26. The average of 1, 3, 5, 7, 9, 11, -------- to 25 terms is
    

1. 1. 125
      

   
   
   

   2. 25
      

   
   
   

   3. 625
      

   
   
   

   4. 50

   
   
   

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1. View Hint View Answer Discuss in Forum

   Sum of first n odd natural numbers = n²
   

   ∴ Their average =	n²	= n
   	n

   
   

   Correct Option: B

   Sum of first n odd natural numbers = n²
   

   ∴ Their average =	n²	= n
   	n

   
   ∴ Required average = 25 , because n = 25
   

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27. If average of 20 observations x₁, x₂, ....., x₂₀ is y, then the average of x₁ – 101, x₂ – 101, x₃ – 101, ....., x₂₀ –101 is
    

1. 1. y – 20
      

   
   
   

   2. y – 101
      

   
   
   

   3. 20y
      

   
   
   

   4. 101y

   
   
   

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1. View Hint View Answer Discuss in Forum

   Given Here , average of 20 observations x₁, x₂, ....., x₂₀ = y
   Now , we have
   

   Required average =	{ ( x1 - 101 ) + ( x2 - 101 ) + . . . . + ( x20 - 101 ) }
   	20

   
   

   Required average =	{ x1 + x2 + . . . . +x20 } - { 101 × 20 }
   	20

   
   

   Correct Option: B

   Given Here , average of 20 observations x₁, x₂, ....., x₂₀ = y
   Now , we have
   

   Required average =	{ ( x1 - 101 ) + ( x2 - 101 ) + . . . . + ( x20 - 101 ) }
   	20

   
   

   Required average =	{ x1 + x2 + . . . . +x20 } - { 101 × 20 }
   	20

   
   

   Required average =	x1 + x2 + . . . . +x20	-	101 × 20
   	20		20

   
   Hence , Required average = y - 101

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28. The average of x numbers is y² and the average of y numbers is x². So the average of all the numbers taken together is

1. 1. x3 + y3

      x + y

   
   
   

   2. xy

   
   
   

   3. x2 + y2

      x + y

   
   
   

   4. xy² + yx²

   
   
   

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1. View Hint View Answer Discuss in Forum

   Total sum of x numbers = xy²
   Total sum of y numbers = yx²
   

   ∴ Required average =	xy2 + yx2
   	x + y

   
   

   ∴ Required average =	xy(y + x)	= xy
   	x + y

   
   Second method to find the required average ,
   Here, n₁ = x, a₁ = y²
   n₂ = y, a₂ = x²
   

   ∴ Average =	n1a1 + n2a2
   	n1 + n2

   
   

   Correct Option: B

   Total sum of x numbers = xy²
   Total sum of y numbers = yx²
   

   ∴ Required average =	xy2 + yx2
   	x + y

   
   

   ∴ Required average =	xy(y + x)	= xy
   	x + y

   
   Second method to find the required average ,
   Here, n₁ = x, a₁ = y²
   n₂ = y, a₂ = x²
   

   ∴ Average =	n1a1 + n2a2
   	n1 + n2

   
   

   Average =	xy2 + yx2
   	x + y

   
   

   Average = xy		x + y		= xy
   		x + y

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29. The average of n numbers x₁, x₂, .... x_n is x . Then the value of ⅀(x_i - x) , ( i = 1 .......... n ) is equal to
    

1. 1. n
      

   
   
   

   2. 0
      

   
   
   

   3. nx
      

   
   
   

   4. x

   
   
   

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1. View Hint View Answer Discuss in Forum

   As we know that ,
   

   x =	x1 + x2 + . . . .xn
   	n

   
   = (x₁ - x) + (x₂ -x) +....... + (x_n - x)
   = (x₁ + x₂ + . . . . . + x_n) - n . x
   

   Correct Option: B

   As we know that ,
   

   x =	x1 + x2 + . . . .xn
   	n

   
   = (x₁ - x) + (x₂ -x) +....... + (x_n - x)
   = (x₁ + x₂ + . . . . . + x_n) - n . x
   

   = n.		x1 + x2 + . . . .xn		- n.x
   		n

   
   = nx - nx = 0

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30. Total weekly emoluments of the workers of a factory is ₹ 1534. Average weekly emolument of a worker is ₹ 118. The number of workers in the factory is :
    

1. 1. 16
      

   
   
   

   2. 14
      

   
   
   

   3. 13
      

   
   
   

   4. 12

   
   
   

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1. View Hint View Answer Discuss in Forum

   Here , Average weekly emolument of a worker = ₹ 118
   Total weekly emoluments of the workers = ₹ 1534
   As we know that ,
   

   Number of workers in the factory =	Total weekly emoluments of the workers
   	Average weekly emolument of a worker

   
   

   Correct Option: C

   Here , Average weekly emolument of a worker = ₹ 118
   Total weekly emoluments of the workers = ₹ 1534
   As we know that ,
   

   Number of workers in the factory =	Total weekly emoluments of the workers
   	Average weekly emolument of a worker

   
   

   Number of workers in the factory =	1534	= 13
   	118

   
   

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