Tag: business mathematics and statistics

Questions Related to business mathematics and statistics

A village road is to be constructed by a team of $250$ workers. After $12$ days it was found that only $2/7^{th}$ part of the work was complete. To complete the rest in another $25$ days, how many more workers should be employed?

business mathematics and statistics applications of calculus revenue functions from marginal revenue functions minimization of cost function and maximization of revenue function and profit function integral calculus – ii

  1. $53$

  2. $52$

  3. $55$

  4. $50$

Show answer
Correct Option: D
Explanation:

Let the total unit of work be $7$ units.
According to question,
$\dfrac {M _{1}D _{1}}{W _{1}} = \dfrac {M _{2}D _{2}}{W _{2}} \Rightarrow \dfrac {250\times 12}{2} = \dfrac {M _{2}\times 25}{(7 - 2)} \Rightarrow M _{2} = 300$
Number of extra workers needed $= 300 - 250 = 50$.

A railway half ticket costs half the full fare but reservation charge is same. One reserved ticket from Ranchi to Howrah cost $Rs.1720$ and one full and one half ticket (both reserved) costs $Rs.2610$. Thus, the reservation charge is

business mathematics and statistics applications of calculus revenue functions from marginal revenue functions minimization of cost function and maximization of revenue function and profit function integral calculus – ii

  1. $Rs.60$

  2. $Rs.30$

  3. $Rs.40$

  4. None

Show answer
Correct Option: A
Explanation:

Let the full ticket cost $ Rs  \ x $and reservation charge be $ y $
As per given statements,
$ x + y = 1720 $    -(1)
And $ x + y + \frac {x}{2} + y = 2610 => \frac {3x}{2} + 2y = 2610 $ --- (2)

Multiplying equation  $ (1) $ with $ 2 $ we get, $ 2x + 2y
= 3440 $ ----- equation $ (3) $

Subtracting equation $ (2) $

from $ (3) $, we get $ \frac{x}{2} = 830 => x = 1660 $

Substituting $ x = 1660 $ in the equation $ (1) $, we get $ 1660 +y = 1720 => y = 60 $ 

Thus reservation charges were Rs $60 $

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 = $157.5$