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TOPICS
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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