CBSE Class 8 Maths Direct and Inverse Variation HOTs

HOTS

Question. A vegetable vender has Rs. 2000 to buy potatoes which are available at the rate of Rs. 8 per kg. If the price of potatoes increases by 25%, find how much potatoes he can purchase with same amount?
Answer: 200kg

Question. The speed of a train 200 m long is 50 km per hour. How much time will it take to pass a platform 550m long?
Answer: 54 sec.

Question. A group of 100 masons can make a road in 3 months while another group of 60 masons can build the same length of road in 4 months. Which group is more efficient in the construction of the road.
Answer: Second Group

Question. If 721 men construct a bridge in 48 days then in how many days 1442 men can do this work.
Answer: 24 Days

Question. Shabnam takes 20 minutes to reach her school at an average speed of 6 Km/hour. If she is required to reach school in 24 minutes, what should be her speed?
Answer: 5 km/h

Question. A train 154 m long is running at 60km/hr. It takes 30 seconds to cross a bridge, find the length of the bridge.
Answer: 346 m

Question. A person cycles at a distance of 64 km. at a speed of 12 km/hr. How much time does he take to cover the distance?
Answer: 5 Hours 20 Minutes

Question. A train is running at a speed of 18 km/hr. If it crosses a pole in 35 seconds find the length of the train.
Answer: 175m

Question. A Hero Honda motor bike is running at 72km/hr. How much distance will it cover in 5 sec.?
Answer: 100 m

Question. In a military camp there is a food for 30 days for 50 soldiers. Assuming that average meal of every soldied is same. If 25 more soldiers join them then how many days this food will lost.
Answer: 20 Days

Question. How long will a boy take to run around a square field of side 35m, if he runs at a speed of 9km/hr?
Answer: 56 sec.

Question. Seema and Riya can do a piece of work in 9 days but Seema alone can do it in 12 days.
How long will Riya take to finish this work alone.
Answer: 36 Days

Question. A gun is fixed at a distance of 1.75 km away from a watch point. The watchman hears the sound of the gun fire after 5 seconds. Find the speed at which sound travels.'
Answer: 350 m/sec.

Question. Arnav weaves 35 seats of chairs in 7 days. How many days will he take to weave 140 similar seats of chairs.
Answer: 28 Days

Question. A is thrice as good a workman as B and therefore is able to finish a job in 60 days less than B.
Working together, find in how many days they can do it?
Answer: 22½ days

Question. 39 persons can repair a road in 12 days working 5 hours a day. In how many days will 30 persons, working 6 hours a day, complete the work?
Answer: 13

Question. In a dairy farm, 40 cows eat 40 bags of husk in 40 days. In how many days will one cow eat one bag of husk?
Answer: 40

Question. A can finish a work in 18 days and B can do the same work in 15 days. B worked for 10 days and left the job. In how many days can A alone finish the remaining work?
Answer: 6

CHALLENGES

1. Given that the volume of a sphere varies as the cube of its radius and its volume is 179.7 cm3, when radius is 3.5 cm. Find the volume when radius is 1.75 cm.

2. The distance through which a body falls from rest varies as square of time it takes to fall that distance. It is known that the body falls 64 cm in 2 seconds. How far does that body fall in 6 seconds?

3. The area of an isosceles right angled triangle varies directly as the square of the length of its leg. If the area is 18 cm² when the length of its leg is 6 cm, find (a) the relation between the area and length, and (b) the area of the triangle when the length of its leg is 5 cm.

4. In the diagram, the number in each rectangle represents its area. Fill the areas of the blank rectangles.

5. The electric current I, in amperes, in a circuit varies directly as the voltage V. When 12 volts are applied, the current is 4 amperes. What is the current when 18 volts are applied?

6. The number of kilograms of water in a person's body varies directly as the person's mass. A person with a mass of 90 kg contains 60 kg of water. How many kilograms of water is there in a person with a mass of 50 kg?

7. If 7 workers can build a trench in 25 hours, how many workers will be required to do the same work in 35 hours ?

8. A farmer has a stock of food enough to feed 28 animals for nine days. He buys 8 more animals which take the same quantity of food. How long would the food last now?

9. Suppose the rate of melting M grams per second of a sphere of ice is inversely proportional to the square of the radius, r cm. When r = 20, M = 0.6.
a. Find the constant of proportionality.
b. Find the rate of melting when r = 40 cm.
c. Find the radius when the rate of melting is 1 gram per second (give your answer correct to 2 decimal places).

10. When a ball is thrown upwards, the time, T seconds during which the ball remains in the air is directly proportional to the square root of the height, h metres, reached. We know T = 4.47 sec, when h = 25 m.
a. Find a formula for T in terms of h.
b. Find the value of T when h = 50 (give your answer correct to 3 decimal places.)
c. If the ball is thrown upwards and remains in the air for 5 seconds, find the height reached (correct to 2 decimal places).

11. Given that the relation is either y∝ x or y∝ 1/x , find which law is followed in the adjacent table. Find x when y=2. Find y when x=5. 

12. Each pair of values form an inverse variation. Find the missing value:
a. (3, 7), (8, y) b. (2, 5), (4, y)
c. (4, 6), (x, 3) d. (2.6, 4.5), (x, 6.3)

13. y varies inversely with the square of x. y=50 when x=4. Find y when x=5.

14. The volume V of gas varies inversely to the pressure P. The volume of a gas is 200 cm³ under pressure of 32 kg/cm². What will be its volume under pressure of 40 kg/cm2?

15. The time it takes to fly from Bangalore to New Delhi varies inversely as the speed of the plane. If the trip takes 3 hours at 800 km/hr, how long would it take at 700 km/hr?

16. A and B together can do a job in 12 days, while B alone can finish it in 30 days. In how many days can A alone finish the work ?

17. A is twice as good a workman as B and together they can finish a job in 24 days. How many days does A alone take to finish the job?

18. B is 60% more efficient than A. If A can finish a job in 15 days, how many days does B need to finish the same job?

19. A can do a piece of work in 14 days while B can do it in 21 days. They begin together and worked at it for 6 days. Then A fell ill and B had to complete the work alone. In how many days was the work completed ?

20. A takes twice as much time as B and thrice as much time as C to complete a work. If all of them work together, they can finish the work in 2 days. How much time do B and C working together need to finish it?

21. Two taps can fill a tub in 5 min and 7 min respectively. A pipe can empty it in 3 min. If all the three are kept open simultaneously, how much time does it take to fill the tub?

22. Two people P and Q can do a work in 20 days. After having worked together for 12 days, P fell ill and Q alone had to complete the work which took him another 10 days. In how many days can each of them finish the work alone?

23. Two taps A and B can fill a tank in 9 min and 12 min respectively. If B starts working two min after A, in how much more time will it fill the tank?

24. Akbar can do 5/8 part of a work in 20 days. He has worked for 2 days when Amar joins him. They work together and complete the work in 20 days. How much time will Amar take to finish the work alone?

25. A work can be done by 10 men and 15 women in 6 days. The same work can be done by 12 men and 27 women in 4 days. How many days will it take for 6 men and 11 women to complete the same work?

26. A tank has a leak at its bottom and gets filled in 15 hours. Had there been no leak, it could have been filled in 12 hours. Despite the leakage, it has been filled to its full capacity. In how much time will the tank be empty again?

27. Two pipes M and N take 24 min and 32 min respectively to fill a tank. If both the pipes are opened together, after how much time should N be closed, so that the tank gets filled in 18 min?

28. In a camp with 250 trainees, there is enough food to run the camp for 15 days. But after 4 days, 25 more trainees join the camp. How early will they have to wind up the camp now?

29. Four people, A, B, C, D together can do a job in 10 days. A and B together can do the same job in 25 days. They start working together. After 4 days, C and D had to leave due to health reasons.
How much more time will A and B require to complete the work?

30. Find the ratio between the time taken by a train 240 m long to cross a signal post and a bridge 80 m long.

31. The distance between two places A and B is 36 km. Two persons P and Q start simultaneously from A and B respectively with respective speeds of 8 km/hr and 4 km/hr. How much time will they take to meet each other if (a) they are moving in the opposite direction, and (b) if they are moving in the same direction. (Note that if they are moving in the same direction, the one walking faster can only catch up with the other; this decides the direction.)

32. Two trains 210 m and 240 m long run on parallel tracks in the same direction. Speed of the first train is 90 km/hr. The second train crosses the first train in 36 seconds. How much time do they take to cross each other if they move in the opposite direction?

33. Two trains 220 m and 180 m long cross each other in 16 seconds when they run in opposite direction. They take 1 minute to cross each other, when they run in the same direction. Find their speeds.

34. In a 100 m race, A beats B by 5 m and B beats C by 10m. By how much distance will A beat C in the race?

35. A stream flows at the rate of 5 km/hr. A boat traverses downstream 3 times the distance traversed by it upstream. Find the speed of the boat in still water.

36. Wind is blowing at a speed of 5 km/hr. A cyclist goes against the wind uptil a certain distance. While returning along the direction of the wind, he travelled double the distance in the same time. What is his speed if the wind is standstill?

37. Two cyclists start from the same place in the opposite directions. One goes at a speed of 18 km/hr and the other at 20 km/hr. After how much time will they be 47.5 km apart?

38. A train 100 m long is moving at a speed of 70 km/hr. A man is running at 10 km/hr in the same direction. How time does the train take to cross the man?

39. When do the hands of a clock coincide between the hours:
a. 3 and 4 b. 10 and 11

40. Find the time when the two hands of a clock make 180° between the hours:
a. 7 and 8 b. 9 and 10

41. Find the time when the two hands of a clock make the angle specified between the hours given:
a. 20° between 3 and 4 b. 40° between 5 and 6.

42. Find the angle between the two hands of a clock at 6 hours and 20 minutes.

43. A clock was set at 12 noon on Saturday. It showed 6 hours 12 minutes on Sunday at 6 a.m. What was the correct time if the clock showed 9 hours on Tuesday morning?

44. Two trains start from stations A and B at the same time and move towards each other with respective speeds 36 km/hr and 42 km/hr. When they meet, it is found that one train has travelled 48 km more than the other. Find the distance between the two stations.

SUMMARY

1. Two quantities are said to be in variation if any increase or decrease in one quantity causes the increase or decrease in other quantity.

2. Two quantities are said to vary directly if the increase (or decrease) in one quantity causes the increase (or decrease) in other quantity.

3. Two quantities x and y are said to be in direct variation if x/y =k, where k is the constant of variation.

4. Two quantities are said to vary inversely if the increase (or decrease) in one quantity causes the decrease (or increase) in other quantity.

5. Two quantities x and y are said to be in inverse variation if xy=k, where k is the constant of variation.

6. In solving problems on time and work.
a. One day's work = 1/Number of days to complete thework
b. Number of days to complete the work = 1/Oneday'swork
c. Time required to do a certain work = Work to be done/One day'swork

ERRORANALYSIS

1. Students get confused in both the varistions.

2. Students forget to write units.

3. Students do not write clearly the final answer obtained.

4. Students forget to write the variation used.

5. One day work has to be considered while doing questions of time and work.

ACTIVITY I

Objective : To show the relation between two things in direct and inverse variation by plotting the data of both on a graph paper.
Material Required : Data, Graph papers, Pencil, Scale
Observe the two tables given below :

Steps for the Activity
1. Take suitable scale, plot the points of Data 1 (Table A) on graph and join the joints (see fig. 1)
2. Take suitable scale, plot the points of Data 2 (Table B) on a different graph paper and join the points (see Fig. 2)
3. Observe and compare both graphs by moving from left to right. 

Observation
For 1st Graph : We observe that as we go from left to right the number of items purchased increases, so the total cost also increases.
For 2nd Graph :We observe that as the number of children increases, the number of days for which the food will last decreases.

Conclusion
We conclude that 1st graph is of direct variation and 2nd graph is of inverse variation.

Result
In direct variations, the two things given are in direct relation i.e, if one increases the other also increases and if one decreases the other also decreases. In inverse variation, the result is inverse i.e, if one increases the other will decrease and vice-versa.

ACTIVITY II

Across

2. Distance ÷ speed
Answer: Time

3. The distance covered by a car varies _____________ as its speed, time remain constant.
Answer: Directly

5. The increase (or decrease) in one quantity causes the increase (or decrease) in the other quantity.
Answer: Direct Variation

6. Time × speed
Answer: Distance

8. We find the value of one unit from the given quantity and then the value of the required quantity.
Answer: Unitary method

Down

1. The increase (or decrease) in one quantity causes the decrease (or increase) in the other quantity
Answer: Inverse variation

4. Distance ÷ Time
Answer: Speed

7. The time taken to finish the piece of work varies ____________ with the number of men at work.
Answer: Inversely