Linear Equations In Two Variables - Holidays H.W

1.  Solve: 47x + 31y = 63, 31x + 47y = 15.
2. The sum of the digits of a two digit number is 12. The number obtained by interchanging the two digits exceeds the given number by 18. Find the number.
3. The monthly incomes of A and B are in the ratio of 5 : 4 and their monthly expenditures are in the ratio of 7 : 5. If each saves Rs. 3000 per month, find the monthly income of each.
4. 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 26days, pays Rs 1180 as hostel charges. Find the fixed charges and the cost of the food per day.
5. Nine times a two-digit number is the same as twice the number obtained by interchanging the digits of the number. If one digit of the number exceeds the other number by 7, find the number.
6. If 4 times the area of a smaller square is subtracted from the area of a larger square, the result is 144 m². The sum of the areas of the two squares is 464 m². Determine the sides of the two squares.
7. Half the perimeter of a rectangular garden, whose length is 4 m more than its breadth is 36 m. Find the dimensions of the garden.
8. A man travels 370 km partly by train and partly by car. If he covers 250 km by train and rest by car, it takes him 4 hours. But if he travels 130 km by train and rest by car, he takes 18 minutes longer. Find the speed of the train and that of the car.
9.   The age of a father is equal to sum of the ages of his 6 children. After 15 years, twice the age of the father will be the sum of ages of his children. Find the age of the father.
10.  The auto fare for the first kilometre is fixed and is different from the rate per km for the remaining distance. A man pays Rs. 57 for the distance of 16 km and Rs. 92 for a distance of 26 km. Find the auto fare for the first kilometre and for each successive kilometre.
11. A two digit number is obtained by either multiplying the sum of the digits by 8 or adding 1, or by multiplying the difference of the digits by 13 and adding 2. Find the number. How many such numbers are there?
12. A man wished to give Rs 12 to each person and found that he fell short of Rs 6 when he wanted to give to all persons. He therefore, distributed Rs 9 to each person and found that Rs 9 were left over. How much money did he have and how many persons were there?
13. A person sells two articles together for Rs. 46, making a profit of 10% on one and 20% on the other. If he had sold each article at 15% profit, the result would have been the same. At what price does he sell each article?
14.   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 breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.
15. A railway half ticket costs half the full fare, but the reservation charges are the same on a half ticket as on a full ticket. One reserved first class ticket from station A to B cots Rs 2530. Also, one reserved first class ticket and one reserved first class half ticket from A to B cost Rs 3810. Find the full first class fare from station A to B and also the reservation charges for a ticket.
16. It takes 12 hours to fill a swimming pool using two pipes. If the pipe of larger diameter is used for 4 hours and the pipe of smaller diameter for 9 hours, only half the pool can be filled. How long would it take for each pipe to fill the pool separately?
17. In a rectangle, if the length is increased and breadth reduced each by 2 meters, the area is reduced by 28 sq. m. If the length is reduced by 1 m and the breadth is increased by 2 m, the area increases by 33 sq. m. Find the length and the breadth of the rectangle.
18.  There are two classrooms A and B containing students. If 5 students are shifted from room A to room B, the resulting number of students in the two rooms become equal. If 5 students are shifted from room B to room A, the resulting number of students in room A becomes double the number of students left in room B. Find the original number of students in the two rooms.
19. Students of a class are made to stand in rows. If four students are extra in each row, there would be two rows less. If 4 students are less in each row, there would be 4 more rows. Find the number of students in the class.
20.  A person invested some amount @ 12% simple interest and some other amount @ 10% simple interest. He received an yearly interest of Rs. 13000. But if he had interchanged the invested amounts, he would have received Rs. 400 more as interest. How much amount did he invest at different rates?