Chapter-2-Linear-Equations-in-One-Variable

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Linear Equations in one Variable

Question 1.
Solve the following equations:

Solution.
1. x-2 = 7

Linear Equations in One Variable Ex 2.2

Question 1. If you subtract 1/2 from a number and multiply the result 1/2 by you get 1/8 What is the number ?

Solution.

Question 2.
The perimeter of a rectangular swimming pool is 154 m. Its length is 2 m more than twice its breadth. What are the length and the breadth of the pool?
Solution.

Question 3.
The base of an isosceles triangle is 4/3cm. The perimeter of the triangle is 4 2/15cm. What is the length of either of the remaining equal sides ?

Solution.

Question 4.
Sum of two numbers is 95. If one exceeds the other by 15, find the numbers.

Solution.

Question 5.
Two numbers are in the ratio 5 :3. If they differ by 18, what are the numbers?

Solution.

Question 6.
Three consecutive integers add up to 51. What are these integers ?

Solution.

Question 7.
The sum of three consecutive multiples of 8 is 888. Find the multiples.

Solution.

Question 8.
Three consecutive integers are such that when they are taken in increasing order and multiplied by 2, 3 and 4 respectively, they add up to 74. Find these numbers.

Solution.

Question 9.
The ages of Rahul and Haroon are in the ratio 5 : 7. Four years later, the sum of their ages will be 56 years. What are their present ages ?

Solution.

Question 10.
The numbers of boys and girls in a class are in the ratio 7 : 5. The number of boys is 8 more than the number of girls. What is the total class strength ?

Solution..

Question 11.
Baichung’s father is 26 years younger than Baichung’s grandfather and 29 years older than Baichung. The sum of the ages of all the three is 135 years. What is the age of each one of them ?

Solution.

Question 12.
Fifteen years from now Ravi’s age will be four times his present age. What is Ravi’s present age ?

Solution.

Question 13.
A rational number is such that when you multiply it by 52 and add 23 to the product, you get−712. What is the number ?

Solution.

Question 14.
Lakshmi is a cashier in a bank. She has currency notes of denominations ? 100, ? 50 and ? 10, respectively. The ratio of the number of these notes is 2 : 3 : 5. The total cash with Lakshmi is ? 4,00,000. How many notes of each denomination does she have ?

Solution.

Question 15.
I have a total of oft 300 in coins of denomination ? 1, ? 2 and ? 5. The number of ? 2 coins is 3 times the number of ? 5 coins. The total number of coins is 160. How many coins of each denomination are with me ?

Solution.

Hence, I have 80, 60, and 20 coins of denomination ? 1, ? 2 and ? 5 respectively.

Question 16.
The organizers of an essay competition decide that a winner in the competition gets a prize of ? 100 and a participant who does not win gets a prize of ? 25. The total prize money distributed is ? 3,000. Find the number of winners, if the total number of participants is 63.

Solution.

Linear Equations in One Variable Ex 2.3

Question 1. Solve the following equations and check your results:

1. 3x=2x+18
2. 5t-3=3t-5
3. 5x+9=5+3x
4. 4z+3=6+2z
5. 2x-1=14-x
6. 8x+4=3(x-1)+7
7. x=4/5(x+10)
8. 2x/3+1=7x/15+3
9. 2y+5/3=26/3?y
10. 3m=5m?8/5

Solution. 1. 3x=2x+18

2. 5t-3=3t-5

3. 5x+9=5+3x

4. 4z+3=6+2z

5. 2x-1=14-x

6. 8x+4=3(x-1)+7

7. x=4/5(x+10)

8. 2x/3+1=7x/15+3

9. 2y+5/3=26/3?y

10. 3m=5m?8/5

Linear Equations in One Variable Ex 2.4

Question 1.
Amina thinks of a number and subtracts 52 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 be x.
Then, according to the question,

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 numbers be x and 5x.
If 21 is added to both the numbers, then first new number = x + 21
and, second new number = 5x + 21

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’s digit of the two-digit number be x.
Then, the ten’s digit of the two-digit number = 9 – x

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 in the original number

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 be x year. Then, the present age of Shobo’s mother = 6x years.
Five years from now
Shobo’s age = (x + 5) years
According to the question,

Hence, their present ages are 5 years and 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 of ? 100 per meter, it will cost the village panchayat ? 75000 to fence the plot. What are the dimensions of the plot?

Solution.
Let the length and breadth of the plot be 11x m and 4x m respectively.
Then, the perimeter of the plot

Question 7.
Hasan buys two kinds of cloth materials for school uniforms, shirt material that costs him ? 50 per meter and trouser material that costs him ? 90 per meter. For every 2 meters of the trouser material, he buys 3 meters of the shirt 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.
Suppose that he bought x meters of trouser material.

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 number of deer in the herd be x.

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 be x years
Then, the present age of grandfather is 10x years
According to the question,

Question 10.
Am.an’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 Aman’s son be x years
Then, the present age of Aman

Linear Equations in One Variable Ex 2.5

Question 1.
Solve the following linear equations:

1. x/2?1/5=x/3+1/4
2. n/2?3n/4+5n/6=21
3. x+7?8x/3=17/6?5x/2
4. x?5/3=x?3/5
5. 3t?2/4?2t+3/3=2/3?t
6. m?m?1/2=1?m?2/3

Solution.

1. x/2?1/5=x/3+1/4

2. n/2?3n/4+5n/6=21

3. x+7?8x/3=17/6?5x/2

4. x?5/3=x?3/5

5. 3t?2/4?2t+3/3=2/3?t

6. m?m?1/2=1?m?2/3
We have m?m?1/2=1?m?2/3

Question 2.
Simplify and solve the following linear equations:
7. 3(t-3)=5(2t+1)
8. 15(y-4)-2(y-9)+5 (y+6) = 0
9. 3(5z-7)-2 (9z-11)=4 (8z-13)-17
10. 0.25(4f-3) = 0.05 (10f-9).

Solution.

7. 3(t-3)=5(2t+1)

8. 15(y-4)-2(y-9)+5 (y+6) = 0

9. 3(5z-7)-2 (9z-11)=4 (8z-13)-17

10. 0.25(4f-3) = 0.05 (10f-9)

Linear Equations in One Variable Ex 2.6

Question 1.
Solve the following equations:

Solution.

1. 8x?3/3x=2

2. 9x/7?6x=15

3. z/z+15=49

4. 3y+4/2?6y=?2/5

5. 7y+4/y+2=?4/3

Question 2.
The ages of Hari and Harry are in the ratio 5: 7. Four years from now the ratio of their ages will be 3 :4. Find their present ages.

Solution.
Let the present ages of Hari and Harry be 5x years and 7x years respectively.

Question 3.
The denominator of a rational number is greater than its numerator by 8. If the numerator is increased by 17 and the denominator is decreased by 1, the number obtained is 32. Find the rational number.

Solution.

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Linear Equations in one Variable

Linear Equations in one Variable

Question 1. Solve the following equations:

Linear Equations in One Variable Ex 2.2

Question 1. If you subtract 1/2 from a number and multiply the result 1/2 by you get 1/8 What is the number ?

Question 2. The perimeter of a rectangular swimming pool is 154 m. Its length is 2 m more than twice its breadth. What are the length and the breadth of the pool? Solution.

Question 3. The base of an isosceles triangle is 4/3cm. The perimeter of the triangle is 4 2/15cm. What is the length of either of the remaining equal sides ?

Question 4. Sum of two numbers is 95. If one exceeds the other by 15, find the numbers.

Question 5. Two numbers are in the ratio 5 :3. If they differ by 18, what are the numbers?

Question 6. Three consecutive integers add up to 51. What are these integers ?

Question 7. The sum of three consecutive multiples of 8 is 888. Find the multiples.

Question 8. Three consecutive integers are such that when they are taken in increasing order and multiplied by 2, 3 and 4 respectively, they add up to 74. Find these numbers.

Question 9. The ages of Rahul and Haroon are in the ratio 5 : 7. Four years later, the sum of their ages will be 56 years. What are their present ages ?

Question 10. The numbers of boys and girls in a class are in the ratio 7 : 5. The number of boys is 8 more than the number of girls. What is the total class strength ?

Question 11. Baichung’s father is 26 years younger than Baichung’s grandfather and 29 years older than Baichung. The sum of the ages of all the three is 135 years. What is the age of each one of them ?

Question 12. Fifteen years from now Ravi’s age will be four times his present age. What is Ravi’s present age ?

Question 13. A rational number is such that when you multiply it by 52 and add 23 to the product, you get−712. What is the number ?

Question 14. Lakshmi is a cashier in a bank. She has currency notes of denominations ? 100, ? 50 and ? 10, respectively. The ratio of the number of these notes is 2 : 3 : 5. The total cash with Lakshmi is ? 4,00,000. How many notes of each denomination does she have ?

Question 15. I have a total of oft 300 in coins of denomination ? 1, ? 2 and ? 5. The number of ? 2 coins is 3 times the number of ? 5 coins. The total number of coins is 160. How many coins of each denomination are with me ?

Question 16. The organizers of an essay competition decide that a winner in the competition gets a prize of ? 100 and a participant who does not win gets a prize of ? 25. The total prize money distributed is ? 3,000. Find the number of winners, if the total number of participants is 63.

Linear Equations in One Variable Ex 2.3

Question 1. Solve the following equations and check your results:

Linear Equations in One Variable Ex 2.4

Question 1. Amina thinks of a number and subtracts 52 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?

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?

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?

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?

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?

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 of ? 100 per meter, it will cost the village panchayat ? 75000 to fence the plot. What are the dimensions of the plot?

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.

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

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

Linear Equations in One Variable Ex 2.5

Question 1. Solve the following linear equations:

Question 1. Solve the following equations:

Question 2. The ages of Hari and Harry are in the ratio 5: 7. Four years from now the ratio of their ages will be 3 :4. Find their present ages.

Question 3. The denominator of a rational number is greater than its numerator by 8. If the numerator is increased by 17 and the denominator is decreased by 1, the number obtained is 32. Find the rational number.

Question 1.
Solve the following equations:

Question 2.
The perimeter of a rectangular swimming pool is 154 m. Its length is 2 m more than twice its breadth. What are the length and the breadth of the pool?
Solution.

Question 3.
The base of an isosceles triangle is 4/3cm. The perimeter of the triangle is 4 2/15cm. What is the length of either of the remaining equal sides ?

Question 4.
Sum of two numbers is 95. If one exceeds the other by 15, find the numbers.

Question 5.
Two numbers are in the ratio 5 :3. If they differ by 18, what are the numbers?

Question 6.
Three consecutive integers add up to 51. What are these integers ?

Question 7.
The sum of three consecutive multiples of 8 is 888. Find the multiples.

Question 8.
Three consecutive integers are such that when they are taken in increasing order and multiplied by 2, 3 and 4 respectively, they add up to 74. Find these numbers.

Question 9.
The ages of Rahul and Haroon are in the ratio 5 : 7. Four years later, the sum of their ages will be 56 years. What are their present ages ?

Question 10.
The numbers of boys and girls in a class are in the ratio 7 : 5. The number of boys is 8 more than the number of girls. What is the total class strength ?

Question 11.
Baichung’s father is 26 years younger than Baichung’s grandfather and 29 years older than Baichung. The sum of the ages of all the three is 135 years. What is the age of each one of them ?

Question 12.
Fifteen years from now Ravi’s age will be four times his present age. What is Ravi’s present age ?

Question 13.
A rational number is such that when you multiply it by 52 and add 23 to the product, you get−712. What is the number ?

Question 14.
Lakshmi is a cashier in a bank. She has currency notes of denominations ? 100, ? 50 and ? 10, respectively. The ratio of the number of these notes is 2 : 3 : 5. The total cash with Lakshmi is ? 4,00,000. How many notes of each denomination does she have ?

Question 15.
I have a total of oft 300 in coins of denomination ? 1, ? 2 and ? 5. The number of ? 2 coins is 3 times the number of ? 5 coins. The total number of coins is 160. How many coins of each denomination are with me ?

Question 16.
The organizers of an essay competition decide that a winner in the competition gets a prize of ? 100 and a participant who does not win gets a prize of ? 25. The total prize money distributed is ? 3,000. Find the number of winners, if the total number of participants is 63.

Question 1.
Amina thinks of a number and subtracts 52 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?

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?

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?

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?

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?

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 of ? 100 per meter, it will cost the village panchayat ? 75000 to fence the plot. What are the dimensions of the plot?

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.

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

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

Question 1.
Solve the following linear equations:

Question 1.
Solve the following equations:

Question 2.
The ages of Hari and Harry are in the ratio 5: 7. Four years from now the ratio of their ages will be 3 :4. Find their present ages.

Question 3.
The denominator of a rational number is greater than its numerator by 8. If the numerator is increased by 17 and the denominator is decreased by 1, the number obtained is 32. Find the rational number.