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 if we only add 1 to the denominator. What is the fraction?

**Solution:**

Let numerator

Let denominator

According to given condition, we have

and

And,

**(1) **And, ** (2)**

So, we have equations **(1)** and **(2)**, multiplying equation **(1)** by 2 we get **(3)**

**(3)**

**(2)**

Subtracting equation **(2)** from **(3)**, we get

Putting value of y in **(1)**, we get

Therefore, 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?

**Solution:**

Let present age of Nuri years

Let present age of Sonu years

5 years ago, age of Nuri = (x-5) years

5 years ago, age of Sonu = (y-5) years

According to given condition, we have

**(1)**

10 years later from present, age of Nuri years

10 years later from present, age of Sonu years

According to given condition, we have

**(2)**

Subtracting equation **(1)** from **(2)**, we get

years

Putting value of **y** in **(1)**, we get

years

Therefore, present age of Nuri years

And, present age of Sonu 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.

**Solution:**

Let digit at ten’s place

Let digit at one’s place

According to given condition, we have

** (1)**

And,

**(2)**

Adding **(1)** and **(2)**, we get

Putting value of **x** in **(1)**, we get

Therefore, number

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

**Solution:**

Let number of Rs 100 notes

Let number of Rs 50 notes

According to given conditions, we have

**(1)**

And,

**(2)**

Subtracting **(2)** from **(1)**, we get

Putting value of **x** in **(1)**, we get

Therefore, number of Rs 100 notes

Number of Rs 50 notes

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

**Solution:**

Let fixed charge for 3 days = Rs

Let additional charge for each day thereafter = Rs

According to given condition, we have

**(1)**

**(2)**

Subtracting **(2)** from **(1)**, we get

Rs 3

Putting value of **y** in **(1)**, we get

Rs 15

Therefore, fixed charge for 3 days **Rs 15**

Additional charge for each day after 3 days **Rs 3**