Question 4 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. (ii)A fraction becomeswhen 1 is subtracted from the numerator and it becomeswhen 8 is added to its denominator. Find the 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? (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? (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.
Class 10 - Math - Pair of Linear Equations in Two Variables Page 62
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(1)
Let x be the fixed charge of the food and y be the charge for food per day.
According to the question,
x + 20y = 1000 ... (i)
x + 26y = 1180 ... (ii)
Subtracting equation (i) from equation (ii), we get
6y = 180
y = 180/6 = 30
Putting this value in equation (i), we get
x + 20 × 30 = 1000
x = 1000 - 600
x = 400
Hence, fixed charge = Rs 400 and charge per day = Rs 30
(2)
Let the fraction be x/y
According to the question,
x-1/y = 1/3
⇒ 3x - y = 3... (i)
x/y+8 = 1/4
⇒ 4x - y = 8 ... (ii)
Subtracting equation (i) from equation (ii), we get
x = 5 ... (iii)
Putting this value in equation (i), we get
15 - y = 3
y = 12
Hence, the fraction is 5/12.
(3)
Let the number of right answers and wrong answers be x and y respectively.
According to the question,
3x - y = 40 ... (i)
4x - 2y = 50
⇒ 2x - y = 25 ... (ii)
Subtracting equation (ii) from equation (i), we get
x = 15 ... (iii)
Putting this value in equation (ii), we get
30 - y = 25
y = 5
Therefore, number of right answers = 15
And number of wrong answers = 5
Total number of questions = 20
(4)
Let the speed of 1st car and 2nd car be u km/h and v km/h.
Respective speed of both cars while they are travelling in same direction = (u - v) km/h
Respective speed of both cars while they are travelling in opposite directions i.e., travelling towards each other = (u + v) km/h
According to the question,
5(u - v) = 100
⇒ u - v = 20 ... (i)
1(u + v) = 100 ... (ii)
Adding both the equations, we get
2u = 120
u = 60 km/h ... (iii)
Putting this value in equation (ii), we obtain
v = 40 km/h
Hence, speed of one car = 60 km/h and speed of other car = 40 km/h
(5)
Let length and breadth of rectangle be x unit and y unit respectively.
Area = xy
According to the question,
(x - 5) (y + 3) = xy - 9
⇒ 3x - 5y - 6 = 0 ... (i)
(x + 3) (y + 2) = xy + 67
⇒ 2x - 3y - 61 = 0 ... (ii)
By cross multiplication, we get
x/305-(-18) = y/-12-(-183) = 1/9-(-10)
x/323 = y/171 = 1/19
x = 17, y = 9
Hence, the length of the rectangle = 17 units and breadth of the rectangle = 9 units.
Let x be the fixed charge of the food and y be the charge for food per day.
According to the question,
x + 20y = 1000 ... (i)
x + 26y = 1180 ... (ii)
Subtracting equation (i) from equation (ii), we get
6y = 180
y = 180/6 = 30
Putting this value in equation (i), we get
x + 20 × 30 = 1000
x = 1000 - 600
x = 400
Hence, fixed charge = Rs 400 and charge per day = Rs 30
(2)
Let the fraction be x/y
According to the question,
x-1/y = 1/3
⇒ 3x - y = 3... (i)
x/y+8 = 1/4
⇒ 4x - y = 8 ... (ii)
Subtracting equation (i) from equation (ii), we get
x = 5 ... (iii)
Putting this value in equation (i), we get
15 - y = 3
y = 12
Hence, the fraction is 5/12.
(3)
Let the number of right answers and wrong answers be x and y respectively.
According to the question,
3x - y = 40 ... (i)
4x - 2y = 50
⇒ 2x - y = 25 ... (ii)
Subtracting equation (ii) from equation (i), we get
x = 15 ... (iii)
Putting this value in equation (ii), we get
30 - y = 25
y = 5
Therefore, number of right answers = 15
And number of wrong answers = 5
Total number of questions = 20
(4)
Let the speed of 1st car and 2nd car be u km/h and v km/h.
Respective speed of both cars while they are travelling in same direction = (u - v) km/h
Respective speed of both cars while they are travelling in opposite directions i.e., travelling towards each other = (u + v) km/h
According to the question,
5(u - v) = 100
⇒ u - v = 20 ... (i)
1(u + v) = 100 ... (ii)
Adding both the equations, we get
2u = 120
u = 60 km/h ... (iii)
Putting this value in equation (ii), we obtain
v = 40 km/h
Hence, speed of one car = 60 km/h and speed of other car = 40 km/h
(5)
Let length and breadth of rectangle be x unit and y unit respectively.
Area = xy
According to the question,
(x - 5) (y + 3) = xy - 9
⇒ 3x - 5y - 6 = 0 ... (i)
(x + 3) (y + 2) = xy + 67
⇒ 2x - 3y - 61 = 0 ... (ii)
By cross multiplication, we get
x/305-(-18) = y/-12-(-183) = 1/9-(-10)
x/323 = y/171 = 1/19
x = 17, y = 9
Hence, the length of the rectangle = 17 units and breadth of the rectangle = 9 units.
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