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Direct and Inverse Proportions Notes | Study Mathematics (Maths) Class 8 - Class 8

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 Page 1


Q u e s t i o n : 1
Observe the tables given below and in each one find whether x and y are proportional:
i
x 3 5 8 11 26
y 9 15 24 33 78
ii
x 2.5 4 7.5 10 14
y 10 16 30 40 42
iii
x 5 7 9 15 18 25
y 15 21 27 60 72 75
S o l u t i o n :
i
Clearly, 
x
y
=
3
9
=
5
15
=
8
24
=
11
33
=
26
78
=
1
3
constant Therefore, x and y are proportional.
ii
 Clearly, 
x
y
=
2.5
10
=
4
16
=
7.5
30
=
10
40
=
1
4
, while 
14
42
=
1
3
i. e. , 
2.5
10
=
4
16
=
7.5
30
=
10
40
is not equal to 
14
42
. Therefore, x and y are not proportional.
iii
 Clearly, 
x
y
=
5
15
=
7
21
=
9
27
=
25
75
=
1
3
, while 
15
60
=
18
72
=
1
4
i. e. , 
5
15
=
7
21
=
9
27
=
25
75
is not equal to 
15
60
 and
18
72
. Therefore, x and y are not proportional.
Q u e s t i o n : 2
If x and y are directly proportional, find the values of x
1
 , x
2
 and y
1
 in the table given below:
x 3 x
1
x
2 10
y 72 120 192 y
1
S o l u t i o n :
Since x and y are directly propotional, we have: 
3
72
=
x 1
120
=
x 2
192
=
10
y 1
Now, 
3
72
=
x 1
120
? x
1
=
120×3
72
 = 5
And, 
3
72
 = 
x 2
192
   ? x
2
 = 
3 × 192
72
 = 8
And, 
3
72
=
10
y 1
? y
1
=
72×10
3
= 240
Therefore, x
1
 = 5, x
2
 = 8 and y
1
 = 240
Q u e s t i o n : 3
A truck covers a distance of 510 km in 34 litres of diesel. How much distance would it cover in 20 litres of diesel?
S o l u t i o n :
Let the required distance be x km. Then, we have:
Quantity of diesel inlitres
 
34 20
Distance inkm 510 x
Clearly, the less the quantity of diesel consumed, the less is the distance covered.
So, this is a case of direct proportion.
Now, 
34
510
=
20
x
?
1
15
=
20
x
? x ×1 = 20 ×15 = 300
Therefore, the required distance is 300 km.
Q u e s t i o n : 4
A taxi charges a fare of Rs 2550 for a journey of 150 km. How much would it charge for a journey of 124 km?
S o l u t i o n :
Let the charge for a journey of 124 km be x.
Pricein 2550 x
Distanceinkm 150 124
More is the distance travelled, more will be the price.
So, it is a case of direct proportion.
?
2550
150
=
x
124
? x =
2550×124
150
= 2108
Thus, the taxi charges 2,108 for the distance of 124 km.
Q u e s t i o n : 5
A loaded truck covers 16 km in 25 minutes. At the same speed, how far can it travel in 5 hours?
S o l u t i o n :
Let the required distance be x km. Then, we have:
1 h = 60 mini. e. , 5 h = 5 ×60 = 300 min
( )
Page 2


Q u e s t i o n : 1
Observe the tables given below and in each one find whether x and y are proportional:
i
x 3 5 8 11 26
y 9 15 24 33 78
ii
x 2.5 4 7.5 10 14
y 10 16 30 40 42
iii
x 5 7 9 15 18 25
y 15 21 27 60 72 75
S o l u t i o n :
i
Clearly, 
x
y
=
3
9
=
5
15
=
8
24
=
11
33
=
26
78
=
1
3
constant Therefore, x and y are proportional.
ii
 Clearly, 
x
y
=
2.5
10
=
4
16
=
7.5
30
=
10
40
=
1
4
, while 
14
42
=
1
3
i. e. , 
2.5
10
=
4
16
=
7.5
30
=
10
40
is not equal to 
14
42
. Therefore, x and y are not proportional.
iii
 Clearly, 
x
y
=
5
15
=
7
21
=
9
27
=
25
75
=
1
3
, while 
15
60
=
18
72
=
1
4
i. e. , 
5
15
=
7
21
=
9
27
=
25
75
is not equal to 
15
60
 and
18
72
. Therefore, x and y are not proportional.
Q u e s t i o n : 2
If x and y are directly proportional, find the values of x
1
 , x
2
 and y
1
 in the table given below:
x 3 x
1
x
2 10
y 72 120 192 y
1
S o l u t i o n :
Since x and y are directly propotional, we have: 
3
72
=
x 1
120
=
x 2
192
=
10
y 1
Now, 
3
72
=
x 1
120
? x
1
=
120×3
72
 = 5
And, 
3
72
 = 
x 2
192
   ? x
2
 = 
3 × 192
72
 = 8
And, 
3
72
=
10
y 1
? y
1
=
72×10
3
= 240
Therefore, x
1
 = 5, x
2
 = 8 and y
1
 = 240
Q u e s t i o n : 3
A truck covers a distance of 510 km in 34 litres of diesel. How much distance would it cover in 20 litres of diesel?
S o l u t i o n :
Let the required distance be x km. Then, we have:
Quantity of diesel inlitres
 
34 20
Distance inkm 510 x
Clearly, the less the quantity of diesel consumed, the less is the distance covered.
So, this is a case of direct proportion.
Now, 
34
510
=
20
x
?
1
15
=
20
x
? x ×1 = 20 ×15 = 300
Therefore, the required distance is 300 km.
Q u e s t i o n : 4
A taxi charges a fare of Rs 2550 for a journey of 150 km. How much would it charge for a journey of 124 km?
S o l u t i o n :
Let the charge for a journey of 124 km be x.
Pricein 2550 x
Distanceinkm 150 124
More is the distance travelled, more will be the price.
So, it is a case of direct proportion.
?
2550
150
=
x
124
? x =
2550×124
150
= 2108
Thus, the taxi charges 2,108 for the distance of 124 km.
Q u e s t i o n : 5
A loaded truck covers 16 km in 25 minutes. At the same speed, how far can it travel in 5 hours?
S o l u t i o n :
Let the required distance be x km. Then, we have:
1 h = 60 mini. e. , 5 h = 5 ×60 = 300 min
( )
.
Distance inkm
 
16 x
Time inmin 25 300
Clearly, the more the time taken, the more will be the distance covered.
So, this is a case of direct proportion.
Now, 
16
25
=
x
300
? x =
16×300
25
? x = 192
Therefore, the required distance is 192 km.
Q u e s t i o n : 6
If 18 dolls cost Rs 630, how many dolls can be bought for Rs 455?
S o l u t i o n :
Let the required number of dolls be x. Then, we have:
 
 No of dolls 18 x
Cost of dolls 
inrupees
630 455
Clearly, the less the amount of money, the less will be the number of dolls bought.
So, this is a case of direct proportion.
Now, 
18
630
=
x
455
?
1
35
=
x
455
? x =
455
35
? x = 13
Therefore, 13 dolls can be bought for Rs 455.
Q u e s t i o n : 7
If 9 kg of sugar costs  238.50, how much sugar can be bought for  371?
S o l u t i o n :
Let the quantity of sugar bought for 371 be x kg.
Quantity inkg 9 x
Pricein 238.50 371
The price increases as the quantity increases. Thus, this is a case of direct proportion.
?
9
238.50
=
x
371
? x =
9×371
238.50
= 14
Thus, the quantity of sugar bought for 371 is 14 kg.
Q u e s t i o n : 8
The cost of 15 metres of a cloth is Rs 981. What length of this cloth can be purchased for Rs 1308?
S o l u t i o n :
Let the length of cloth be x m. Then, we have:
Length of cloth inmetres 15 x
Cost of cloth inrupees 981 1308
Clearly, more length of cloth can be bought by more amount of money.
So, this is a case of direct proportion.
Now, 
15
981
=
x
1308
? x =
15×1308
981
? x = 20
Therefore, 20 m of cloth can be bought for Rs 1,308.
Q u e s t i o n : 9
In a model of a ship, the mast is 9 cm high, while the mast of the actual ship is 15m high. If the length of the ship is 35 metres, how long is the model ship?
S o l u t i o n :
Let x m be the length of the model of the ship. Then, we have:
1 m = 100 cmTherefore, 15 m = 1500 cm35 m = 3500 cm
 
 Length of the mast incm Length of the  ship incm
Actual ship 1500 3500
Model of the
ship
9 x
Clearly, if the length of the actual ship is more, then the length of the model ship will also be more.
So, this is a case of direct proportion.
Now, 
1500
9
=
3500
x
? x =
3500×9
1500
? x = 21 cm
Therefore, the length of the model of the ship is 21 cm.
Q u e s t i o n : 1 0
In 8 days, the earth picks up (6.4 × 10
7
) kg of dust from the atmosphere. How much dust will it pick up in 15 days?
S o l u t i o n :
Let x kg be the required amount of dust. Then, we have:
 
No. of days 8 15
Dust inkg 6. 4 ×10
7
x
( )
Page 3


Q u e s t i o n : 1
Observe the tables given below and in each one find whether x and y are proportional:
i
x 3 5 8 11 26
y 9 15 24 33 78
ii
x 2.5 4 7.5 10 14
y 10 16 30 40 42
iii
x 5 7 9 15 18 25
y 15 21 27 60 72 75
S o l u t i o n :
i
Clearly, 
x
y
=
3
9
=
5
15
=
8
24
=
11
33
=
26
78
=
1
3
constant Therefore, x and y are proportional.
ii
 Clearly, 
x
y
=
2.5
10
=
4
16
=
7.5
30
=
10
40
=
1
4
, while 
14
42
=
1
3
i. e. , 
2.5
10
=
4
16
=
7.5
30
=
10
40
is not equal to 
14
42
. Therefore, x and y are not proportional.
iii
 Clearly, 
x
y
=
5
15
=
7
21
=
9
27
=
25
75
=
1
3
, while 
15
60
=
18
72
=
1
4
i. e. , 
5
15
=
7
21
=
9
27
=
25
75
is not equal to 
15
60
 and
18
72
. Therefore, x and y are not proportional.
Q u e s t i o n : 2
If x and y are directly proportional, find the values of x
1
 , x
2
 and y
1
 in the table given below:
x 3 x
1
x
2 10
y 72 120 192 y
1
S o l u t i o n :
Since x and y are directly propotional, we have: 
3
72
=
x 1
120
=
x 2
192
=
10
y 1
Now, 
3
72
=
x 1
120
? x
1
=
120×3
72
 = 5
And, 
3
72
 = 
x 2
192
   ? x
2
 = 
3 × 192
72
 = 8
And, 
3
72
=
10
y 1
? y
1
=
72×10
3
= 240
Therefore, x
1
 = 5, x
2
 = 8 and y
1
 = 240
Q u e s t i o n : 3
A truck covers a distance of 510 km in 34 litres of diesel. How much distance would it cover in 20 litres of diesel?
S o l u t i o n :
Let the required distance be x km. Then, we have:
Quantity of diesel inlitres
 
34 20
Distance inkm 510 x
Clearly, the less the quantity of diesel consumed, the less is the distance covered.
So, this is a case of direct proportion.
Now, 
34
510
=
20
x
?
1
15
=
20
x
? x ×1 = 20 ×15 = 300
Therefore, the required distance is 300 km.
Q u e s t i o n : 4
A taxi charges a fare of Rs 2550 for a journey of 150 km. How much would it charge for a journey of 124 km?
S o l u t i o n :
Let the charge for a journey of 124 km be x.
Pricein 2550 x
Distanceinkm 150 124
More is the distance travelled, more will be the price.
So, it is a case of direct proportion.
?
2550
150
=
x
124
? x =
2550×124
150
= 2108
Thus, the taxi charges 2,108 for the distance of 124 km.
Q u e s t i o n : 5
A loaded truck covers 16 km in 25 minutes. At the same speed, how far can it travel in 5 hours?
S o l u t i o n :
Let the required distance be x km. Then, we have:
1 h = 60 mini. e. , 5 h = 5 ×60 = 300 min
( )
.
Distance inkm
 
16 x
Time inmin 25 300
Clearly, the more the time taken, the more will be the distance covered.
So, this is a case of direct proportion.
Now, 
16
25
=
x
300
? x =
16×300
25
? x = 192
Therefore, the required distance is 192 km.
Q u e s t i o n : 6
If 18 dolls cost Rs 630, how many dolls can be bought for Rs 455?
S o l u t i o n :
Let the required number of dolls be x. Then, we have:
 
 No of dolls 18 x
Cost of dolls 
inrupees
630 455
Clearly, the less the amount of money, the less will be the number of dolls bought.
So, this is a case of direct proportion.
Now, 
18
630
=
x
455
?
1
35
=
x
455
? x =
455
35
? x = 13
Therefore, 13 dolls can be bought for Rs 455.
Q u e s t i o n : 7
If 9 kg of sugar costs  238.50, how much sugar can be bought for  371?
S o l u t i o n :
Let the quantity of sugar bought for 371 be x kg.
Quantity inkg 9 x
Pricein 238.50 371
The price increases as the quantity increases. Thus, this is a case of direct proportion.
?
9
238.50
=
x
371
? x =
9×371
238.50
= 14
Thus, the quantity of sugar bought for 371 is 14 kg.
Q u e s t i o n : 8
The cost of 15 metres of a cloth is Rs 981. What length of this cloth can be purchased for Rs 1308?
S o l u t i o n :
Let the length of cloth be x m. Then, we have:
Length of cloth inmetres 15 x
Cost of cloth inrupees 981 1308
Clearly, more length of cloth can be bought by more amount of money.
So, this is a case of direct proportion.
Now, 
15
981
=
x
1308
? x =
15×1308
981
? x = 20
Therefore, 20 m of cloth can be bought for Rs 1,308.
Q u e s t i o n : 9
In a model of a ship, the mast is 9 cm high, while the mast of the actual ship is 15m high. If the length of the ship is 35 metres, how long is the model ship?
S o l u t i o n :
Let x m be the length of the model of the ship. Then, we have:
1 m = 100 cmTherefore, 15 m = 1500 cm35 m = 3500 cm
 
 Length of the mast incm Length of the  ship incm
Actual ship 1500 3500
Model of the
ship
9 x
Clearly, if the length of the actual ship is more, then the length of the model ship will also be more.
So, this is a case of direct proportion.
Now, 
1500
9
=
3500
x
? x =
3500×9
1500
? x = 21 cm
Therefore, the length of the model of the ship is 21 cm.
Q u e s t i o n : 1 0
In 8 days, the earth picks up (6.4 × 10
7
) kg of dust from the atmosphere. How much dust will it pick up in 15 days?
S o l u t i o n :
Let x kg be the required amount of dust. Then, we have:
 
No. of days 8 15
Dust inkg 6. 4 ×10
7
x
( )
Clearly, more amount of dust will be collected in more number of days.
So, this is a case of direct proportion.
Now, 
8
6.4×10
7
=
15
x
? x =
15×6.4×10
7
8
? x = 12 ×10
7
Therefore, 12,00,00,000 kg of dust will be picked up in 15 days.
Q u e s t i o n : 1 1
A car is travelling at the average speed of 50 km/hr. How much distance would it travel in 1 hour 12 minutes?
S o l u t i o n :
Let x km be the required distance. Then, we have:
1 h = 60 mini. e. , 1h 12 min = (60 +12) min = 72 min
 
Distance covered inkm 50 x
Time inmin 60 72
Clearly, more distance will be covered in more time.
So, this is a case of direct proportion.
Now, 
50
60
=
x
72
? x =
50×72
60
? x = 60
Therefore, the distance travelled by the car in 1 h 12 min is 60 km.
Q u e s t i o n : 1 2
Ravi walks at the uniform rate of 5 km/hr. What distance would he cover in 2 hours 24 minutes?
S o l u t i o n :
Let x km be the required distance covered by Ravi in 2 h 24 min.
Then, we have:
1 h = 60mini. e. , 2 h 24 min = (120 +24) min = 144 min
 
Distance covered 
inkm
5 x
Time inmin 60 144
Clearly, more distance will be covered in more time.
So, this is a case of direct proportion.
Now,
5
60
=
x
144
? x =
5×144
60
? x = 12
Therefore, the distance covered by Ravi in 2 h 24 min is 12 km.
Q u e s t i o n : 1 3
If the thickness of a pile of 12 cardboards is 65 mm, find the thickness of a pile of 312 such cardboards.
S o l u t i o n :
Let x mm be the required thickness. Then, we have:
 
Thickness of cardboard inmm 65 x
No. of cardboards 12 312
Clearly, when the number of cardboard is more, the thickness will also be more.
So, it is a case of direct proportion.
Now, 
65
12
=
x
312
? x =
65×312
12
? x = 1690
Therefore, the thickness of the pile of 312 cardboards is 1690 mm.
Q u e s t i o n : 1 4
11 men can dig 6
3
4
-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?
S o l u t i o n :
Let x be the required number of men.
Now, 6
3
4
 m =
27
4
 m
Then, we have:
Number of men 11 x
Length of trench inmetres
27
4
27
Clearly, the longer the trench, the greater will be the number of men required.
So, it is a case of direct proportion.
Now, 
11
27
4
=
x
27
?
11×4
27
=
x
27
? x = 44
Therefore, 44 men should be employed to dig a trench of length 27 m.
Q u e s t i o n : 1 5
Page 4


Q u e s t i o n : 1
Observe the tables given below and in each one find whether x and y are proportional:
i
x 3 5 8 11 26
y 9 15 24 33 78
ii
x 2.5 4 7.5 10 14
y 10 16 30 40 42
iii
x 5 7 9 15 18 25
y 15 21 27 60 72 75
S o l u t i o n :
i
Clearly, 
x
y
=
3
9
=
5
15
=
8
24
=
11
33
=
26
78
=
1
3
constant Therefore, x and y are proportional.
ii
 Clearly, 
x
y
=
2.5
10
=
4
16
=
7.5
30
=
10
40
=
1
4
, while 
14
42
=
1
3
i. e. , 
2.5
10
=
4
16
=
7.5
30
=
10
40
is not equal to 
14
42
. Therefore, x and y are not proportional.
iii
 Clearly, 
x
y
=
5
15
=
7
21
=
9
27
=
25
75
=
1
3
, while 
15
60
=
18
72
=
1
4
i. e. , 
5
15
=
7
21
=
9
27
=
25
75
is not equal to 
15
60
 and
18
72
. Therefore, x and y are not proportional.
Q u e s t i o n : 2
If x and y are directly proportional, find the values of x
1
 , x
2
 and y
1
 in the table given below:
x 3 x
1
x
2 10
y 72 120 192 y
1
S o l u t i o n :
Since x and y are directly propotional, we have: 
3
72
=
x 1
120
=
x 2
192
=
10
y 1
Now, 
3
72
=
x 1
120
? x
1
=
120×3
72
 = 5
And, 
3
72
 = 
x 2
192
   ? x
2
 = 
3 × 192
72
 = 8
And, 
3
72
=
10
y 1
? y
1
=
72×10
3
= 240
Therefore, x
1
 = 5, x
2
 = 8 and y
1
 = 240
Q u e s t i o n : 3
A truck covers a distance of 510 km in 34 litres of diesel. How much distance would it cover in 20 litres of diesel?
S o l u t i o n :
Let the required distance be x km. Then, we have:
Quantity of diesel inlitres
 
34 20
Distance inkm 510 x
Clearly, the less the quantity of diesel consumed, the less is the distance covered.
So, this is a case of direct proportion.
Now, 
34
510
=
20
x
?
1
15
=
20
x
? x ×1 = 20 ×15 = 300
Therefore, the required distance is 300 km.
Q u e s t i o n : 4
A taxi charges a fare of Rs 2550 for a journey of 150 km. How much would it charge for a journey of 124 km?
S o l u t i o n :
Let the charge for a journey of 124 km be x.
Pricein 2550 x
Distanceinkm 150 124
More is the distance travelled, more will be the price.
So, it is a case of direct proportion.
?
2550
150
=
x
124
? x =
2550×124
150
= 2108
Thus, the taxi charges 2,108 for the distance of 124 km.
Q u e s t i o n : 5
A loaded truck covers 16 km in 25 minutes. At the same speed, how far can it travel in 5 hours?
S o l u t i o n :
Let the required distance be x km. Then, we have:
1 h = 60 mini. e. , 5 h = 5 ×60 = 300 min
( )
.
Distance inkm
 
16 x
Time inmin 25 300
Clearly, the more the time taken, the more will be the distance covered.
So, this is a case of direct proportion.
Now, 
16
25
=
x
300
? x =
16×300
25
? x = 192
Therefore, the required distance is 192 km.
Q u e s t i o n : 6
If 18 dolls cost Rs 630, how many dolls can be bought for Rs 455?
S o l u t i o n :
Let the required number of dolls be x. Then, we have:
 
 No of dolls 18 x
Cost of dolls 
inrupees
630 455
Clearly, the less the amount of money, the less will be the number of dolls bought.
So, this is a case of direct proportion.
Now, 
18
630
=
x
455
?
1
35
=
x
455
? x =
455
35
? x = 13
Therefore, 13 dolls can be bought for Rs 455.
Q u e s t i o n : 7
If 9 kg of sugar costs  238.50, how much sugar can be bought for  371?
S o l u t i o n :
Let the quantity of sugar bought for 371 be x kg.
Quantity inkg 9 x
Pricein 238.50 371
The price increases as the quantity increases. Thus, this is a case of direct proportion.
?
9
238.50
=
x
371
? x =
9×371
238.50
= 14
Thus, the quantity of sugar bought for 371 is 14 kg.
Q u e s t i o n : 8
The cost of 15 metres of a cloth is Rs 981. What length of this cloth can be purchased for Rs 1308?
S o l u t i o n :
Let the length of cloth be x m. Then, we have:
Length of cloth inmetres 15 x
Cost of cloth inrupees 981 1308
Clearly, more length of cloth can be bought by more amount of money.
So, this is a case of direct proportion.
Now, 
15
981
=
x
1308
? x =
15×1308
981
? x = 20
Therefore, 20 m of cloth can be bought for Rs 1,308.
Q u e s t i o n : 9
In a model of a ship, the mast is 9 cm high, while the mast of the actual ship is 15m high. If the length of the ship is 35 metres, how long is the model ship?
S o l u t i o n :
Let x m be the length of the model of the ship. Then, we have:
1 m = 100 cmTherefore, 15 m = 1500 cm35 m = 3500 cm
 
 Length of the mast incm Length of the  ship incm
Actual ship 1500 3500
Model of the
ship
9 x
Clearly, if the length of the actual ship is more, then the length of the model ship will also be more.
So, this is a case of direct proportion.
Now, 
1500
9
=
3500
x
? x =
3500×9
1500
? x = 21 cm
Therefore, the length of the model of the ship is 21 cm.
Q u e s t i o n : 1 0
In 8 days, the earth picks up (6.4 × 10
7
) kg of dust from the atmosphere. How much dust will it pick up in 15 days?
S o l u t i o n :
Let x kg be the required amount of dust. Then, we have:
 
No. of days 8 15
Dust inkg 6. 4 ×10
7
x
( )
Clearly, more amount of dust will be collected in more number of days.
So, this is a case of direct proportion.
Now, 
8
6.4×10
7
=
15
x
? x =
15×6.4×10
7
8
? x = 12 ×10
7
Therefore, 12,00,00,000 kg of dust will be picked up in 15 days.
Q u e s t i o n : 1 1
A car is travelling at the average speed of 50 km/hr. How much distance would it travel in 1 hour 12 minutes?
S o l u t i o n :
Let x km be the required distance. Then, we have:
1 h = 60 mini. e. , 1h 12 min = (60 +12) min = 72 min
 
Distance covered inkm 50 x
Time inmin 60 72
Clearly, more distance will be covered in more time.
So, this is a case of direct proportion.
Now, 
50
60
=
x
72
? x =
50×72
60
? x = 60
Therefore, the distance travelled by the car in 1 h 12 min is 60 km.
Q u e s t i o n : 1 2
Ravi walks at the uniform rate of 5 km/hr. What distance would he cover in 2 hours 24 minutes?
S o l u t i o n :
Let x km be the required distance covered by Ravi in 2 h 24 min.
Then, we have:
1 h = 60mini. e. , 2 h 24 min = (120 +24) min = 144 min
 
Distance covered 
inkm
5 x
Time inmin 60 144
Clearly, more distance will be covered in more time.
So, this is a case of direct proportion.
Now,
5
60
=
x
144
? x =
5×144
60
? x = 12
Therefore, the distance covered by Ravi in 2 h 24 min is 12 km.
Q u e s t i o n : 1 3
If the thickness of a pile of 12 cardboards is 65 mm, find the thickness of a pile of 312 such cardboards.
S o l u t i o n :
Let x mm be the required thickness. Then, we have:
 
Thickness of cardboard inmm 65 x
No. of cardboards 12 312
Clearly, when the number of cardboard is more, the thickness will also be more.
So, it is a case of direct proportion.
Now, 
65
12
=
x
312
? x =
65×312
12
? x = 1690
Therefore, the thickness of the pile of 312 cardboards is 1690 mm.
Q u e s t i o n : 1 4
11 men can dig 6
3
4
-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?
S o l u t i o n :
Let x be the required number of men.
Now, 6
3
4
 m =
27
4
 m
Then, we have:
Number of men 11 x
Length of trench inmetres
27
4
27
Clearly, the longer the trench, the greater will be the number of men required.
So, it is a case of direct proportion.
Now, 
11
27
4
=
x
27
?
11×4
27
=
x
27
? x = 44
Therefore, 44 men should be employed to dig a trench of length 27 m.
Q u e s t i o n : 1 5
Reenu types 540 words during half an hour. How many words would she type in 8 minutes?
S o l u t i o n :
Let Reenu type x words in 8 minutes.
 
No. of words 540 x
Time taken inmin 30 8
Clearly, less number of words will be typed in less time. 
So, it is a case of direct proportion.
Now,
540
30
=
x
8
? x =
540×8
30
? x = 144
Therefore, Reenu will type 144 words in 8 minutes.
Q u e s t i o n : 1 6
Observe the tables given below and in each case find whether x and y are inversely proportional:
i
x 6 10 14 16
y 9 15 21 24
ii
x 5 9 15 3 45
y 18 10 6 30 2
iii
x 9 3 6 36
y 4 12 9 1
S o l u t i o n :
i
Clearly, 6 ×9 ? 10 ×15 ? 14 ×21 ? 16 ×24Therefore, x and y are not inversely proportional.
ii
Clearly, 5 ×18 = 9 ×10 = 15 ×6 = 3 ×30 = 45 ×2 = 90 = (consant)Therefore, x and y are inversely proportional.
iii
Clearly, 9 ×4 = 3 ×12 = 36 ×1 = 36, while 6 ×9 = 54i. e. , 9 ×4 = 3 ×12 = 36 ×1 ? 6 ×9Therefore, x and y are not inversely proportional.
Q u e s t i o n : 1 7
If x and y are inversely proportional, find the values of x
1
, x
2
, y
1
 and y
2
 in the table given below:
x 8 x
1 16 x
2 80
y y
1 4 5 2 y
2
S o l u t i o n :
 Since x and y are inversely proportional, xy must be a constant.
Therefore, 8 ×y
1
= x
1
×4 = 16 ×5 = x
2
×2 = 80 ×y
2
Now, 16 ×5 = 8 ×y
1
?
80
8
= y
1
? y
1
= 1016 ×5 = x
1
×4 ?
80
4
= x
1
? x
1
= 2016 ×5 = x
2
×2 ?
80
2
= x
2
? x
2
= 4016 ×5 = 80 ×y
2
?
80
80
= y
2
? y
2
=
Q u e s t i o n : 1 8
If 35 men can reap a field in 8 days, in how many days can 20 men reap the same field?
S o l u t i o n :
Let x be the required number of days. Then, we have:
 
No. of days 8 x
No. of men 35 20
Clearly, less men will take more days to reap the field.
So, it is a case of inverse proportion.
Now, 8 × 35 = x × 20 ?
8 × 35
20
= x ? 14 = x
Therefore, 20 men can reap the same field in 14 days.
Q u e s t i o n : 1 9
12 men can dig a pond in 8 days. How many men can dig it in 6 days?
S o l u t i o n :
Let x be the required number of men. Then, we have:
 
No. of days 8 6
No. of men 12 x
Clearly, more men will require less number of days to dig the pond.
So, it is a case of inverse proportion.
Now, 8 × 12 = 6 × x ? x =
8 × 12
6
 ? x = 16
Therefore, 16 men can dig the pond in 6 days.
Page 5


Q u e s t i o n : 1
Observe the tables given below and in each one find whether x and y are proportional:
i
x 3 5 8 11 26
y 9 15 24 33 78
ii
x 2.5 4 7.5 10 14
y 10 16 30 40 42
iii
x 5 7 9 15 18 25
y 15 21 27 60 72 75
S o l u t i o n :
i
Clearly, 
x
y
=
3
9
=
5
15
=
8
24
=
11
33
=
26
78
=
1
3
constant Therefore, x and y are proportional.
ii
 Clearly, 
x
y
=
2.5
10
=
4
16
=
7.5
30
=
10
40
=
1
4
, while 
14
42
=
1
3
i. e. , 
2.5
10
=
4
16
=
7.5
30
=
10
40
is not equal to 
14
42
. Therefore, x and y are not proportional.
iii
 Clearly, 
x
y
=
5
15
=
7
21
=
9
27
=
25
75
=
1
3
, while 
15
60
=
18
72
=
1
4
i. e. , 
5
15
=
7
21
=
9
27
=
25
75
is not equal to 
15
60
 and
18
72
. Therefore, x and y are not proportional.
Q u e s t i o n : 2
If x and y are directly proportional, find the values of x
1
 , x
2
 and y
1
 in the table given below:
x 3 x
1
x
2 10
y 72 120 192 y
1
S o l u t i o n :
Since x and y are directly propotional, we have: 
3
72
=
x 1
120
=
x 2
192
=
10
y 1
Now, 
3
72
=
x 1
120
? x
1
=
120×3
72
 = 5
And, 
3
72
 = 
x 2
192
   ? x
2
 = 
3 × 192
72
 = 8
And, 
3
72
=
10
y 1
? y
1
=
72×10
3
= 240
Therefore, x
1
 = 5, x
2
 = 8 and y
1
 = 240
Q u e s t i o n : 3
A truck covers a distance of 510 km in 34 litres of diesel. How much distance would it cover in 20 litres of diesel?
S o l u t i o n :
Let the required distance be x km. Then, we have:
Quantity of diesel inlitres
 
34 20
Distance inkm 510 x
Clearly, the less the quantity of diesel consumed, the less is the distance covered.
So, this is a case of direct proportion.
Now, 
34
510
=
20
x
?
1
15
=
20
x
? x ×1 = 20 ×15 = 300
Therefore, the required distance is 300 km.
Q u e s t i o n : 4
A taxi charges a fare of Rs 2550 for a journey of 150 km. How much would it charge for a journey of 124 km?
S o l u t i o n :
Let the charge for a journey of 124 km be x.
Pricein 2550 x
Distanceinkm 150 124
More is the distance travelled, more will be the price.
So, it is a case of direct proportion.
?
2550
150
=
x
124
? x =
2550×124
150
= 2108
Thus, the taxi charges 2,108 for the distance of 124 km.
Q u e s t i o n : 5
A loaded truck covers 16 km in 25 minutes. At the same speed, how far can it travel in 5 hours?
S o l u t i o n :
Let the required distance be x km. Then, we have:
1 h = 60 mini. e. , 5 h = 5 ×60 = 300 min
( )
.
Distance inkm
 
16 x
Time inmin 25 300
Clearly, the more the time taken, the more will be the distance covered.
So, this is a case of direct proportion.
Now, 
16
25
=
x
300
? x =
16×300
25
? x = 192
Therefore, the required distance is 192 km.
Q u e s t i o n : 6
If 18 dolls cost Rs 630, how many dolls can be bought for Rs 455?
S o l u t i o n :
Let the required number of dolls be x. Then, we have:
 
 No of dolls 18 x
Cost of dolls 
inrupees
630 455
Clearly, the less the amount of money, the less will be the number of dolls bought.
So, this is a case of direct proportion.
Now, 
18
630
=
x
455
?
1
35
=
x
455
? x =
455
35
? x = 13
Therefore, 13 dolls can be bought for Rs 455.
Q u e s t i o n : 7
If 9 kg of sugar costs  238.50, how much sugar can be bought for  371?
S o l u t i o n :
Let the quantity of sugar bought for 371 be x kg.
Quantity inkg 9 x
Pricein 238.50 371
The price increases as the quantity increases. Thus, this is a case of direct proportion.
?
9
238.50
=
x
371
? x =
9×371
238.50
= 14
Thus, the quantity of sugar bought for 371 is 14 kg.
Q u e s t i o n : 8
The cost of 15 metres of a cloth is Rs 981. What length of this cloth can be purchased for Rs 1308?
S o l u t i o n :
Let the length of cloth be x m. Then, we have:
Length of cloth inmetres 15 x
Cost of cloth inrupees 981 1308
Clearly, more length of cloth can be bought by more amount of money.
So, this is a case of direct proportion.
Now, 
15
981
=
x
1308
? x =
15×1308
981
? x = 20
Therefore, 20 m of cloth can be bought for Rs 1,308.
Q u e s t i o n : 9
In a model of a ship, the mast is 9 cm high, while the mast of the actual ship is 15m high. If the length of the ship is 35 metres, how long is the model ship?
S o l u t i o n :
Let x m be the length of the model of the ship. Then, we have:
1 m = 100 cmTherefore, 15 m = 1500 cm35 m = 3500 cm
 
 Length of the mast incm Length of the  ship incm
Actual ship 1500 3500
Model of the
ship
9 x
Clearly, if the length of the actual ship is more, then the length of the model ship will also be more.
So, this is a case of direct proportion.
Now, 
1500
9
=
3500
x
? x =
3500×9
1500
? x = 21 cm
Therefore, the length of the model of the ship is 21 cm.
Q u e s t i o n : 1 0
In 8 days, the earth picks up (6.4 × 10
7
) kg of dust from the atmosphere. How much dust will it pick up in 15 days?
S o l u t i o n :
Let x kg be the required amount of dust. Then, we have:
 
No. of days 8 15
Dust inkg 6. 4 ×10
7
x
( )
Clearly, more amount of dust will be collected in more number of days.
So, this is a case of direct proportion.
Now, 
8
6.4×10
7
=
15
x
? x =
15×6.4×10
7
8
? x = 12 ×10
7
Therefore, 12,00,00,000 kg of dust will be picked up in 15 days.
Q u e s t i o n : 1 1
A car is travelling at the average speed of 50 km/hr. How much distance would it travel in 1 hour 12 minutes?
S o l u t i o n :
Let x km be the required distance. Then, we have:
1 h = 60 mini. e. , 1h 12 min = (60 +12) min = 72 min
 
Distance covered inkm 50 x
Time inmin 60 72
Clearly, more distance will be covered in more time.
So, this is a case of direct proportion.
Now, 
50
60
=
x
72
? x =
50×72
60
? x = 60
Therefore, the distance travelled by the car in 1 h 12 min is 60 km.
Q u e s t i o n : 1 2
Ravi walks at the uniform rate of 5 km/hr. What distance would he cover in 2 hours 24 minutes?
S o l u t i o n :
Let x km be the required distance covered by Ravi in 2 h 24 min.
Then, we have:
1 h = 60mini. e. , 2 h 24 min = (120 +24) min = 144 min
 
Distance covered 
inkm
5 x
Time inmin 60 144
Clearly, more distance will be covered in more time.
So, this is a case of direct proportion.
Now,
5
60
=
x
144
? x =
5×144
60
? x = 12
Therefore, the distance covered by Ravi in 2 h 24 min is 12 km.
Q u e s t i o n : 1 3
If the thickness of a pile of 12 cardboards is 65 mm, find the thickness of a pile of 312 such cardboards.
S o l u t i o n :
Let x mm be the required thickness. Then, we have:
 
Thickness of cardboard inmm 65 x
No. of cardboards 12 312
Clearly, when the number of cardboard is more, the thickness will also be more.
So, it is a case of direct proportion.
Now, 
65
12
=
x
312
? x =
65×312
12
? x = 1690
Therefore, the thickness of the pile of 312 cardboards is 1690 mm.
Q u e s t i o n : 1 4
11 men can dig 6
3
4
-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?
S o l u t i o n :
Let x be the required number of men.
Now, 6
3
4
 m =
27
4
 m
Then, we have:
Number of men 11 x
Length of trench inmetres
27
4
27
Clearly, the longer the trench, the greater will be the number of men required.
So, it is a case of direct proportion.
Now, 
11
27
4
=
x
27
?
11×4
27
=
x
27
? x = 44
Therefore, 44 men should be employed to dig a trench of length 27 m.
Q u e s t i o n : 1 5
Reenu types 540 words during half an hour. How many words would she type in 8 minutes?
S o l u t i o n :
Let Reenu type x words in 8 minutes.
 
No. of words 540 x
Time taken inmin 30 8
Clearly, less number of words will be typed in less time. 
So, it is a case of direct proportion.
Now,
540
30
=
x
8
? x =
540×8
30
? x = 144
Therefore, Reenu will type 144 words in 8 minutes.
Q u e s t i o n : 1 6
Observe the tables given below and in each case find whether x and y are inversely proportional:
i
x 6 10 14 16
y 9 15 21 24
ii
x 5 9 15 3 45
y 18 10 6 30 2
iii
x 9 3 6 36
y 4 12 9 1
S o l u t i o n :
i
Clearly, 6 ×9 ? 10 ×15 ? 14 ×21 ? 16 ×24Therefore, x and y are not inversely proportional.
ii
Clearly, 5 ×18 = 9 ×10 = 15 ×6 = 3 ×30 = 45 ×2 = 90 = (consant)Therefore, x and y are inversely proportional.
iii
Clearly, 9 ×4 = 3 ×12 = 36 ×1 = 36, while 6 ×9 = 54i. e. , 9 ×4 = 3 ×12 = 36 ×1 ? 6 ×9Therefore, x and y are not inversely proportional.
Q u e s t i o n : 1 7
If x and y are inversely proportional, find the values of x
1
, x
2
, y
1
 and y
2
 in the table given below:
x 8 x
1 16 x
2 80
y y
1 4 5 2 y
2
S o l u t i o n :
 Since x and y are inversely proportional, xy must be a constant.
Therefore, 8 ×y
1
= x
1
×4 = 16 ×5 = x
2
×2 = 80 ×y
2
Now, 16 ×5 = 8 ×y
1
?
80
8
= y
1
? y
1
= 1016 ×5 = x
1
×4 ?
80
4
= x
1
? x
1
= 2016 ×5 = x
2
×2 ?
80
2
= x
2
? x
2
= 4016 ×5 = 80 ×y
2
?
80
80
= y
2
? y
2
=
Q u e s t i o n : 1 8
If 35 men can reap a field in 8 days, in how many days can 20 men reap the same field?
S o l u t i o n :
Let x be the required number of days. Then, we have:
 
No. of days 8 x
No. of men 35 20
Clearly, less men will take more days to reap the field.
So, it is a case of inverse proportion.
Now, 8 × 35 = x × 20 ?
8 × 35
20
= x ? 14 = x
Therefore, 20 men can reap the same field in 14 days.
Q u e s t i o n : 1 9
12 men can dig a pond in 8 days. How many men can dig it in 6 days?
S o l u t i o n :
Let x be the required number of men. Then, we have:
 
No. of days 8 6
No. of men 12 x
Clearly, more men will require less number of days to dig the pond.
So, it is a case of inverse proportion.
Now, 8 × 12 = 6 × x ? x =
8 × 12
6
 ? x = 16
Therefore, 16 men can dig the pond in 6 days.
Q u e s t i o n : 2 0
6 cows can graze a field in 28 days. How long would 14 cows take to graze the same field?
S o l u t i o n :
Let x be the number of days. Then, we have:
 
No. of days 28 x
No. of cows 6 14
Clearly, more number of cows will take less number of days to graze the field.
So, it is a case of inverse proportion.
Now, 28 × 6 = x × 14 ? x =
28 × 6
14
 ? x = 12
Therefore, 14 cows will take 12 days to graze the field.
Q u e s t i o n : 2 1
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?
S o l u t i o n :
Let x h be the required time taken. Then, we have:
 
Speed inkm/h 60 75
Time inh 5 x
Clearly, the higher the speed, the lesser will be the the time taken.
So, it is a case of inverse proportion.
Now, 60 ×5 = 75 ×x ? x =
60×5
75
? x = 4
Therefore, the car will reach its destination in 4 h if it travels at a speed of 75 km/h.
Q u e s t i o n : 2 2
A factory requires 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?
S o l u t i o n :
Let x be the number of machines required to produce same number of articles in 48.
Then, we have:
 
No. of machines 42 x
No. of days 56 48
Clearly, less number of days will require more number of machines.
So, it is a case of inverse proportion.
Now, 42 ×56 = x ×48 ? x =
42×56
48
? x = 49
Therefore, 49 machines would be required to produce the same number of articles in 48 days.
Q u e s t i o n : 2 3
7 teps 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?
S o l u t i o n :
Let x be the required number of taps. Then, we have:
1 h = 60 min
i.e., 1 h 36 min = 60 +36
min = 96 min
 
No. of taps 7 8
Time inmin 96 x
Clearly, more number of taps will require less time to fill the tank.
So, it is a case of inverse proportion.
Now, 7 ×96 = 8 ×x ? x =
7×96
8
? x = 84
Therefore, 8 taps of the same size will take 84 min or 1 h 24 min to fill the tank.
Q u e s t i o n : 2 4
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?
S o l u t i o n :
Let x min be the required number of time. Then, we have:
No. of taps 8 6
Time inmin 27 x
Clearly, less number of taps will take more time to fill the tank .
So, it is a case of inverse proportion.
Now, 8 ×27 = 6 ×x ? x =
8×27
6
? x = 36
Therefore, it will take 36 min to fill the tank.
Q u e s t i o n : 2 5
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?
S o l u t i o n :
Let x be the required number of days. Then, we have:
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