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Exercise - 2.4

Linear Equations in One Variable

**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?

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?

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?

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?

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?

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?

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?

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.

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.

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.

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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