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

**(i) A part of monthly hostel charges is fixed and the remaining depends on the number of days one has taken food in the mess. When a student A takes food for 20 days she has to pay Rs 1000 as hostel charges whereas a student B, who takes food for 26 days, pays Rs 1180 as hostel charges. Find the fixed charges and the cost of food per day.**

**Solution:**

Let fixed monthly charge = Rs x

Let charge of food for one day = Rs y

According to given conditions, we have

**(1)**, **and**

** (2)**

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

**Rs 30**

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

Rs 400

**Therefore, fixed monthly charges = Rs 400**

**And, charges of food for one day = Rs 30**

**(ii) A fraction becomes when 1 is subtracted from the numerator and it becomes when 8 is added to its denominator. Find the fraction.**

**Solution:**

Let numerator

Let denominator

According to given condition ,we have

**(1)** ** (2)**

**(1) ** **(2)**

**(1)** ** (2)**

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

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

Therefore, numerator = 5

And, denominator = 12

## It means fraction

**(iii) Yash scored 40 marks in a test, getting 3 marks for each right answer and losing 1 mark for each wrong answer. Had 4 marks been awarded for each correct answer and 2 marks been deducted for each incorrect answer, then Yash would have scored 50 marks. How many questions were there in the test?**

**Solution:**

Let number of correct answers

Let number of wrong answers

According to given conditions, we have

** (1)**

And, **(2)**

From equation **(1)**, we can say

Putting this in **(2)**, we get

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

Therefore, number of correct answers

Number of wrong answers

Total questions

**(iv) Places A and B are 100 km apart on a highway. One car starts from A and another from B at the same time. If the cars travel in the same direction at different speeds, they meet in 5 hours. If they travel towards each other, they meet in 1 hour. What are the speeds of the two cars?**

**Solution:**

Let speed of car which starts from part A = x km/hr

Let speed of car which starts from part B = y km/hr

According to given conditions, we have

** (Assuming x>y)**

** (1)**

And,

**(2)**

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

km/hr

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

km/hr

**Therefore, speed of car starting from point A = 60 km/hr**

**And, Speed of car starting from point B = 40 km/hr **

**(v) The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.**

**Solution:**

Let length of rectangle = x units

Let breadth of rectangle = y units

Area

According to given conditions, we have

**(1)**

And,

**(2)**

From equation **(1)**, we can say that

Putting this in **(2)**, we get

units

Putting value of **y** in **(2)**, we get

units

**Therefore, length= 17 units**

**And, Breadth = 9 units**