CBSE Class 8 Maths Linear Equations in One Variable HOTs

HOTS

Question.  A banana costs ₹ 6, while an apple costs ₹ 5. How many apples and bananas can be purchased for a total of ₹ 32?
Answer : 4 apples and 2 bananas

Question. Simplify the expression x/2 + x/3 + x/6
Answer : x

Question. Perimeter of the top of a table in the conference hall is 32cm. If the length of the table is 3 times its breadth, how long is the table?
Answer : 12m

Question. If sum of two numbers is 35 and one of them is 23. Form an equation for finding another number.
Answer : 23 + x = 35

Question. A train is moving at the speed of x km/hour. What distance will it cover in 15 hours if it stops for 1 hour at two stations.
Answer : 14x

Question. A man has Rs. x with him. He gave half to his wife (1/3)^rd to his son and rest of Rs. 1500 to his daughter. Form an equation to find x.
Answer : x/2 + x/3 + 1500 = x

Question. Find the value of P from the equation x + 2 = 1/2 (P + 4 ) where x = P - 4 
Answer : 8

Question. Form an expression : when twice a number x is added to thrice its reciprocal.
Answer : 2x + 3/x

Question. Ratio of three sides of a triangle are 1 : 3 : 5 and perimeter of the triangle is 270m. Find the sides.
Answer : 30m, 90m, 150m

Question. Find 3x – 2 when x = y + 1.
Answer : 3y + 1.

Question. How many variables are there in x2 + 4x + 1 = 0.
Answer : One

Question. Form an equation for a multiple of 5 added to 19 is 54.
Answer : 5x + 19 = 54

Question. 48 sweets are to be distributed among three friends A, B and C in such a way that B gets 5 sweets more than A and C gets 7 sweets more than A. Form an equation.
Answer : 3x + 12 = 48.

Question. Sum of two consecutive odd numbers is 56, form an equation.
Answer : (2x – 1) + (2x + 1) = 56

Question. Sum of ages of 5 friends is x. What is sum of their ages after 3 years?
Answer : x + 15

Question. Some monkeys were playing in two groups. In one group there were 5 more than (1/3)^rd of total monkeys and in other group (1/4)^th of the total monkeys. Write in the form of an equation.
Answer : x/3 + 5 + x/4 = x

Question. There are some benches in a classroom. If 4 students sit on each bench, three benches are left vacant; and if 3 students sit on each bench, 3 students are left standing. What is the total number of students in the class?
Answer : 48

Question. Devesh's school is 15 km from his home. He covers this distance in 3 hours, partly by walking and partly by cycling. If he walks at 3 km/h and cycles at 9 km/h, how much distance does he cover by cycling?
Answer : 9 km/hr

Question. Complete the magic square using linear equations so that the sum of each row, column and diagonal is equal. 

Answer : 

CHALLENGES

1. The sum of two numbers is 45 and their ratio is 7:8. Find the numbers.

2. Shona's mother is four times as old as Shona. After five years, her mother will be three times as old as Shona (at that time). What are their present ages?

3. The sum of three consecutive even numbers is 336. Find them.

4. Two friends A and B start a joint business with a capital ₹ 60,000. If A's share is twice that of B, how much has each invested?

5. Find the number from which 40 is subtracted to give one-third of the original number.

6. Find the number whose sixth part exceeds its eighth part by 3.

7. A house and a garden together cost ₹ 8,40,000. The price of the garden is 5/7 times the price of the house. Find the price of the house and the garden. 

8. Two farmers A and B together own a stock of grocery. They agree to divide it by its value. Farmer A takes 72 bags while farmer B takes 92 bags and gives ₹ 8,000 to A. What is the cost of each bag?

9. A father's age is four times that of his son. After 5 years, it will be three times that of his son. How many more years will it take if father's age is to be twice that of his son?

10. Find a number which when multiplied by 7 is as much above 132 as it was originally below it.

11. A person buys 25 pens worth ₹ 250, each of equal cost. He wants to keep 5 pens for himself and sell the remaining to recover his money. What should be the price of each pen?

12. The sum of the digits of a two-digit number is 12. If the number formed by reversing the digits is greater than the original number by 18, find the original number. Check your solution.

13. The distance between two stations is 340 km. Two trains start simultaneously from these stations on parallel tracks and cross each other. The speed of one of the them is greater than that of the other by 5km/hr. If the distance between two trains after 2 hours of their start is 30 km, find the speed of each train.

14. A steamer goes downstream and covers the distance between two ports in 4 hours, while it covers the same distance upstream in 5 hours. If the speed of the steamer upstream is 2 km/hr, find the speed of steamer in still water.

15. The numerator of a rational number is less than its denominator by 3. If the numerator becomes three times and the denominator is increased by 20, the new number becomes 1/8. Find the original number.

16. The digit at the tens place of a two digit number is three times the digit at the units place. If the sum of this number and the number formed by reversing its digits is 88, find the number.

17. The altitude of a triangle is five-thirds the length of its corresponding base. If the altitude is increased by 4 cm and the base decreased by 2 cm, the area of the triangle would remain the same. Find the base and altitude of the triangle.

18. One of the angles of a triangle is equal to the sum of the other two angles. If the ratio of the other two angles of the triangle is 4:5, find the angles of the triangle.

19. Three students Anita, Myra and Sahana are room mates. Together they have 80 books. Myra has 5 more than Sahana and Anita has 10 more than Myra. Find the number of books each one has.

20. Kirti needs 2/3 a cup of sugar to make a sweet. But she has only 3/8 a cup of sugar. She borrows the rest from her neighbour. If s is the amount of sugar that she borrows, write an algebraic equation for s.

21. The distance around a rectangular swimming pool is 90 metres. The rectangular boundary of the swimming pool is painted red. The cost of painting is ` c per metre. It is found that the cost of painting the length is twice the cost of painting the width. Find the dimensions of the swimming pool.

22. Sumitra has two bags; bag 1 containing red marbles and bag 2 containing green marbles. It is given that 1/5 of the total number of marbles are green in colour. It is also given that the number of red marbles is 120 more than the number of green marbles. Now she transfers some red marbles to the bag containing green marbles. After the transfer, bag 2 contains 3/10 of the total number of marbles. Find the number of red marbles in bag1 after the transfer.

23. A gardener plants N hibiscus plants, of which 12 did not grow at all. If 1/6 of the remaining hibiscus plants gave red flowers, write an equation for the number of hibiscus plants that gave red flowers. Find the number of plants that gave red flowers when N=48.

24. The distance from a place P to Q is 3(1)/2 km. Two persons A and B start together from P; A travels by a cycle at 6 km/hr and B walks at 3 km/hr. If A rests for 15 minutes at Q and returns, then at what distance from P does he meet B?

25. On the railway track AB, 220 km long, three trains P, Q, R travel at the rates 25 km, 20 km and 30 km, respectively. The trains P and Q leave A at 7 am and 8.15 am, respectively, while R leaves B at 10.30 am.When and where will P be equidistant from Q and R?

SUMMARY

1. An equation consisting of only linear polynomials is called a linear equation.

2. The solution of a linear equation is the value of the variable which satisfies the equation.

3. Rules for solving a linear equation:
a. Same number can be added on both the sides of the equation without affecting the equality.
b. Same number can be subtracted from both the sides of the equation without affecting the equality.
c. Both sides of the equation can be multiplied by the same number ‘m’ (m≠ 0) without affecting the equality.
d. Both sides of the equation can be divided by the same number ‘m’ (m≠ 0) without affecting the equality.

4. Any term of the equation can be transposed to the other side by changing its sign.

5. To solve equations of the form px q/n = rx + t /m , cross multiply to get a linear equation, i.e., m(px + q)=n(rx +t) and solve.

ERROR ANALYSIS

1. Students make mistake
a. informing relationship between a set of variables or unknown in mathematical equation.
b. in solving equations having brackets –
i. 2(x+3)=4x+2
2x+3=4x+2
Error
(It should be 2x+6)
ii. 3(x–5)=6x+3
3x+15=6x+3
Error
(It should be 3x – 15)
2. Error in transposition (Do not change the sign).

ACTIVITY

Objective: To trace the relationship in one variable
Example:
Let your age be 15 years and your teacher's age be 35 years.
What relationship can you think of between the two ages?
Let's try...
The obvious relationship is that your teacher's age is 5 years more than twice your age. In terms of one variable, if your age = x years, your teacher's age = (2x+5) years
But you already know that to solve for x, you need two conditions.
The second relationship is that the sum of your ages is 50.
i.e., x + 2x + 5 = 50
3x + 5 = 50
3x = 50–5 = 45
x= = 15
Isn't it fun to form equations using relations!
Repeat this activity and try forming age relationships between you and your:
a. father
b. mother
c. brother/sister