multiplicative processes into additive ones.


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.


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.


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.


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)


Examples for Characteristic

Number Characteristic

0.00507

0.000670

-3

-4


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


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


For Example If log 61720=4.7904 Then 61720=anti log 4.7904


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”.


Example-2 Write 42 = 16 in logarithmic form.

Solution:

log4 16 = 2

Read as: “the log base 4 of 16 is equal to 2”.


Write 2− 3 =18

in logarithmic form.

log2

18= − 3

Solution:

1Read as: "the log base 2 of is equal to -3".8


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


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


Example continued

Solve: log8 (x2 −14) = log8 (5x)Solution:

x = 7 or x = −2


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.


Example If log a bc=X, log bca=y, log cab=z prove that

11z

11y

11x

1=

++

++

+


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


log abc a+ log abc b + log abc c

=log abc abc =1

abcabcabc cba log1

log1

log1

++


Multiple Choice Questions


1________ is the mean proportional between 12x2 and 27y2.

A) 18xy B) 81 xy C) 8 xy D) 19.5 xy


2.log 32/4 is equal to A) log 32/log4 B) log 32 – log4 C)23

D) None of these


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


4.The value of (8/27)1/3 is A) 2/3 B) 3/2 C) 2/9 D) None of these


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.


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


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


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


8.If p: q = r: s, implies q: p = s: r, then the process is called

A) Componendo B) Invertendo C) Alternendo. D) Dividendo


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


10. The triplicate ratio of 4: 5 is ________. A) 125: 64 B)16:25 C)64:125 D) None of these

THE END


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