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# Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method:

(i) If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes 1/2 if we only add 1 to the denominator. What is the fraction?

(ii) Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu?

(iii) The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.

(iv) Meena went to a bank to withdraw ₹ 2000. She asked the cashier to give her ₹ 50 and ₹ 100 notes Meena got 25 notes in all. Find how many notes of ₹ 50 and ₹ 100 she received.

(v) A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid ₹ 27 for a book kept for seven days, while Susy paid ₹ 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day.

**Solution:**

(i) If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes 1/2 if we only add 1 to the denominator. What is the fraction?

A fraction has two parts numerator and denominator. So, assume the numerator as x, and the denominator as y. We will be framing two linear equations using the given situation.

Let the numerator = x

And the denominator = y

Then the fraction = x/y

When 1 is added to the numerator and 1 is subtracted from the denominator, the fraction reduces to 1 as shown below.

(x + 1)/(y - 1) = 1

x + 1 = y - 1

x - y + 1 + 1 = 0

x - y + 2 = 0 ...(1)

When 1 is added to the denominator, the fraction becomes 1/2 as shown below.

x/(y + 1) = 1/2

2x = y + 1

2x - y - 1 = 0 ...(2)

By subtracting equation (2) from equation (1)

(x - y + 2) - (2x - y - 1) = 0

x - y + 2 - 2x + y + 1 = 0

- x + 3 = 0

x = 3

Substitute x = 3 in equation (1)

3 - y + 2 = 0

y = 5

Equations are x - y + 2 = 0 and 2x - y -1 = 0 where the numerator of the fraction is x, and denominator is y.

The fraction is 3/5.

(ii) Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu?

Assuming the present age of Nuri as x years and Sonu as y years, two linear equations can be formed for the given situation.

Let the present age of Nuri = x years

And the present age of Sonu = y years

5 years ago,

Nuri’s age = (x - 5) years

Sonu’s age = (y - 5) years

x - 5 = 3(y - 5)

x - 5 = 3y - 15

x - 3y - 5 + 15 = 0

x - 3y + 10 = 0 ...(1)

10 years later,

Nuri’s age = (x + 10) years

Sonu’s age = (y + 10) years

x + 10 = 2(y + 10)

x + 10 = 2y + 20

x - 2y + 10 - 20 = 0

x - 2y - 10 = 0 ...(2)

By subtracting equation (2) from equation (1)

(x - 3y +10) - ( x - 2y -10) = 0

x - 3y +10 - x + 2y +10 = 0

- y + 20 = 0

y = 20

Now, substitute y = 20 in equation (1)

x - 3 × 20 + 10 = 0

x - 60 + 10 = 0

x - 50 = 0

x = 50

Linear equations are x - 3y +10 = 0 and x - 2 y -10 = 0 where the present age of Nuri is x and Sonu is y

The age of Sonu is 20 years and age of Nuri is 50 years.

(iii) The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.

A two-digit number’s form is 10y + x where y and x are ten’s and one’s digit respectively.

Let the one’s place = x

And the ten’s place = y

Then the number = 10y + x

Sum of the digits of the number;

x + y = 9 ...(1)

By reversing the order of the digits, the number = 10x + y

Hence,

9(10y + x) = 2 (10x + y )

90y + 9x = 20x + 2 y

20x + 2y - 90y - 9x = 0

11x - 88y = 0

11( x - 8 y ) = 0

x - 8y = 0 ...(2)

By subtracting equation (2) from equation (1)

( x + y ) - ( x - 8y ) = 9 - 0

x + y - x + 8y = 9

9y = 9

y = 1

Substitute y = 1 in equation (1)

x + 1 = 9

x = 9 - 1

x = 8

Equations are x + y = 9 and x - 8y = 0 where y and x are ten’s and one’s digit respectively.

The two-digit number is 18.

(iv) Meena went to a bank to withdraw ₹ 2000. She asked the cashier to give her ₹ 50 and ₹ 100 notes Meena got 25 notes in all. Find how many notes of ₹ 50 and ₹ 100 she received.

Assuming the number of notes of ₹ 50 as x and ₹ 100 as y, two linear equations can be formed for the given situation.

Let number of notes of ₹ 50 = x

Number of notes of ₹ 100 = y

Meena got 25 notes in all;

x + y = 25 ...(1)

Meena withdrew ₹ 2000;

50x +100y = 2000

50(x + 2y) = 2000

x + 2y = 2000/50

x + 2y = 40 ...(2)

By subtracting equation (1) from equation (2)

(x + 2y) - (x + y) = 40 - 25

x + 2y - x - y = 15

y = 15

Substituting, y = 15 in equation (1)

x + 15 = 25

x =10

Equations are x + y = 25 and x + 2y = 40 where number of ₹ 50 and ₹ 100 notes are x and y respectively.

The number of ₹ 50 notes is 10 and the number of ₹ 100 notes is 15.

(v) A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid ₹ 27 for a book kept for seven days, while Susy paid ₹ 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day.

Assuming fixed charges as ₹ x and the additional charges for each extra day as ₹ y, two linear equations can be formed for the given situation.

Let the fixed charge = x

Charge per extra day = y

Saritha paid ₹ 27 for a book kept for 7 days;

We know that the lending library has a fixed charge for the first three days and an additional charge for each day thereafter.

Therefore,

x + (7 - 3) y = 27

x + 4y = 27 ...(1)

Susy paid ₹ 21 for a book kept for 5 days;

x + (5 - 3) y = 21

x + 2y = 21 ...(2)

By subtracting equation (2) from equation (1)

(x + 4 y) - (x + 2 y) = 27 - 21

x + 4y - x - 2y = 6

2y = 6

y = 6/2

y = 3

Substituting y = 3 in equation (1)

x + 4 × 3 = 27

x + 12 = 27

x = 27 - 12

x = 15

Equations are x + 2y = 21 and x + 4y = 27 where fixed charge is ₹ x and charge for each extra day is ₹ y.

The fixed charge is ₹ 15 and the charge for each extra day is ₹ 3.

**☛ Check: **NCERT Solutions Class 10 Maths Chapter 3

**Video Solution:**

## Form the pair of linear equations in the following problems, and find their solutions (if they exist) by the elimination method:(i) If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes 1/2 if we only add 1 to the denominator. What is the fraction?(ii) Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. How old are Nuri and Sonu?(iii) The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. Find the number.(iv) Meena went to a bank to withdraw ₹ 2000. She asked the cashier to give her ₹ 50 and ₹ 100 notes Meena got 25 notes in all. Find how many notes of ₹ 50 and ₹ 100 she received.(v) A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid ₹ 27 for a book kept for seven days, while Susy paid ₹ 21 for the book she kept for five days. Find the fixed charge and the charge for each extra day.

NCERT Solutions for Class 10 Maths - Chapter 3 Exercise 3.4 Question 2

**Summary:**

1) If we add 1 to the numerator and subtract 1 from the denominator, a fraction reduces to 1. It becomes 1/2 if we only add 1 to the denominator, the fraction is 3/5. 2) Five years ago, Nuri was thrice as old as Sonu. Ten years later, Nuri will be twice as old as Sonu. The age of Sonu is 20 years and age of Nuri is 50 years. 3) The sum of the digits of a two-digit number is 9. Also, nine times this number is twice the number obtained by reversing the order of the digits. The two-digit number is 18. 4) Meena went to a bank to withdraw ₹ 2000. She asked the cashier to give her ₹ 50 and ₹ 100 notes Meena got 25 notes in all. The number of ₹ 50 notes is 10 Number of ₹ 100 notes is 15. 5) A lending library has a fixed charge for the first three days and an additional charge for each day thereafter. Saritha paid ₹ 27 for a book kept for seven days, while Susy paid ₹ 21 for the book she kept for five days. The fixed charge is ₹ 15 and the charge for each extra day is ₹ 3.

**☛ Related Questions:**

- Solve the following pair of linear equations by the substitution method. (i) x + y = 14; x - y = 4 (ii) s - t = 3; s/3 + t/2 = 6 (iii) 3x - y = 3; 9x - 3y = 9 (iv) 0.2x + 0.3y = 1.3; 0.4x + 0.5 y = 2.3 (v) √2x + √3y = 0; √3x - √8y = 0 (vi) (3x/2) - (5y/3) = -2; x/3 + y/2 = 13/6
- Solve 2x + 3y =11 and 2x - 4 y = -24, hence find the value of ‘m’ for which y = mx +3.
- Form the pair of linear equations for the following problems and find their Solution by substitution method. (i) The difference between two numbers is 26 and one number is three times the other. Find them. (ii) The larger of two supplementary angles exceeds the smaller by 18 degrees. Find them.
- Solve the following pair of linear equations by the elimination method and the substitution method: (i) x + y = 5 and 2x - 3y = 4 (ii) 3x + 4 y = 10 and 2x - 2y = 2 (iii) 3x - 5y - 4 = 0 and 9x = 2y + 7 (iv) x/2 + 2y/3 = -1 and x - y/3 = 3

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