Direct And Inverse Variable
Class 8th Mathematics RD Sharma Solution
- Explain the concept of direct variation.
- Which of the following quantities vary directly with each other? (i) Number of…
- In which of the following tables x and y vary directly? (i) A 7 9 13 21 25 B 21…
- Fill in the blanks in each of the following so as to make the statement true:…
- Complite the following tables given that x varies directly as y. (i) X 2.5 . .…
- Find the constant of variation from the table given below: x 3 5 7 9 y 12 20 28…
- Rohit bought 12 registers for Rs 156, find the cost of 7 such registers.…
- Anupama takes 125 minutes in walding a distance of 100 metre. What distance…
- If the cost of 93 m of a certain kind of plastic sheet Rs 1395, then what would…
- Suneeta types 1080 words in one hour. What is her GWAM (gross words a minute…
- A car is travelling at the average speed of 50 km/hr. How much distance would…
- 68 boxes of certain commodity require a shelf-length of 13.6 m. How many boxes…
- In a library 136 copies of a certain book require a shelf-length of 3.4 metre.…
- The second class railway fare for 240 km of Journey is Rs 1500. What would be…
- If the thickness of a pile of 12 cardboards is 35 mm, find the thickness of a…
- The cost of 97 metre of cloth is Rs 242.50 What length of this can be…
- 11 men can dig 6 3/4 metre long trench in one day. How many men should be…
- A worker is paid Rs 210 for 6 days work. If his total income of the moth is Rs…
- A worker is paid Rs 200 for 8 days work. If he works for 20 days, how much…
- The anount of extension in an elastic string varies directly as the weight…
- The anount of extension in an elastic spring varies directly with the weight…
- In 10 days, the earth picks up 2.6 x 10^8 pounds of dust from the atmosphere.…
- In 15 days, the earth picks up 1.2 x 10^8 kg of dust from the atmosphere. In…
- In which of the following tables x and y vary inversely: (i) X 4 3 12 1 Y 6 8 2…
- It x and y vary inversely, fill in the following blanks: (i) X 12 16 .. 8 Y . 6…
- Which of the following quantities vary inversely as each other? (i) The number…
- It is known that for a given mass of gas, the volume v varies inversely as the…
- If 36 men can do a piece of work in 25 days, in how many days will 15 men do…
- A work force of 50 men with a contractor can finish a piece of work in 5…
- A work-force of 420 men with a contractor can finish a certain piece of work in…
- 1200 men can finish a stock of food in 35 days. How many more men should join…
- In a hostel of 50 girls, there are food provisions for 40 days. If 30 more…
- A car can finish a certain journey in 10 hours at the speed of 48 km/hr. By…
- 1200 soldiers in a fort had enough food for 28 days. After 4 days, some…
- Three spraying machines working together can finish painting a house in 60…
- A group of 3 friends staying together, consume 54 kg of wheat every month.…
- 55 cows can graze a field in 16 days. How many cows will graze the same field…
- 18 men can reap a field in 35 days. For reaping the same field in 15 days, how…
- A person has money to buy 25 cycles worth Rs. 500 each. How many cycles he…
- Raghu has enough money to buy 75 machines worth Rs. 200 each. How many…
- If x and y vary inversely as each other and (i) x = 3 when y = 8, find y when…
Exercise 10.1
Question 1.Explain the concept of direct variation.
Answer:
In case of direct variation both the quantities vary in the ratio i.e if one quantity is doubled then other will also get double. If one quantity gets half the other will also get half.
Question 2.
Which of the following quantities vary directly with each other?
(i) Number of articles (x) and their price (y)
(ii) Weight of articles (x) and their cost (y).
(iii) Distance x and time y, speed remaining the same.
(iv) Wages (y) and number of hours (x) of work.
(v) Speed (x) and time (y) distance covered remaining the same).
(vi) Area of a land (x) and its cost (y).
Answer:
(i) If number of articles is increasing then cost will also increase. So it is a case of direst proportion
(ii) Cost will also increase when weight of the article is increasing. So it is a case of direst proportion
(iii) Time will increase when distance will increase, if speed remaining constant. So it is a case of direst proportion
(iv) Wages will increase if the number of working hours is increasing. So it is a case of direst proportion
(v) For same distance time taken will reduce if speed is increased. So it is not a case of direst proportion
(vi) Cost for the land will increase if its area will increase. So it is a case of direst proportion
Question 3.
In which of the following tables x and y vary directly?
(i)
(ii)
(iii)
(iv)
Answer:
(i) Directly proportional.
Eplanation: In this table value of ‘b’ is thrice the value of ‘a’ in all the columns. Therefore ‘a’ and ‘b’ are directly proportional.
(ii) Directly proportional.
Eplanation: In this table value of ‘b’ is half of the value of ‘a’ in all the the columns. Therefore ‘a’ and ‘b’ are directly proportional.
(iii) Not directly proportional.
Eplanation: In this table value of ‘b’ is not thrice the value of ‘a’ in all the columns. Therefore ‘a’ and ‘b’ are not directly proportional.
(iv) Not directly proportional.
Eplanation: In this table value of ‘b’ is not varying in the same ratio as the value of ‘a’ in all the columns. Therefore ‘a’ and ‘b’ are not directly proportional.
Question 4.
Fill in the blanks in each of the following so as to make the statement true:
(i) Two quantites are said to very _______ with each other if they increase (decrease) together in such a way that the raio of the corresoponding values remains same
(ii) x and y are said to very directly with each if for some positive number k, _______ = k.
Answer:
(i) Two quantites are said to very directly with each other if they increase (decrease) together in such a way that the raio of the corresoponding values remains same
(ii) x and y are said to very directly with each if for some positive number k, wher kR = k.
Question 5.
Complite the following tables given that x varies directly as y.
(i)
(ii)
(iii)
(iv)
(v)
Answer:
(i)
On cross multiplication, we get
On cross multiplication, we get
On cross multiplication, we get
(ii)
On cross multiplication, we get
On cross multiplication, we get
On cross multiplication, we get
On cross multiplication, we get
On cross multiplication, we get
(iii)
On cross multiplication, we get
On cross multiplication, we get
On cross multiplication, we get
(iv)
On cross multiplication, we get
On cross multiplication, we get
On cross multiplication, we get
On cross multiplication, we get
On cross multiplication, we get
(v)
On cross multiplication, we get
On cross multiplication, we get
Question 6.
Find the constant of variation from the table given below:
Set up table and solve the following problems. Use unitary method to verfy the answer.
Answer:
The constant of variation in the given table is 4.
Explanation: For First Column: 
For Second Column: 
Similarly for other columns also y is four times x.
Question 7.
Rohit bought 12 registers for Rs 156, find the cost of 7 such registers.
Answer:
Let the cost of 7 registers is Rs 
On cross multiplication, we get
Therefore cost of 7 registers is Rs 91
Question 8.
Anupama takes 125 minutes in walding a distance of 100 metre. What distance would she cover in 315 minutes?
Answer:
Let the distance covered in 315 Min be 
Meter
On cross multiplication, we get
Therefore distance covered is 252 meter.
Question 9.
If the cost of 93 m of a certain kind of plastic sheet Rs 1395, then what would it cost to buy 105 m of such plastic sheet?
Answer:
Let the cost of 105 m plastic sheet be Rs
On cross multiplication, we get
Therefore cost is Rs 1575
Question 10.
Suneeta types 1080 words in one hour. What is her GWAM (gross words a minute rate)?
Answer:
Let the number of words typed in one minute be
On cross multiplication, we get
Therefore No of words typed in one minute are 18
Question 11.
A car is travelling at the average speed of 50 km/hr. How much distance would it travel in 12 minutes?
Answer:
One hour = 60 minutes
Let the the distance travelled be meter
On cross multiplication, we get
Therefore the distance covered is 10 km
Question 12.
68 boxes of certain commodity require a shelf-length of 13.6 m. How many boxes of the same commodity would occupy a shelf length of 20.4m?
Answer:
Let the number of boxes be
On cross multiplication, we get
Therefore number of boxes is 102
Question 13.
In a library 136 copies of a certain book require a shelf-length of 3.4 metre. How many copies of the same book would occupy a shelf length of 5.1 metres?
Answer:
Let the number of copies be
On cross multiplication, we get
Therefore number of copies is 204
Question 14.
The second class railway fare for 240 km of Journey is Rs 1500. What would be the fare for a Journey of 139.2 km?
Answer:
Let the fare be Rs
On cross multiplication, we get
Therefore fare is Rs 870
Question 15.
If the thickness of a pile of 12 cardboards is 35 mm, find the thickness of a pile of 294 cardboards.
Answer:
Let the fare be Rs
On cross multiplication, we get
Therefore thickness of the cardboard is 857.5 mm
Question 16.
The cost of 97 metre of cloth is Rs 242.50 What length of this can be purchased for Rs 302.50?
Answer:
Let the length of the cloth be meter
On cross multiplication, we get
Therefore length of the cloth is 121 M
Question 17.
11 men can dig
metre long trench in one day. How many men should be employed for digging 27 metre long trench of the same type in one day?
Answer:
Let the number of men required is
On cross multiplication, we get
Therefore number of men required is 44
Question 18.
A worker is paid Rs 210 for 6 days work. If his total income of the moth is Rs 875, for how many days did he work?
Answer:
Let the number of working days are
On cross multiplication, we get
Therefore number of days of work is 25 days
Question 19.
A worker is paid Rs 200 for 8 days work. If he works for 20 days, how much will he get?
Answer:
Let the income is Rs
On cross multiplication, we get
Therefore amount paid is Rs 500
Question 20.
The anount of extension in an elastic string varies directly as the weight hung on it. If a weight of 150 gm produces an extension of 2.9 cm, then what weight would produce an extension of 17.4 cm?
Answer:
Let the weight required is gm
On cross multiplication, we get
Therefore weight is 900 gm
Question 21.
The anount of extension in an elastic spring varies directly with the weight hung on it, If a weight of 250 gm produces an extension of 3.5 cm, find the extension produced by the wight of 700 gm.
Answer:
Let the extension produced is cm
On cross multiplication, we get
Therefore extension produced is 9.8 cm
Question 22.
In 10 days, the earth picks up 
pounds of dust from the atmosphere. How much dust will it pick up in 45 days?
Answer:
Let the weight of dust picked up is pounds
On cross multiplication, we get
Therefore the weight of dust picked up is 11.7 × 108 pounds
Question 23.
In 15 days, the earth picks up
kg of dust from the atmosphere. In how many days it will pick up
kg of dust?
Answer:
Let the number of days is
On cross multiplication, we get
Therefore the number of days required to pick up the dust is 60 days
Exercise 10.2
Question 1.In which of the following tables x and y vary inversely:
(i)
(ii)
(iii)
(iv)
Answer:
(i) x varies inversely as y since one quantiy is inceasing and other is decreasing in the same proportion.
(ii) x varies inversely as y since one quantiy is inceasing and other is decreasing in the same proportion.
(iii) x doesn’t varies inversely as y since the quanties are not varying in same proportion
(iv) x doesn’t varies inversely as y since the quanties are not varying in same proportion
Question 2.
It x and y vary inversely, fill in the following blanks:
(i)
(ii)
(iii)
Answer:
(i)
P = 
P = 
q = 
(ii)
(iii)
Question 3.
Which of the following quantities vary inversely as each other?
(i) The number of x men hired to construct a wall and the time y taken to finish the job.
(ii) The length x of a journey by bus and price y of the ticket.
(iii) Journey (x km) undertaken by a car and the petrol (y litres) consumed by it.
Answer:
(i) Varies inversely.
Since the time taken to construct the wall varies with the number of men hired. If the number of men hired is increased then time required for constructing wall shall decrease and vice-versa.
(ii) Varies directly.
Since the cost of bus journey will vary with distance. If the distance of journey will increase then cost of ticket will increase and vice-versa.
(iii) Varies directly.
Since the quantity of petrol required varies with the distance. If the distance of journey will increase then petrol required will also increase and vice-versa.
Question 4.
It is known that for a given mass of gas, the volume v varies inversely as the pressure p. Fill in the missing entries in the following table :
Answer:
Explanation:
Question 5.
If 36 men can do a piece of work in 25 days, in how many days will 15 men do it?
Answer:
Let the number of days required is
Therefore the number of days required to complete the work is 60 days
Question 6.
A work force of 50 men with a contractor can finish a piece of work in 5 months. In how many months the same work can be completed by 125 men?
Answer:
Let the number of months required is x.
As number of men increases the days to complete the work decreases.
Hence it is a inverse proportion.
Therefore the number of months required to complete the work is 2 months
Question 7.
A work-force of 420 men with a contractor can finish a certain piece of work in 9 months. How many extra men must he employ to complete the job in 7 months?
Answer:
Let the number of men required is
Therefore the number of extra men required to complete the work = 540 – 420 = 120 Men
Question 8.
1200 men can finish a stock of food in 35 days. How many more men should join them so that the same stock may last for 25 days?
Answer:
Let the number of men joined is
Therefore the number of extra men required to finish the stock = 1680 – 1200 = 480 Men
Question 9.
In a hostel of 50 girls, there are food provisions for 40 days. If 30 more girls join the hostel, how long will these provisions last?
Answer:
Let the number of days is 
Total number of girls = 50 + 30 = 80
Therefore the number of days the food will last is 25 days
Question 10.
A car can finish a certain journey in 10 hours at the speed of 48 km/hr. By how much should its speed be increased so that it may take only 8 hours to cover the same distance?
Answer:
Let the speed of car be km/hr
Therefore increased speed = 60 – 48 = 12 km/hr
Question 11.
1200 soldiers in a fort had enough food for 28 days. After 4 days, some soldiers were transferred to another fort and thus the food lasted now for 32 more days. How many soldiers left the fort?
Answer:
Let the number of soldiers transferred to another fort be
After four days number of soldiers left in the fort = 1200 – x
And for remaining soldiers (1200 – x ) the food lasted for 32 more days
Therefore 300 soldiers left for another fort.
Question 12.
Three spraying machines working together can finish painting a house in 60 minutes. How long will it take for 5 machines of the same capacity to do the same job?
Answer:
Let the time required be minutes
Therefore the time taken by five spraying machines is 36 Minutes.
Question 13.
A group of 3 friends staying together, consume 54 kg of wheat every month. Some more friends join this group and they find that the same amount of wheat lasts for 18 days. How many new members are there in this group now?
Answer:
Let the number of friends joined be
Therefore the number of friends joined the group is 5 – 3 = 2.
Question 14.
55 cows can graze a field in 16 days. How many cows will graze the same field in 10 days?
Answer:
Let the number of cows be
Therefore the number of cows is 88
Question 15.
18 men can reap a field in 35 days. For reaping the same field in 15 days, how many men are required?
Answer:
Let the number of men be
Therefore the number of men is 42
Question 16.
A person has money to buy 25 cycles worth Rs. 500 each. How many cycles he will be able to buy if each cycle is costing Rs. 125 more?
Answer:
Let the number of cycles be
Increased cost of one cycles is = 500 + 125 = Rs 625
Therefore the number of cycles is 20
Question 17.
Raghu has enough money to buy 75 machines worth Rs. 200 each. How many machines can he buy if he gets a discount of Rs. 50 on each machine?
Answer:
Let the number of machines be x.
Decreased cost of one machines is = 200 - 50 = Rs 150
Therefore the number of machines is 100.
Question 18.
If x and y vary inversely as each other and
(i) x = 3 when y = 8, find y when x = 4
(ii) x = 5, when y = 15, find x when y = 12
(iii) x = 30, find y when constant of variation = 900
(iv) y = 35, find x when constant of variation = 7
Answer:
(i) Let 
Therefore y is 6
(ii) Let 
Therefore x is 
(iii) Let y is p
Constant of variation is 900
Therefore p is 30
(iv) Let x is p
Constant of variation is 7
Therefore x is 