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Construction of index numbers - class-XI

Description: construction of index numbers
Number of Questions: 23
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Laspeyre's index $= 110$, Paasche's index $= 108$, then Fisher's Ideal index is equal to: 

business mathematics and statistics index numbers weighted methods to calculate index numbers construction of index numbers applied statistics

  1. $110$

  2. $108$

  3. $100$

  4. $109$

Show answer
Correct Option: D
Explanation:

Laspeyre's Index $(L.I.)$ $= 110$

Paasche's Index $(P.I.)$ $= 108$

Fisher's Ideal Index $= \sqrt{L.I. \times P.I.}$
                                 $= \sqrt{110\times 108}$
                                 $=108.995 \approx 109$

If all the values are not of equal importance the index number is called: 

business mathematics and statistics index numbers weighted methods to calculate index numbers construction of index numbers applied statistics

  1. Simple

  2. Unweighted

  3. Weighted

  4. None

Show answer
Correct Option: C
Explanation:

$\Rightarrow$  If all the values are not of equal importance the index number is called : $Weighted$

$\Rightarrow$  The ratio of the sum of weighted prices of current and base time periods multiplied by 100 is called weighted aggregate price index.
$\Rightarrow$  This index is calculated after allocating weight to each commodity on the basis of their relative importance.
$\Rightarrow$  Weight of these commodities are then multiplied by the prices of base and current time periods. these prices are called weighted price.

The aggregative expenditure method and family budget method always give:

business mathematics and statistics applied statistics weighted methods to calculate index numbers construction of index numbers index numbers

  1. Different results

  2. Approximate results

  3. Same results

  4. None of them

Show answer
Correct Option: C
Explanation:

$\Rightarrow$  The aggregative expenditure method and family budget method always give : $Same\,\,result.$

$\Rightarrow$  Aggregate Expenditure Method - In this method, the quantities of commodities consumed by the particular group in the base year are estimated and these figures or their proportions are used as weights.
$P _{0n}=\dfrac{\sum P _n q _0}{\sum P _0 q _0}\times 100$
Here, $P _n$ Represent the price of the current year,
$P _0$  Represents the price of the base year and
$q _0$  Represents the quantities consumed in the base year.
$\Rightarrow$  Family Budget Method - In this method, the family budgets of a large number of people are carefully studied and the aggregate expenditure of the average family for various items is estimated. These values are used as weights.
$P _{0n}=\dfrac{\sum WI}{\sum W}$    Here, $I=\dfrac{P _n}{P _0}\times 100$  and $W=P _0 q _0$

A factory uses three raw materials A ,B and C in the manufacturing process.The price of material were as shown below: Calculate a simple aggregate index for $2005$.

Commodity Price in Rs in 1995 Price in Rs in 2005
A 4 5
B 60 57
C 36 42      
business mathematics and statistics applied statistics weighted methods to calculate index numbers construction of index numbers index numbers

  1. 119

  2. 106

  3. 104

  4. 108

Show answer
Correct Option: C
Explanation:

<

 Commodity Price in Rs in $1995$ $({p} _{0})$ Price in Rs in $2005$ $({p} _{1})$
Commodity
 A  $4$  $5$
 B  $60$  $57$
 C  $36$ $42$


$\sum { {p} _{0} }$ = $100$ , $\sum {{p} _{1} }$= $104$
price index number ${p} _{01}$ = $\dfrac {\sum {{p} _{1} }}{\sum { {p} _{0} } } \times 100$
=$\dfrac{104}{100} \times 100$
= $104$

The most appropriate average in averaging the price relatives is:

business mathematics and statistics applied statistics weighted methods to calculate index numbers construction of index numbers index numbers

  1. Median

  2. Harmonic mean

  3. Arithmetic mean

  4. Geometric mean

Show answer
Correct Option: D
Explanation:

$\Rightarrow$  The most appropriate average in averaging the price relative is : $Geometric\,\, mean.$

$\Rightarrow$  The geometric mean is the average of a set of products, the calculation of which is commonly used to determine the performance results of an investment or portfolio. It is technically defined as "the 'n'th root product of 'n' numbers.
$\Rightarrow$  The geometric mean must be used when working with percentages, which are derived from values.

Using simple aggregate method, calculate price index number from the following data:

Commodity A B C D E
1993 prices (in Rs) 50 40 10 5 2
1995 prices(in Rs) 80 60 20 10 6
business mathematics and statistics applied statistics weighted methods to calculate index numbers construction of index numbers index numbers

  1. 164.69

  2. 154.75

  3. 162.69

  4. 152.75

Show answer
Correct Option: A
Explanation:
 $Commodity$  $Price\, in\, 1993$  $( In\, Rs.)$ $P _0$  $Price\, in\, 1995$$(In\, Rs.)$ $P _1$
 $A$  $50$ $80$
 $B$ $40$  $60$ 
 $C$ $10$  $20$ 
$D$  $5$  $10$
 $E$ $2$  $6$ 
 $Total$  $\sum\,P _0=107$  $\sum\,P _1=176$

$\Rightarrow$   Here, $\sum\,P _0=107$ and $\sum\,P _1=176$

$\therefore$      Simple Aggregate Price Index $P _{01}=\dfrac{\sum P _1}{\sum P _0}\times 100=\dfrac{176}{107}\times 100=164.48$

Calculate the index number for the year 2006 with 1996 as the base year by the weighted average of price relatives method from the following data.

Commodity A B C D E
Weight 40 25 5 20 10
Price(Rs) per unit 1996 32 80 1 10.24 4
Price(Rs) per unit 2006 40 120 1 15.36 3
business mathematics and statistics index numbers weighted methods to calculate index numbers construction of index numbers applied statistics

  1. 130

  2. 133.34

  3. 138.34

  4. 139.45

Show answer
Correct Option: A
Explanation:
 $Commodity$ $Weight$ $w$ $Price\,in\,1996$$(Rs.)\,\,P _0$  $Price\,in\,2006$$(Rs.)\,\,P _1$  $Price\,relative$$I=\dfrac{P _1}{P _0}\times 100$  $I.w$ 
$A$  $40$  $32$  $40$  $125$  $5000$ 
$B$  $25$  $80$  $120$  $150$  $3750$ 
$C$  $5$  $1$  $1$  $100$  $500$ 
$D$  $20$  $10.24$  $15.36$  $150$  $3000$ 
$E$  $10$  $4$  $3$  $75$  $750$ 
 $Total$  $100$       $13000$

$\Rightarrow$  $P _{01}=\dfrac{\sum Iw}{\sum w}=\dfrac{13000}{100}=130$

Calculate price index for the following by using price relative method.

Material Cement Timber Steel Bricks
Price in 1969 (in Rs) 5 9.5 35 12
Price in 1970 (in Rs) 8 14.3 42 24
business mathematics and statistics applied statistics weighted methods to calculate index numbers construction of index numbers index numbers

  1. 152.34

  2. 135.5

  3. 157.5

  4. 154.25

Show answer
Correct Option: C
Explanation:
 $Material$  $Price\, in\, 1969$$(in\,Rs.)\,[P _0]$  $Price\,in\,1970$$(in\,Rs.)\,[P _1]$ $Price\, Relative$$\dfrac{P _1}{P _0}\times 100$ 
$Cement$  $5$ $8$   $160.00$
$Timber$ $9.5$  $14.3$   $150.52$
$Steel$  $35$  $42$   $120.00$
$Bricks$ $12$  $24$   $200.00$
$Total$      $630.52$

$\Rightarrow$  $P _{01}=\dfrac{\dfrac{P _1}{P _0}\times 100}{N}=\dfrac{630.52}{4}=157.5$

$\therefore$    Price index for 1970, taking 1969 for base year = $157.5$

Construct a composite index number as a weighted mean from the following data:

Index Number 122 145 101 98 137 116
Weight 7 2 4 1 6 5
business mathematics and statistics index numbers weighted methods to calculate index numbers construction of index numbers applied statistics

  1. 120

  2. 122

  3. 130

  4. 132

Show answer
Correct Option: B
Explanation:
 $Index\,number$        $I$ $Weight$$w$  $I.w$ 
 $122$ $7$  $854$ 
$145$  $2$  $290$ 
$101$  $4$  $404$ 
$98$  $1$  $98$ 
$137$  $6$  $822$ 
$116$  $5$  $580$ 
$Total$  $\sum w=25$  $\sum Iw=3048$ 

$\Rightarrow$  Composite index number = $\dfrac{\sum Iw}{\sum w}=\dfrac{3048}{25}=121.92\approx 122$.

A firm uses three raw materials E ,F ,G in processing . The price per kg of these materials are as shown:

Item 1957 1967
E 4 3
F 60 57
G 36 42

Calculate simple aggregate price index for 1967 using 1957 as the base year.

business mathematics and statistics applied statistics weighted methods to calculate index numbers construction of index numbers index numbers

  1. 104

  2. 98

  3. 100

  4. 102

Show answer
Correct Option: D
Explanation:
 $Item$  $Price\,in\,1957$         $P _0$  $Price\,in\,1967$         $P _1$
 $E$ $4$  $3$ 
$F$  $60$  $57$ 
$G$  $36$  $42$ 
$Total$ $\sum P _0=100$  $\sum P _1=102$

$\Rightarrow$ $\sum P _{01}=\dfrac{\sum P _1}{\sum P _0}\times=\dfrac{102}{100}\times 100=102$

$\Rightarrow$  The price index for year $1967$, taking $1957$ base year is $102$.

Compute a price index for the following by simple aggregate method.

Commodity A B C D E F
Price in 1986 (Rs) 20 30 10 25 40 50
Price in 1991 (Rs) 25 30 15 35 45 55
business mathematics and statistics applied statistics weighted methods to calculate index numbers construction of index numbers index numbers

  1. 117.14

  2. 118.13

  3. 119.13

  4. 107.13

Show answer
Correct Option: A
Explanation:
 commodity  price in $1986$ $({p} _{0})$ price in $1991$ $({p} _{1})$ 
 A  $20$  $25$
 B  $30$  $30$
 C  $10$  $15$
 D  $25$  $35$
 E  $40$  $45$
 F  $50$  $50$


$\sum { {p} _{0} }$ = $175$ , $\sum {{p} _{1} }$= $200$
price index number ${p} _{01}$ = $\dfrac {\sum {{p} _{1} }}{\sum { {p} _{0} } } \times 100$
=$\dfrac{200}{175} \times 100$

=$ \dfrac{20000}{175}$

= $117.14$

Compute the consumer price index for 1990 taking 1989 as the base year.

Commodity Price in 1989 Price in 1990
Butter 20 21
Cheese 16 12
Milk 3 3
Eggs 2.80 2.80
business mathematics and statistics applied statistics weighted methods to calculate index numbers construction of index numbers index numbers

  1. 93

  2. 94

  3. 95

  4. 96

Show answer
Correct Option: A
Explanation:
 $Commodity$ $Price\,in\,1989$$P _0$  $Price\,in\,1990$$P _1$ 
 $Butter$ $20$  $21$ 
$Cheese$  $16$  $12$ 
$Milk$  $3$  $3$ 
$Eggs$  $2.80$  $2.80$ 
 $Total$ $\sum P _0=41.8$  $\sum P _1=38.8$ 

$\therefore$   By using simple aggregate method,

$\Rightarrow$  $P _{01}=\dfrac{\sum P _1}{\sum P _0}\times 100=\dfrac{38.8}{41.8}\times 100 =92.82 \approx 93$

Calculate cost of living index from the following table of prices and weights.

Commodity Weight Price index
Food 35 108.5
Rent 9 102.6
Clothes 10 97
Fuel 7 100.9
MIscellaneous 39 103.7
business mathematics and statistics index numbers weighted methods to calculate index numbers construction of index numbers applied statistics

  1. 104.4

  2. 106.5

  3. 126.5

  4. 128.5

Show answer
Correct Option: A
Explanation:
 $Commodity$ $Weight$$w$  $Price\,index$$I$  $I.w$ 
 $Food$ $35$  $108.5$  $3797.5$ 
$Rent$  $9$  $102.6$  $923.4$ 
$Clothes$  $10$  $97$  $970$ 
$Fuel$  $7$  $100.9$  $706.3$ 
$Miscellaneous$  $39$  $103.7$  $4044.3$ 
$Total$  $\sum w=100$    $\sum I.w=10441.5$ 

$\Rightarrow$   Cost of living index = $\dfrac{\sum I.w}{\sum w}=\dfrac{10441.5}{100}=104.4$

Calculate weighted index number for 2001 from the following data:

Item A B C
Quantity 20 15 10
Price in 2000 200 100 20
Price in 2001 320 120 28
business mathematics and statistics index numbers weighted methods to calculate index numbers construction of index numbers applied statistics

  1. 134.56

  2. 142.22

  3. 148.77

  4. 150.78

Show answer
Correct Option: B
Explanation:
 $Item$ $Quantity$$w$  $Price\,in\,2000$$P _0$  $Price\,in\,2001$$P _1$  $I=\dfrac{P _1}{P _0}\times100$ $Iw$ 
 $A$ $20$  $ 200$ $320$  $160$  $3200 $
$B$ $15$  $100$  $120$  $120$  $1800$ 
$C$  $10$  $20$  $28$  $140$  $1400$ 
 $Total$ $\sum w=45$        $\sum Iw=6400$

$\therefore$   By using weighted average price relative method.

$\Rightarrow$  $P _{01}=\dfrac{\sum Iw}{\sum w}=\dfrac{6400}{45}=142.22$ 

Taking 1975 as the base year with an index number 100 , calculate an index number for 1985 based on weighted average of price relatives.

Commodity A B C D
weight 20 30 10 40
Price per unit in 1975 10 20 5 40
Price per unit in 1985 30 35 10 80
business mathematics and statistics index numbers weighted methods to calculate index numbers construction of index numbers applied statistics

  1. 212.5

  2. 217.5

  3. 219.5

  4. 345.65

Show answer
Correct Option: A
Explanation:
$Commodity$ $Weight$$w$  $Price\,in\,1975$$P _0$  $Price\,in\,1985$ $P _1$ $Price\,relative $$I=\dfrac{P _1}{P _0}\times 100$ $I.w$ 
 $A$ $20$  $10$  $30$  $300$  $6000$ 
$B$  $30$  $20$  $35$  $175$  $5250$
$C$ $10$  $5$  $10$  $200$  $2000$ 
$D$ $40$  $40$  $80$  $200$  $8000$ 
 $Total$ $\sum w=100$        $\sum I.w=21250$

$\Rightarrow$  By using weighted average of price relative method,

$\Rightarrow$  $P _{01}=\dfrac{\sum I.w}{\sum w}=\dfrac{21250}{100}=212.5$

The quotations for four different commodities for the years 2000 and 2005 are given below. Calculate the index number for 2005 , with 2000 as base year by using weighted average of price relatives method.

business mathematics and statistics index numbers weighted methods to calculate index numbers construction of index numbers applied statistics

  1. $164.05$

  2. $161.05$

  3. $154.05$

  4. $151.05$

Show answer
Correct Option: A

Calculate the cost of living index(approximately) from the following data:

Group Weights Group Index No.
Food 47 247
Fuel and Lightning 7 293
Clothing 8 289
House Rent 13 100
Miscellaneous 14 236
business mathematics and statistics index numbers weighted methods to calculate index numbers construction of index numbers applied statistics

  1. $231.2$

  2. $265.4$

  3. $245.7$

  4. $123.78$

Show answer
Correct Option: A
Explanation:

Cost of living index $=\cfrac { \sum _{ i=1 }^{ 5 }{ { \left( \text{weight }\right)  } _{ i } } \times { \left( \text{Index no.} \right)  } _{ i } }{ \sum _{ i=1 }^{ 5 }{ { \left( \text{weight} \right)  } _{ i } }  } $
$=\cfrac { 47\times 247+7\times 293+8\times 289+13\times 100+14\times 236 }{ 47+7+8+13+14 } $
$=\cfrac { 11609+2051+2312+1300+3304 }{ 89 } $
$=\cfrac { 20756 }{ 89 } =231.19$
$\simeq 231.2$

Using simple aggregate method, calculate price index number from the following data:

Commodity A B C D
Price in 1997 90 40 90 30
Price in 1998 95 60 110 35
business mathematics and statistics applied statistics weighted methods to calculate index numbers construction of index numbers index numbers

  1. 110

  2. 120

  3. 130

  4. 140

Show answer
Correct Option: B
Explanation:
 $Commodity$  $Price\, in\, 1997$          $P _0$  $Price\, in\, 1998$       $P _1$
 $A$ $90$ $95$
 $B$ $40$  $60$ 
 $C$ $90$  $110$
 $D$ $30$ $35$ 
 $Total$  $\sum P _0=250$  $\sum P _1=300$

$\Rightarrow$  Price index number $(P _{01})$ = $\dfrac{\sum P _1}{\sum P _0}\times 100=\dfrac{300}{250}\times=120$

Construct a composite index number from the following index numbers and weights:

Index Numbers 127 142 186 172 115
Weight 5 4 3 6 8
business mathematics and statistics index numbers weighted methods to calculate index numbers construction of index numbers applied statistics

  1. $134$

  2. $145$

  3. $143$

  4. $149$

Show answer
Correct Option: C
Explanation:

Composite index no. $=\cfrac { \sum _{ i=1 }^{ 5 }{ { \left( \text{index no. }\right)  } _{ i } } \times { \left( \text{Index no.Weight} \right)  } _{ i } }{ \sum _{ i=1 }^{ 5 }{ { \left(\text{ weight }\right)  } _{ i } }  } $
$=\cfrac { 127\times 5+142\times 4+186\times 3+172\times 6+115\times 8 }{ 5+4+3+6+8 } $
$=\cfrac { 635+568+558+1038+920 }{ 26 } $
$=\cfrac { 3713 }{ 26 } =142.8$
$\simeq 143$

The following commodities have the given price indices relative to a base of $100$. The weights are also given:

Commodity Relative Index Weight
Butter 181 4
Bread 116 12
Tea 110 3
Bacon 152 7

Calculate the new index for this set of commodities

business mathematics and statistics index numbers weighted methods to calculate index numbers construction of index numbers applied statistics

  1. $132$

  2. $133$

  3. $134$

  4. $135$

Show answer
Correct Option: D
Explanation:

New index $=\cfrac { \sum _{ i=1 }^{ 4 }{ { \left( \text{Relative index }\right)  } _{ i } } \times { \left( \text{Index no. Weight }\right)  } _{ i } }{ \sum _{ i=1 }^{ 4 }{ { \left( \text{weight }\right)  } _{ i } }  } $
$=\cfrac { 181\times 4+116\times 12+110\times 3+152\times 7 }{ 4+12+3+7 } $
$=\cfrac { 724+1392+330+1064 }{ 26 } $
$=\cfrac { 3510 }{ 26 } =\cfrac { 1755 }{ 13 } $
$=135$

If all the values are of equal importance, the index numbers are called: 

business mathematics and statistics applied statistics weighted methods to calculate index numbers construction of index numbers index numbers

  1. Weighted

  2. Unweighted

  3. Composite

  4. Value index

Show answer
Correct Option: B
Explanation:

$\Rightarrow$  If all the values are of equal importance, the index numbers are called: $Unweighted$.

$\Rightarrow$  There are two methods of constructing unweighted index numbers: $(1)$ Simple Aggregative Method $(2)$ Simple Average of Relative method.
$\Rightarrow$  Simple Aggregative Method - In this method, the total price of commodities in a given (current) year is divided by the total price of commodities in a base year and expressed as percentage.
$\Rightarrow$  Simple Average of Relative method - In this method, we compute price relatives or link relatives of the given commodities and then use one of the averages such as the arithmetic mean, geometric mean, median, etc.

Using $2005$ as base year , the price of a commodity in $2007$ is 125. Calculate the index number for 2007 if 2006 is taken as the base year.

business mathematics and statistics applied statistics weighted methods to calculate index numbers construction of index numbers index numbers

  1. 105.93

  2. 104.23

  3. 103.93

  4. 112.3

Show answer
Correct Option: A

Using $2005$ as base year , the price of a commodity in $2006$ are 118. Calculate the index number for 2005 if 2006 is taken as the base year.

business mathematics and statistics applied statistics weighted methods to calculate index numbers construction of index numbers index numbers

  1. 84.75

  2. 82.18

  3. 81.18

  4. 78.07

Show answer
Correct Option: A
Explanation:

$N=1$ (Since only one commodity)

Take price $=Rs.100$  as base price for $2005$
$\implies p _1=100$
$p _0=118$ (given)
$\implies $ index no. $=\cfrac{1}{N}(\sum\cfrac{p _1}{p _0}\times 100)$
$\implies\cfrac{1}{1}(\sum\cfrac{100}{118}\times 100)=84.775\approx 84.75$

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