Direct And Inverse Proportions
Class 8th Mathematics RS Aggarwal Solution
- Observe the tables given below and in each one find whether x and y are…
- If x and y are directly proportional, find the value of x1, x2 and y2 in the…
- If truck covers a distance of 510 km in 34 litres of diesel. How much distance…
- A taxi charges a fare of Rs.2550 for journey of 150 km. How much would it…
- A loaded truck covers 16 km in 25 minutes. At the same speed, how far can it…
- If 18 dolls cost Rs.630, how many dolls can be bought for Rs.455?…
- If 9 kg of sugar costs Rs. 238.50, how much sugar can be bought for Rs.371?…
- The cost of 15 metres of cloth is Rs.981. What length of this cloth can be…
- In a model of a ship, the mast is 9 cm high, while the mast of the actual ship…
- In 8 days, the earth picks up (6.4 x 10^7) kg of dust from the atmosphere. How…
- A cars is travelling at the average speed of 50km/hr. How much distance would…
- Ravi walks at the uniform rate of 5 km/hr. What distance would he cover in 2…
- If the thickness of a pile of 12 cardboard is 65 mm, find the thickness of a…
- 11 men can dig 6 3/4 -metre-long trench in one day. How many men should be…
- Reenu type 540 words during half an hour. How many words would she type in 8…
- Observe the tables given below and in each case find whether x and y are…
- If x and y are inversely proportional, find the values of x1, x2, y1 and y2 in…
- If 35 men can reap a field in 8 days, in how many days can 20 men reap the same…
- 12 men can dig a pond in 8 days. How many men can dig it in 6 days?…
- 6 cows can graze a field in 28 days. How long would 14 cows take to graze the…
- A car takes 5 hours to reach a destination by travelling at the speed of 60…
- A factory required 42 machines to produce a given number of articles in 56…
- 7 taps of the same size fill a tank in 1 hour 36 minutes. How long will 8 taps…
- 8 taps of the same size fill a tank in 27 minutes. If two taps go out of order,…
- A farmer has enough food to feed 28 animals in his cattle for 9 days. How long…
- A garrison of 900 men had provisions for 42 days. However, a reinforcement of…
- In a hostel, 75 students had food provision for 24 days. If 15 students leave…
- A school has 9 periods a day each of 40 minutes duration. How long would each…
- If x and y vary inversely and x = 15 when y = 6, find y when x = 9.…
- If x and y vary inversely and x = 18 when y = 8, find x when y = 16.…
- If 14 kg of pulses cost Rs.882, what is the cost of 22 kg of pulses?A. Rs.1254…
- If 8 orange cost Rs.52, how many oranges can be bought for Rs.169?A. 13 B. 18…
- A machine fills 420 bottles in 3 hours. How many bottles will it fill in 5…
- A car is travelling at a uniform speed of 75 km/hr. How much distance will it…
- The weight of 12 sheets of a thick paper is 40 grams. How many sheets would…
- A pole 14 m high casts a shadow of 10 m. At the same time, what will be the…
- A photograph of a bacteria enlarged 50000 times attains a length of 5 cm. The…
- 6 Pipes fill a tank in 120 minutes, then 5 pipes will fill it inA. 100 min B.…
- 3 persons can build a wall in 4 days, then 4 persons can build it inA. 5 1/3…
- A car takes 2 hours to reach a destination by travelling at 60 km/hr. How long…
- 350 boxes can be placed in 25 cartons. How many boxes can be placed in 16 cartons?…
- The cost of 140 tennis balls is Rs.4900. Find the cost of 2 dozen such balls.…
- The railway fare for 61 km is Rs.183. Find the fare fare for 53 km.…
- 10 people can dig a trench in 6 days. How many people can dig it in 4 days?…
- 30 men can finish a piece of work in 28 days. How many days will be taken by 21 men to…
- A garrison of 200 men had provisions for 45 days. After 15 days, 40 more men join the…
- 6 pipes can fill a tank in 24 minutes. One pipe can fill it inA. 4 minutes B. 30…
- 14 workers can build a wall in 42 days. One worker can build it inA. 3 days B. 147 days…
- 35 men can reap a field in 8 days. In how many days can 20 men reap it?A. 14 days B. 28…
- A car is travelling at an average speed of 60 km per hour. How much distance will it…
- Rashmi types 510 words in half an hour. How many words would she type in 10 minutes?A.…
- x and y vary directly. When x = 3, then y = 36. What will be the value of x when y =…
- x and y vary inversely. When x = 15, then y = 6. What will be the value of y when x =…
- Fill in the blanks. (i) If 3 persons can do a piece of work in 4 days, then 4 persons…
Exercise 12a
Question 1.Observe the tables given below and in each one find whether x and y are proportional:
(i)
(ii)
(iii)
Answer:
Checking the 
ratio here
(i) 
, 
, 
, 
, 
; all are equal
(ii) 
, 
, 
, 
, 
; unequal
(iii) , 
, 
, 
, 
, 
; unequal
Question 2.
If x and y are directly proportional, find the value of x1, x2 and y2 in the table given below:
Answer:
We use the relation 
Here x1 = 5, y1 = 210 and x2 = 2
Here, 
⇒ x1 × 72 = 3 × 120
⇒ x1 = 
= 5
Now, 
⇒ x2 × 72 = 3 × 192
⇒ x2 = 
= 8
And 
⇒ y2 × 3 = 10 × 72
⇒ y2 = 
= 240
Question 3.
If truck covers a distance of 510 km in 34 litres of diesel. How much distance would it cover in 20 litres of diesel?
Answer:
Distance covered by truck increases, diesel required also increases. So it is a direct proportion.
Let required distance be x km, 
⇒ 34 × x = 510 × 20
⇒ x = 
= 300 km
Question 4.
A taxi charges a fare of Rs.2550 for journey of 150 km. How much would it charge for a journey of 124 km?
Answer:
Fare increases as the distance of the journey increases. So it is a direct proportion.
Let required fare be Rs x,
⇒ 50 × x = 2550 × 124
⇒ x = 
= Rs. 2108
Question 5.
A loaded truck covers 16 km in 25 minutes. At the same speed, how far can it travel in 5 hours?
Answer:
At the same speed, more the distance travelled more will be the time taken. So it is a direct proportion.
Let required distance be x km, but unit of time is different so we will write 25 min = 
⇒ 
× x = 16 × 5
⇒ x = 
= 192 km
Question 6.
If 18 dolls cost Rs.630, how many dolls can be bought for Rs.455?
Answer:
More the dolls, more will be the cost. So it is a direct proportion.
Let no. of dolls be x, 
⇒ 630 × x = 18 × 455
⇒ x = 
= 13
Question 7.
If 9 kg of sugar costs Rs. 238.50, how much sugar can be bought for Rs.371?
Answer:
More the amount of sugar, more will be the cost. So it is a direct proportion.
Let the amount of sugar be x kg, 
⇒ 238.50 × x = 9 × 371
⇒ x = 
= 14kg
Question 8.
The cost of 15 metres of cloth is Rs.981. What length of this cloth can be purchased for Rs.1308?
Answer:
More the length of cloth, more will be the cost. So it is a direct proportion.
Let the length of cloth be x metres 
⇒ 981 × x = 15 × 1308
⇒ x = 
= 20 meters
Question 9.
In a model of a ship, the mast is 9 cm high, while the mast of the actual ship is 15 m high. If
the length of the ship is 35 metres, how long is the model ship?
Answer:
The length and the height of ship and model should be proportional.
Height of mast of actual ship = 15 m
Height of model ship = 9 m
Length of ship = 35 m
Length of model = x
So,
15/35 = 9/x
Cross multiplying, we get,
x = (9 × 35)/15
x = 21 m
Length of model of the ship is 21 m.
Question 10.
In 8 days, the earth picks up
kg of dust from the atmosphere. How much dust will it pick up in 15 days?
Answer:
More the no. of days, more will be the dust picked by the earth. So it is a direct proportion.
Let the amount of dust be x kg,
⇒ 8 × x = 15 × 6.4 × 107
⇒ x = 
= 12 × 107 kg = 1.2 × 108 kg
Question 11.
A cars is travelling at the average speed of 50km/hr. How much distance would it travel in 1 hour 12 minutes?
Answer:
Average Speed = 
Let distance be x km, time = (1 + 
) hr = (1 + 
)hr = 
hr
⇒ 50km/hr = 
⇒ x = 50 × = 60 km
Question 12.
Ravi walks at the uniform rate of 5 km/hr. What distance would he cover in 2 hours 24 minutes?
Answer:
Uniform Speed =
Let distance be x km, time = 
⇒ 5km/hr = 
⇒ x = 5 × 
= 12 km
Question 13.
If the thickness of a pile of 12 cardboard is 65 mm, find the thickness of a pile of 312 such cardboards.
Answer:
More the no. of cardboards, more will be the thickness. So it is a direct proportion.
Let the thickness be x mm, 
⇒ 12 × x = 65 × 312
⇒ x = 
= 1690 mm = 1m 690mm = 1m 69 cm
Question 14.
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:
More the length of the trench, more will be the no. of men required to finish it in a day. So it is a direct proportion.
6
m = 
m
Let the no. of men be x, 
⇒ 
× x = 27 × 11
⇒ x = 
= 44 men
Question 15.
Reenu type 540 words during half an hour. How many words would she type in 8 minutes?
Answer:
More the time, more will be the no. of words typed. So it is a direct proportion.
Half an hour = 30 minutes
Let the no. of words be x, 
⇒ 30 × x = 540 × 8
⇒ x = 
= 144 words
Exercise 12b
Question 1.Observe the tables given below and in each case find whether x and y are inversely proportional:
(i)
(ii)
(iii)
Answer:
Here we check the values of x × y
(i) 6 × 9 = 54, 10 × 15 = 150, 14 × 21 = 294, 16 × 24 = 384 ; unequal
(i) 5 × 18 = 90, 9 × 10 = 90, 15 × 6 = 90, 3 × 30 = 90,45 × 2 = 90; equal
(i) 9 × 4 = 36, 3 × 12 = 36, 6 × 6 = 36, 6 × 9 = 54, 36 × 1 = 36 ;unequal
Question 2.
If x and y are inversely proportional, find the values of x1, x2, y1 and y2 in the table given below:
Answer:
8 × y1 = 16 × 5
⇒ y1 = 10
x1 × 4 = 16 × 5
⇒ x1 = 20
x2 × 2 = 16 × 5
⇒ x2 = 40
80 y2 = 16 × 5
⇒ y2 = 1
Question 3.
If 35 men can reap a field in 8 days, in how many days can 20 men reap the same field?
Answer:
More the number of men, lesser the days required. So, it is an inverse proportion.
Let the required no. of days be x.
⇒ 35 × 8 = 20 × x
⇒ x = 
= 14 days
Question 4.
12 men can dig a pond in 8 days. How many men can dig it in 6 days?
Answer:
More the number of men, lesser the days required. So, it is an inverse proportion.
Let the required no. of men be x.
⇒ 12 × 8 = 6 × x
⇒ x = 
= 16 men
Question 5.
6 cows can graze a field in 28 days. How long would 14 cows take to graze the same field?
Answer:
More the number of cows, lesser the days required to graze a field. So, it is an inverse proportion.
Let the required no. of days be x.
⇒ 6 × 28 = 14 × x
⇒ x = 
= 12 days
Question 6.
A car takes 5 hours to reach a destination by travelling at the speed of 60 km/hr. How long will it take when the car travels at the speed of 75 km/hr?
Answer:
More the speed of car, lesser the time required. So, it is an inverse proportion.
Let the required time be x hours.
⇒ 5 × 60 = 75 × x
⇒ x = 
= 4 hours
Question 7.
A factory required 42 machines to produce a given number of articles in 56 days. How many machines would be required to produce the same number of articles in 48 days?
Answer:
More the number of machines, lesser the days required to produce a given number of articles. So, it is an inverse proportion.
Let the required no. of machines be x.
⇒ 42 × 56 = 48 × x
⇒ x = 
= 49 machines
Question 8.
7 taps of the same size fill a tank in 1 hour 36 minutes. How long will 8 taps of the same size take to fill the tank?
Answer:
More the number of taps, lesser the time required to fill a tank. So, it is an inverse proportion.
Let the required time be x minutes.
⇒ 7 × (60 + 36) = 8 × x
⇒ x = 
= 84 minutes = 1 hour 24 minutes
Question 9.
8 taps of the same size fill a tank in 27 minutes. If two taps go out of order, how long would the remaining taps take to fill the tank?
Answer:
More the number of taps, lesser the time required to fill a tank. So, it is an inverse proportion.
Let the required time be x minutes.
⇒ 8 × 27 = 6 × x
⇒ x = 
= 36 minutes
Question 10.
A farmer has enough food to feed 28 animals in his cattle for 9 days. How long would the food last, if there were 8 more animals in his cattle?
Answer:
More the number of animals, lesser the days to feed them by given food. So, it is an inverse proportion.
Let the required days be x.
⇒ 28 × 9 = 36 × x
⇒ x = 
= 7 days
Question 11.
A garrison of 900 men had provisions for 42 days. However, a reinforcement of 500 men arrived. For how many days will the food last now?
Answer:
More the number of men, lesser the days to feed them by given food. So, it is an inverse proportion.
Let the required days be x.
⇒ 900 × 42 = 1400 × x
⇒ x = 
= 27 days
Question 12.
In a hostel, 75 students had food provision for 24 days. If 15 students leave the hostel, for how many days would the food provision last?
Answer:
More the number of men, lesser the days to feed them by given food. So, it is an inverse proportion.
Let the required days be x.
⇒ 75 × 24 = 60 × x
⇒ x = 
= 30 days
Question 13.
A school has 9 periods a day each of 40 minutes duration. How long would each period be, if the school has 8 periods a day, assuming the number of school hours to be the same?
Answer:
Lesser the number of periods in a day, more the duration of them. So, it is an inverse proportion.
Let the required duration be x.
⇒ 9 × 40 = 8 × x
⇒ x = 
= 45 days
Question 14.
If x and y vary inversely and x = 15 when y = 6, find y when x = 9.
Answer:
⇒ 15 × 6 = 9 × y
⇒ y = 
= 10
Question 15.
If x and y vary inversely and x = 18 when y = 8, find x when y = 16.
Answer:
⇒ 18 × 8 = x × 16
⇒ x = 
= 9
Exercise 12c
Question 1.If 14 kg of pulses cost Rs.882, what is the cost of 22 kg of pulses?
A. Rs.1254
B. Rs.1298
C. Rs.1342
D. 1386
Answer:
More the amount of pulses, more will be the cost. So it is a direct proportion.
Let the cost be x, 
⇒ x = 
= Rs 1386
Question 2.
If 8 orange cost Rs.52, how many oranges can be bought for Rs.169?
A. 13
B. 18
C. 26
D. 24
Answer:
More the amount of oranges, more will be the cost. So it is a direct proportion.
Let the amount be x, 
⇒ x = 
= Rs 26
Question 3.
A machine fills 420 bottles in 3 hours. How many bottles will it fill in 5 hours?
A. 252
B. 700
C. 504
D. 300
Answer:
More the no of bottles, more will be the time. So it is a direct proportion.
Let the bottles be x, 
⇒ x = 
= 700
Question 4.
A car is travelling at a uniform speed of 75 km/hr. How much distance will it cover in 20 minutes?
A. 25 km
B. 15 km
C. 30 km
D. 20 km
Answer:
Speed = 
Let distance be x km, time = 
hr = 
hr
⇒ 75km/hr = 
⇒ x = 75 × = 25 km
Question 5.
The weight of 12 sheets of a thick paper is 40 grams. How many sheets would weight 1 kg?
A. 480
B. 360
C. 300
D. none of these
Answer:
More the no of sheets, more will be the weight. So it is a direct proportion.
Let the sheets be x, 
⇒ x = 
= 300
Question 6.
A pole 14 m high casts a shadow of 10 m. At the same time, what will be the height of a tree, the length of whose shadow is 7 metres?
A. 20 m
B. 9.8 m
C. 5 m
D. none of these
Answer:
More the height, more will be the length of shadow. So it is a direct proportion.
Let the height of tree be x m, 
⇒ x = 
= 9.8m
Question 7.
A photograph of a bacteria enlarged 50000 times attains a length of 5 cm. The actual length of bacteria is
A. 1000 cm
B. 10-3 cm
C. 10-4 cm
D. 10-2 cm
Answer:
Let the actual length be x cm. When the bacteria is enlarged this much its length becomes 5 cm.
Then, 
Question 8.
6 Pipes fill a tank in 120 minutes, then 5 pipes will fill it in
A. 100 min
B. 144 min
C. 140 min
D. 108 min
Answer:
More the no. of pipes, lesser the time to fill the tank. So, it is an inverse proportion.
Let the required duration be x min.
⇒ 6 × 120 = 5 × x
⇒ x = 
= 144 minutes
Question 9.
3 persons can build a wall in 4 days, then 4 persons can build it in
A. 
days
B. 3 days
C.
days
D. none of these
Answer:
More the no. of persons, lesser the days to build. So, it is an inverse proportion.
Let the required duration be x days.
⇒ 3 × 4 = 4 × x
⇒ x = 
= 3 days
Question 10.
A car takes 2 hours to reach a destination by travelling at 60 km/hr. How long will it take while travelling at 80 km/hr?
A. 1 hr 30 min
B. 1hr 40 min
C. 2hrs 40 min
D. none of these
Answer:
More the speed, lesser the time to travel. So, it is an inverse proportion.
Let the required time be x hr.
⇒ 2 × 60 = 80 × x
⇒ x = 
= 1.5 hours = 1hour 30 minutes
Cce Test Paper-12
Question 1.350 boxes can be placed in 25 cartons. How many boxes can be placed in 16 cartons?
Answer:
More the no of boxes, more will be the cartons required. So it is a direct proportion.
Let the boxes be x, 
⇒ x = 
= 224
Question 2.
The cost of 140 tennis balls is Rs.4900. Find the cost of 2 dozen such balls.
Answer:
More the no of tennis balls, more will be the cost. So it is a direct proportion.
Let the cost be Rs x, 
⇒ x = 
= Rs 840
Question 3.
The railway fare for 61 km is Rs.183. Find the fare fare for 53 km.
Answer:
More the distance, more will be the fare. So it is a direct proportion.
Let the fare be Rs x, 
⇒ x = 
= Rs 159
Question 4.
10 people can dig a trench in 6 days. How many people can dig it in 4 days?
Answer:
More the no. of people, lesser the days. So, it is an inverse proportion.
Let the required time be x days.
⇒ 10 × 6 = 4 × x
⇒ x = 
= 15 days
Question 5.
30 men can finish a piece of work in 28 days. How many days will be taken by 21 men to finish.
Answer:
More the no. of men, lesser the days. So, it is an inverse proportion.
Let the required time be x days.
⇒ 30 × 28 = 21 × x
⇒ x = 
= 40 days
Question 6.
A garrison of 200 men had provisions for 45 days. After 15 days, 40 more men join the garrison. Find the number of days for which the remaining food will last.
Answer:
More the no. of men, lesser the days for which food last. So, it is an inverse proportion.
200 men had provisions for 45 days. After 15 days,200 men had provisions for 30days.
Now,
i) Number of men (x1) = 200 Provisions finished (y1) = 30 days
ii) after 15 days number of men joined are 40. Therefore,
Number of men after 15 days (x2) = 240
Let food last in number of days = y2
y2 = 
= 25 days
Question 7.
6 pipes can fill a tank in 24 minutes. One pipe can fill it in
A. 4 minutes
B. 30 minutes
C. 72 minutes
D. 144 minutes
Answer:
It is an inverse proportion.
If 6 pipes can do it in 24 minutes
Then time taken by 1 pipe = 24 × 6 = 144 minutes
Question 8.
14 workers can build a wall in 42 days. One worker can build it in
A. 3 days
B. 147 days
C. 294 days
D. 588 days
Answer:
It is an inverse proportion.
If 14 workers can do it in 42 days.
Then time taken by 1 worker = 14 × 42 = 588 days
Question 9.
35 men can reap a field in 8 days. In how many days can 20 men reap it?
A. 14 days
B. 28 days
C. 
D. none of these
Answer:
More the no. of men, lesser the days required. So, it is an inverse proportion.
Let the required no. be x days.
⇒ 35 × 8 = 20 × x
⇒ x = = 14 days
Question 10.
A car is travelling at an average speed of 60 km per hour. How much distance will it cover in 1 hour 12 minutes?
A. 50 km
B. 72 km
C. 63 km
D. 67.2 km
Answer:
Average Speed =
Let distance be x km, time = (1 + ) hr = (1 + )hr = hr
⇒ 60 km/hr =
⇒ x = 60 × = 72 km
Question 11.
Rashmi types 510 words in half an hour. How many words would she type in 10 minutes?
A. 85
B. 150
C. 170
D. 153
Answer:
More the time, more will be the no. of words typed. So it is a direct proportion.
Half an hour = 30 minutes
Let the no. of words be x, 
⇒ 30 × x = 510 × 10
⇒ x = 
= 170 words
Question 12.
x and y vary directly. When x = 3, then y = 36. What will be the value of x when y = 96?
A. 18
B. 12
C. 8
D. 4
Answer:
We use the relation Here x1 = 3, y1 = 36 and y2 = 96
Here, 
⇒ x1 × 36 = 3 × 96
⇒ x1 = 
= 8
Question 13.
x and y vary inversely. When x = 15, then y = 6. What will be the value of y when x = 9?
A. 10
B. 15
C. 54
D. 135
Answer:
⇒ 15 × 6 = 9 × y
⇒ y = = 10
Question 14.
Fill in the blanks.
(i) If 3 persons can do a piece of work in 4 days, then 4 persons can do it in……days.
(ii) If 5 pipes can fill a tank in 144 minutes, then 6 pipes can fill it in ……minutes.
(iii) A car covers a certain distance in 1 hr 30 minutes at 60 km per hour. If it moves at 45 km per hour, it will take……hours.
(iv) If 8 oranges cost Rs. 20.80, the cost of 5 oranges is Rs……
(v) The weight of 12 sheets of paper is 50 grams. How many sheets will weigh 500 grams?
Answer:
(i) By Inverse proportion
3 × 4 = 4 × (no. of days required)
(No. of days required) = 
= 3 days
(ii) By Inverse proportion
5 × 144 = 6 × (time required)
(Time required) = 
= 120 minutes
(iii) By Inverse proportion
90 minutes × 60 km/hr = 45 km/hr × (time taken in minutes)
(No. of days required) = 
= 120 minutes = 2 hours
(iv) More the oranges more will be the cost. So it is a direct proportion.
Let the cost be Rs x, 
⇒ 8 × x = 20.80 × 5
x = 
= Rs 13
(v) More the no. of sheets more will be the weight of them. So it is a direct proportion.
Let the no. of sheets be x, 
50 × x = 500 × 12
x = 
= 120 sheets