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Exercise - 2.4

Question-1 :-  Amina thinks of a number and subtracts 5/2 from it. She multiplies the result by 8. The result now obtained is 3 times the same number she thought of. What is the number?

Solution :-
  Let the number  = x
  According to question :
  8 (x - 5/2) = 3x
  8x - 8 x 5/2 = 3x 
  8x - 4 x 5 = 3x
  8x - 20 = 3x
  8x - 3x = 20
       5x = 20
        x = 20/5
        x = 4
  Think of number = x = 4
    

Question-2 :-  A positive number is 5 times another number. If 21 is added to both the numbers, then one of the new numbers becomes twice the other new number. What are the numbers?

Solution :-
  Let the another number = x
  the positive number = 5x
  According to question :
  (5x + 21) = 2(x + 21) 
  5x + 21 = 2x + 42
  5x - 2x = 42 - 21
       3x = 21
        x = 21/3
        x = 7
  The another number = x = 7
  The positive number = 5x = 5 x 7 = 35 
    

Question-3 :- Sum of the digits of a two-digit number is 9. When we interchange the digits, it is found that the resulting new number is greater than the original number by 27. What is the two-digit number?

Solution :-
  Let the unit place digit of a two-digit number = x
      the tens place digit = 9 - x
  Two-digit number = 10 x tens place digit + unit place digit 
  original number = 10(9 - x) + x 
  According to question :
  New number = Original number + 27 
  10x + (9 - x) = 10 (9 - x) + x + 27
  10x + 9 - x = 90 - 10x + x + 27
  10x + 10x - x - x = 90 + 27 - 9
  20x - 2x = 117 - 9
       18x = 108
         x = 108/18
         x = 6   
  The unit place digit of a two-digit number = x = 6
  The tens place digit = 9 - x = 9 - 6 = 3
  Two-digit number = 10(9 - x) + x  
                   = 90 - 10x + x 
                   = 90 - 9x 
                   = 90 - 9 x 6 
                   = 90 - 54 
                   = 36
    

Question-4 :-  One of the two digits of a two digit number is three times the other digit. If you interchange the digits of this two-digit number and add the resulting number to the original number, you get 88. What is the original number?

Solution :-
  Let the unit place digit of a two-digit number = x
      the tens place digit = 3x
  Two-digit number = 10 x tens place digit + unit place digit 
  original number = 10 (3x) + x = 30x + x = 31x 
  According to question :
  New number + Original number = 88
  10x + 3x + 31x = 88
  44x = 88
    x = 88/44
    x = 2 
  The unit place digit of a two-digit number = x = 2   
  The tens place digit = 3x = 3 x 2 = 6
  Two-digit number = 31x = 31 x 2 = 62
    

Question-5 :- Shobo’s mother’s present age is six times Shobo’s present age. Shobo’s age five years from now will be one third of his mother’s present age. What are their present ages?

Solution :-
  Let the present age of Shobo = x years
      the present age of shobo's mother = 6x years
  According to question :
  x + 5 = 1/3 (6x)
  x + 5 = 2x
      5 = 2x - x
      5 = x
      x = 5
  The present age of Shobo = x years = 5 years
  The present age of shobo's mother = 6x years = 6 x 5 = 30 years
    

Question-6 :-  There is a narrow rectangular plot, reserved for a school, in Mahuli village. The length and breadth of the plot are in the ratio 11:4. At the rate ₹ 100 per metre it will cost the village panchayat ₹ 75000 to fence the plot. What are the dimensions of the plot?

Solution :-
  Let the length of the rectangular plot = 11x m
      the breadth of the rectangular plot = 4x m
  Total cost = ₹ 75000
  Per meter cost = ₹ 100 
  Perimeter of rectangular plot = Total cost ÷ Cost of per meter = 75000/100 = 750 m
  According to question :
  Perimeter of rectangle = 2 (length + breadth)
  750 = 2 (11x + 4x)
  750/2 = 15x
  375 = 15x
  375/15 = x
  x = 25 
  The length of the rectangular plot = 11x = 11 x 25 = 275 m
  The breadth of the rectangular plot = 4x = 4 x 25 = 100 m
  

Question-7 :-  Hasan buys two kinds of cloth materials for school uniforms, shirt material that costs him ₹ 50 per metre and trouser material that costs him ₹ 90 per metre. For every 3 meters of the shirt material he buys 2 metres of the trouser material. He sells the materials at 12% and 10% profit respectively. His total sale is ₹ 36,600. How much trouser material did he buy?

Solution :-
  Let ratio between shirt material and trouser material = 3x:2x
  The cost of shirt material = 50 (3x) = ₹ 150x 
  The selling price at 12% gain = [(100 + P%) x C.P.] ÷ 100
                                = [(100 + 12) x 150x] ÷ 100
                                = [112 x 150x] ÷ 100
                                = 16800x/100
                                = ₹ 168x 
  The cost of trouser material = 90 x 2x = ₹ 180x
  The selling price at 12% gain = [(100 + P%) x C.P.] ÷ 100
                                = [(100 + 10) x 180x] ÷ 100
                                = [110 x 180x] ÷ 100
                                = 19800x/100
                                = ₹ 198x 
  According to the question :
  168x + 198x = 36600
         366x = 36600
            x = 36600/366
            x = 100 m
  The material of shirt = 3x = 3 x 100 = 300 m
  The material of trouser = 2x = 2 x 100 = 200 m 
    

Question-8 :-  Half of a herd of deer are grazing in the field and three fourths of the remaining are playing nearby. The rest 9 are drinking water from the pond. Find the number of deer in the herd.

Solution :-
  Let the toal number of dear = x
  Drinking of water deer = 9
  According to question :
  x = x/2 + 3/4 (x - x/2) + 9
  x - x/2 = 3x/4 - 3x/8 + 9
  (2x - x)/2 = (6x - 3x)/8 + 9
  x/2 = 3x/8 + 9
  x/2 - 3x/8 = 9
  (4x - 3x)/8 = 9
  x/8 = 9 
  x = 9 x 8
  x = 72   
  Total number of dear = x = 72       
    

Question-9 :-  A grandfather is ten times older than his granddaughter. He is also 54 years older than her. Find their present ages.

Solution :-
  Let the present age of granddaughter  = x years
      the present age of Grandfather = 10x years
  According to question : 
  10x = x + 54
  10x - x = 54
       9x = 54
        x = 54/9
        x = 6  
  The present age of granddaughter  = x years = 6 years
  The present age of Grandfather = 10x years = 10 x 6 = 60 years
    

Question-10 :-  Aman’s age is three times his son’s age. Ten years ago he was five times his son’s age. Find their present ages.

Solution :-
  Let the present age of Amon’s son = x years
      the present age of Aman’s age  = 3x years
  According to question :
  3x - 10 = 5 (x - 10)
  3x - 10 = 5x - 50
  3x - 5x = -50 + 10
      -2x = -40
       2x = 40
        x = 40/2
        x = 20  
  The present age of Amon’s son = x years = 20 years
  The present age of Aman’s age  = 3x years = 3 x 20 = 60 years
    
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