Direct and Inverse Variations - Exercise 10.1 | Mathematics (Maths) Class 8 PDF Download

 Page 1


Q u e s t i o n : 1
Explain the concept of direct variation.
S o l u t i o n :
When two variables are connected to each other in such a way that if we increase the value of one variable, the value of other variable also increases and vice -versa. Similarly, if we decrease
Q u e s t i o n : 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) distancecoveredremainingthesame
.
vi
Area of a land (x) and its cost (y).
S o l u t i o n :
i
The number of articles is directly related to the price. Therefore, they will vary directly with each other.
ii
The number of articles is directly related to the weight of the articles. Therefore, they will vary directly with each other.
iii
Speed is constant. Therefore, distance and time does not vary directly.
iv
The number of hours is directly related to the wages. Therefore, it is a direct variation.
v
Distance is constant. Therefore, speed and time does not vary directly.
vi
If the area of a land is large, its cost will also be high. Thus, it is a direct variation.
Thus, the respective values in i
, ii
, iv
and vi
vary directly with each other.
Q u e s t i o n : 3
In which of the following tables x and y vary directly?
i
a 7 9 13 21 25
b 21 27 39 63 75
ii
a 10 20 30 40 46
b 5 10 15 20 23
iii
a 2 3 4 5 6
b 6 9 12 17 20
iv
a 1
2
2
2
3
2
4
2
5
2
b 1
3
2
3
3
3
4
3
5
3
S o l u t i o n :
If x and y vary directly, the ratio of the corresponding values of x and y remains constant. i
x
y
=
7
21
=
1
3
x
y
=
9
27
=
1
3
x
y
=
13
39
=
1
3
x
y
=
21
63
=
1
3
x
y
=
25
75
=
1
3
In all the cases, the ratio is the same. Therefore,
Q u e s t i o n : 4
Fill in the blanks in each of the following so as to make the statement true:
i
Two quantities are said to vary.... with each other if they increase decrease
together in such a way that the ratio of the corresponding values remains same.
ii
x and y are said to vary directly with each other if for some positive number k,...... = k.
iii
If u = 3 v, then u and v vary .... with each other.
S o l u t i o n :
i directly ii x and y are said to vary directly with each other if 
x
y
= k, where k is a positive number iii because u = 3v, u and y vary directly with each other
Q u e s t i o n : 5
Complite the following tables given that x varies directly as y.
i
( )
( ) ( ) ( )
Page 2


Q u e s t i o n : 1
Explain the concept of direct variation.
S o l u t i o n :
When two variables are connected to each other in such a way that if we increase the value of one variable, the value of other variable also increases and vice -versa. Similarly, if we decrease
Q u e s t i o n : 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) distancecoveredremainingthesame
.
vi
Area of a land (x) and its cost (y).
S o l u t i o n :
i
The number of articles is directly related to the price. Therefore, they will vary directly with each other.
ii
The number of articles is directly related to the weight of the articles. Therefore, they will vary directly with each other.
iii
Speed is constant. Therefore, distance and time does not vary directly.
iv
The number of hours is directly related to the wages. Therefore, it is a direct variation.
v
Distance is constant. Therefore, speed and time does not vary directly.
vi
If the area of a land is large, its cost will also be high. Thus, it is a direct variation.
Thus, the respective values in i
, ii
, iv
and vi
vary directly with each other.
Q u e s t i o n : 3
In which of the following tables x and y vary directly?
i
a 7 9 13 21 25
b 21 27 39 63 75
ii
a 10 20 30 40 46
b 5 10 15 20 23
iii
a 2 3 4 5 6
b 6 9 12 17 20
iv
a 1
2
2
2
3
2
4
2
5
2
b 1
3
2
3
3
3
4
3
5
3
S o l u t i o n :
If x and y vary directly, the ratio of the corresponding values of x and y remains constant. i
x
y
=
7
21
=
1
3
x
y
=
9
27
=
1
3
x
y
=
13
39
=
1
3
x
y
=
21
63
=
1
3
x
y
=
25
75
=
1
3
In all the cases, the ratio is the same. Therefore,
Q u e s t i o n : 4
Fill in the blanks in each of the following so as to make the statement true:
i
Two quantities are said to vary.... with each other if they increase decrease
together in such a way that the ratio of the corresponding values remains same.
ii
x and y are said to vary directly with each other if for some positive number k,...... = k.
iii
If u = 3 v, then u and v vary .... with each other.
S o l u t i o n :
i directly ii x and y are said to vary directly with each other if 
x
y
= k, where k is a positive number iii because u = 3v, u and y vary directly with each other
Q u e s t i o n : 5
Complite the following tables given that x varies directly as y.
i
( )
( ) ( ) ( )
x 2.5 ... ... 15
y 5 8 12 ...
ii
x 5 ... 10 35 25 ...
y 8 12 ... ... ... 32
iii
x 6 8 10 ... 20
y 15 20 ... 40 ...
iv
x 4 9 ... ... 3 ...
y 16 ... 48 36 ... 4
v
x 3 5 7 9
y ... 20 28 ...
S o l u t i o n :
Here, x and y vary directly. ? x = ky i x = 2. 5 and y = 5i. e. ,  2. 5 = k × 5 ? k = 
2.5
5
 = 0. 5For y = 8 and k = 0. 5, we have: x = ky ? x = 8 × 0. 5 = 4For y = 12 and k = 0. 5,
Q u e s t i o n : 6
Find the constant of variation from the table given below:
x 3 5 7 9
y 12 20 28 36
Set up a table and solve the following problems. Use unitary method to verify the answer.
S o l u t i o n :
Since it is a direct variation, 
x
y
 = k. For x = 3 and y = 12, we have: k = 
3
12
 = 
1
4
Thus, in all cases, k = 
1
4
Q u e s t i o n : 7
Rohit bought 12 registers for Rs 156, find the cost of 7 such registers.
S o l u t i o n :
Let the cost of 7 registers be ? Rs x.
 
Register 12 7
CostinRs. 156 x
If he buys less number of registers, the cost will also be less. Therefore, it is a direct variation. We get: 12: 7 = 156: x ?
12
7
 = 
156
x
Applying cross muliplication, we get: x = 
156×7
12
= 91Thus
Q u e s t i o n : 8
Anupama takes 125 minutes in walking a distance of 100 metre. What distance would she cover in 315 minutes?
S o l u t i o n :
 Let the distance travelled in 315 minutes be x km.
 
Time inminute 125 315
Distanceinmetre 100 x
If the distance travelled is more, the time needed to cover it will also be more. Therefore, it is a direct variation. We get: 125: 315 = 100: x ?
125
315
 = 
100
x
Applying cross muliplication, we get
Q u e s t i o n : 9
If the cost of 93 m of a certain kind of plastic sheet is Rs 1395, then what would it cost to buy 105 m of such plastic sheet?
S o l u t i o n :
Length of
plastic sheet 
inmetre
93 105
Cost inRs 1395 x
Let the cost of the plastic sheet per metre be Rs x.
If more sheets are bought, the cost will also be more. Therefore, it is a direct variation. We get: 93: 105 = 1395: x ?
93
105
 = 
1395
x
Applying cross muliplication, we get: x = 
105 × 1395
93
= 1575Thus
Q u e s t i o n : 1 0
Suneeta types 1080 words in one hour. What is her GWAM grosswordsaminuterate
?
S o l u t i o n :
Number of words 1080 x
Time inminute 60 1
Let x be her GWAM.
If the time taken is less, GWAM will also be less. Therefore, it is a direct variation. 1080: x = 60: 1 ?
1080
x
 = 
60
1
Applying cross muliplication, we get: x = 
1080×1
60
= 18Thus, her GWAM will
( )
Page 3


Q u e s t i o n : 1
Explain the concept of direct variation.
S o l u t i o n :
When two variables are connected to each other in such a way that if we increase the value of one variable, the value of other variable also increases and vice -versa. Similarly, if we decrease
Q u e s t i o n : 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) distancecoveredremainingthesame
.
vi
Area of a land (x) and its cost (y).
S o l u t i o n :
i
The number of articles is directly related to the price. Therefore, they will vary directly with each other.
ii
The number of articles is directly related to the weight of the articles. Therefore, they will vary directly with each other.
iii
Speed is constant. Therefore, distance and time does not vary directly.
iv
The number of hours is directly related to the wages. Therefore, it is a direct variation.
v
Distance is constant. Therefore, speed and time does not vary directly.
vi
If the area of a land is large, its cost will also be high. Thus, it is a direct variation.
Thus, the respective values in i
, ii
, iv
and vi
vary directly with each other.
Q u e s t i o n : 3
In which of the following tables x and y vary directly?
i
a 7 9 13 21 25
b 21 27 39 63 75
ii
a 10 20 30 40 46
b 5 10 15 20 23
iii
a 2 3 4 5 6
b 6 9 12 17 20
iv
a 1
2
2
2
3
2
4
2
5
2
b 1
3
2
3
3
3
4
3
5
3
S o l u t i o n :
If x and y vary directly, the ratio of the corresponding values of x and y remains constant. i
x
y
=
7
21
=
1
3
x
y
=
9
27
=
1
3
x
y
=
13
39
=
1
3
x
y
=
21
63
=
1
3
x
y
=
25
75
=
1
3
In all the cases, the ratio is the same. Therefore,
Q u e s t i o n : 4
Fill in the blanks in each of the following so as to make the statement true:
i
Two quantities are said to vary.... with each other if they increase decrease
together in such a way that the ratio of the corresponding values remains same.
ii
x and y are said to vary directly with each other if for some positive number k,...... = k.
iii
If u = 3 v, then u and v vary .... with each other.
S o l u t i o n :
i directly ii x and y are said to vary directly with each other if 
x
y
= k, where k is a positive number iii because u = 3v, u and y vary directly with each other
Q u e s t i o n : 5
Complite the following tables given that x varies directly as y.
i
( )
( ) ( ) ( )
x 2.5 ... ... 15
y 5 8 12 ...
ii
x 5 ... 10 35 25 ...
y 8 12 ... ... ... 32
iii
x 6 8 10 ... 20
y 15 20 ... 40 ...
iv
x 4 9 ... ... 3 ...
y 16 ... 48 36 ... 4
v
x 3 5 7 9
y ... 20 28 ...
S o l u t i o n :
Here, x and y vary directly. ? x = ky i x = 2. 5 and y = 5i. e. ,  2. 5 = k × 5 ? k = 
2.5
5
 = 0. 5For y = 8 and k = 0. 5, we have: x = ky ? x = 8 × 0. 5 = 4For y = 12 and k = 0. 5,
Q u e s t i o n : 6
Find the constant of variation from the table given below:
x 3 5 7 9
y 12 20 28 36
Set up a table and solve the following problems. Use unitary method to verify the answer.
S o l u t i o n :
Since it is a direct variation, 
x
y
 = k. For x = 3 and y = 12, we have: k = 
3
12
 = 
1
4
Thus, in all cases, k = 
1
4
Q u e s t i o n : 7
Rohit bought 12 registers for Rs 156, find the cost of 7 such registers.
S o l u t i o n :
Let the cost of 7 registers be ? Rs x.
 
Register 12 7
CostinRs. 156 x
If he buys less number of registers, the cost will also be less. Therefore, it is a direct variation. We get: 12: 7 = 156: x ?
12
7
 = 
156
x
Applying cross muliplication, we get: x = 
156×7
12
= 91Thus
Q u e s t i o n : 8
Anupama takes 125 minutes in walking a distance of 100 metre. What distance would she cover in 315 minutes?
S o l u t i o n :
 Let the distance travelled in 315 minutes be x km.
 
Time inminute 125 315
Distanceinmetre 100 x
If the distance travelled is more, the time needed to cover it will also be more. Therefore, it is a direct variation. We get: 125: 315 = 100: x ?
125
315
 = 
100
x
Applying cross muliplication, we get
Q u e s t i o n : 9
If the cost of 93 m of a certain kind of plastic sheet is Rs 1395, then what would it cost to buy 105 m of such plastic sheet?
S o l u t i o n :
Length of
plastic sheet 
inmetre
93 105
Cost inRs 1395 x
Let the cost of the plastic sheet per metre be Rs x.
If more sheets are bought, the cost will also be more. Therefore, it is a direct variation. We get: 93: 105 = 1395: x ?
93
105
 = 
1395
x
Applying cross muliplication, we get: x = 
105 × 1395
93
= 1575Thus
Q u e s t i o n : 1 0
Suneeta types 1080 words in one hour. What is her GWAM grosswordsaminuterate
?
S o l u t i o n :
Number of words 1080 x
Time inminute 60 1
Let x be her GWAM.
If the time taken is less, GWAM will also be less. Therefore, it is a direct variation. 1080: x = 60: 1 ?
1080
x
 = 
60
1
Applying cross muliplication, we get: x = 
1080×1
60
= 18Thus, her GWAM will
( )
Number of Boxes 68 x
Shelf-length inm 13.6 20.4
Number of copies 136 x
Length the shelf inm 3.4 5.1
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 12 minutes?
S o l u t i o n :
Distance inkm 50 x
Time inminute 60 12
Let the distance be x km.
If the time taken is less, the distance covered will also be less. Therefore, it is a direct variation. 50: x = 60: 12 ?
50
x
 = 
60
12
Applying cross muliplication, we get: x =
50×12
60
= 10Thus, the required
Q u e s t i o n : 1 2
68 boxes of a certain commodity require a shelf-length of 13.6 m. How many boxes of the same commodity would occupy a shelf length of 20.4 m?
S o l u t i o n :
Let x be the number of boxes that occupy a shelf-length of 20.4 m.
If the length of the shelf increases, the number of boxes will also increase. Therefore, it is a case of direct variation.
68
x
 = 
13.6
20.4
? 68 × 20. 4 = x × 13. 6 ? x = 
68×20.4
13.6
= 
1387.2
13.6
= 102Thus,
Q u e s t i o n : 1 3
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?
S o l u t i o n :
Let x be the number of copies that would occupy a shelf-length of 5.1 m.
Since the number of copies and the length of the shelf are in direct variation, we have:
136
x
 = 
3.4
5.1
? 136 × 5. 1 = x × 3. 4 ? x = 
136×5.1
3.4
= 204Thus, 204 copies will occupy a shelf of length
Q u e s t i o n : 1 4
The second class railway fare for 240 km of Journey is Rs 15.00. What would be the fare for a journey of 139.2 km?
S o l u t i o n :
Let Rs x be the fare for a journey of 139.2 km.
Distance inkm 240 139.2
Fare inRs. 15 x
Since the distance travelled and the fare are in direct variation, we have:
240
139.2
 = 
15
x
? 240 × x = 15 × 139. 2 ? x = 
15 × 139.2
240
= 
2088
240
= 8. 7Thus, the fare for a journey of 139. 2 km will be 
Q u e s t i o n : 1 5
If the thickness of a pile of 12 cardboards is 35 mm, find the thickness of a pile of 294 cardboards.
S o l u t i o n :
Let x cm be the thickness of a pile of 294 cardboards.
 
Thickness incm 3.5 x
Cardboard  12 294
Since the pile of the cardboards and its thickness are in direct variation, we have:
3.5
x
 = 
12
294
? 3. 5 × 294 = x × 12 ? x = 
3.5 × 294
12
= 
1029
12
= 85. 75 cmThus, the thickness of a pile of 294 
Q u e s t i o n : 1 6
The cost of 97 metre of cloth is Rs 242.50. What length of this can be purchased for Rs 302.50?
S o l u t i o n :
Let x metre be the length of the cloth that can be purchased for Rs 302.50.
 
Length inm 97 x
Cost inRs 242.50 302.50
Since the length of the cloth and its cost are in direct variation, we have:
97
x
 = 
242.50
302.50
? 97 × 302. 50 = x × 242. 50 ? x = 
97 × 302.50
242.50
= 
29342.50
242.50
= 121Thus, the required length will be 121 metre
Q u e s t i o n : 1 7
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 number of men required to dig a trench of 27 metre.
 
Number of men 11 x
Length inm
27
4
27
Since the length of the trench and the number of men are in direct variation, we have:
11
x
 = 
27/4
27
? 11 × 27 = x × 
27
4
? x = 
11 × 27 × 4
27
= 44Thus, 44 men will be required to dig a trench of 27
Q u e s t i o n : 1 8
Page 4


Q u e s t i o n : 1
Explain the concept of direct variation.
S o l u t i o n :
When two variables are connected to each other in such a way that if we increase the value of one variable, the value of other variable also increases and vice -versa. Similarly, if we decrease
Q u e s t i o n : 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) distancecoveredremainingthesame
.
vi
Area of a land (x) and its cost (y).
S o l u t i o n :
i
The number of articles is directly related to the price. Therefore, they will vary directly with each other.
ii
The number of articles is directly related to the weight of the articles. Therefore, they will vary directly with each other.
iii
Speed is constant. Therefore, distance and time does not vary directly.
iv
The number of hours is directly related to the wages. Therefore, it is a direct variation.
v
Distance is constant. Therefore, speed and time does not vary directly.
vi
If the area of a land is large, its cost will also be high. Thus, it is a direct variation.
Thus, the respective values in i
, ii
, iv
and vi
vary directly with each other.
Q u e s t i o n : 3
In which of the following tables x and y vary directly?
i
a 7 9 13 21 25
b 21 27 39 63 75
ii
a 10 20 30 40 46
b 5 10 15 20 23
iii
a 2 3 4 5 6
b 6 9 12 17 20
iv
a 1
2
2
2
3
2
4
2
5
2
b 1
3
2
3
3
3
4
3
5
3
S o l u t i o n :
If x and y vary directly, the ratio of the corresponding values of x and y remains constant. i
x
y
=
7
21
=
1
3
x
y
=
9
27
=
1
3
x
y
=
13
39
=
1
3
x
y
=
21
63
=
1
3
x
y
=
25
75
=
1
3
In all the cases, the ratio is the same. Therefore,
Q u e s t i o n : 4
Fill in the blanks in each of the following so as to make the statement true:
i
Two quantities are said to vary.... with each other if they increase decrease
together in such a way that the ratio of the corresponding values remains same.
ii
x and y are said to vary directly with each other if for some positive number k,...... = k.
iii
If u = 3 v, then u and v vary .... with each other.
S o l u t i o n :
i directly ii x and y are said to vary directly with each other if 
x
y
= k, where k is a positive number iii because u = 3v, u and y vary directly with each other
Q u e s t i o n : 5
Complite the following tables given that x varies directly as y.
i
( )
( ) ( ) ( )
x 2.5 ... ... 15
y 5 8 12 ...
ii
x 5 ... 10 35 25 ...
y 8 12 ... ... ... 32
iii
x 6 8 10 ... 20
y 15 20 ... 40 ...
iv
x 4 9 ... ... 3 ...
y 16 ... 48 36 ... 4
v
x 3 5 7 9
y ... 20 28 ...
S o l u t i o n :
Here, x and y vary directly. ? x = ky i x = 2. 5 and y = 5i. e. ,  2. 5 = k × 5 ? k = 
2.5
5
 = 0. 5For y = 8 and k = 0. 5, we have: x = ky ? x = 8 × 0. 5 = 4For y = 12 and k = 0. 5,
Q u e s t i o n : 6
Find the constant of variation from the table given below:
x 3 5 7 9
y 12 20 28 36
Set up a table and solve the following problems. Use unitary method to verify the answer.
S o l u t i o n :
Since it is a direct variation, 
x
y
 = k. For x = 3 and y = 12, we have: k = 
3
12
 = 
1
4
Thus, in all cases, k = 
1
4
Q u e s t i o n : 7
Rohit bought 12 registers for Rs 156, find the cost of 7 such registers.
S o l u t i o n :
Let the cost of 7 registers be ? Rs x.
 
Register 12 7
CostinRs. 156 x
If he buys less number of registers, the cost will also be less. Therefore, it is a direct variation. We get: 12: 7 = 156: x ?
12
7
 = 
156
x
Applying cross muliplication, we get: x = 
156×7
12
= 91Thus
Q u e s t i o n : 8
Anupama takes 125 minutes in walking a distance of 100 metre. What distance would she cover in 315 minutes?
S o l u t i o n :
 Let the distance travelled in 315 minutes be x km.
 
Time inminute 125 315
Distanceinmetre 100 x
If the distance travelled is more, the time needed to cover it will also be more. Therefore, it is a direct variation. We get: 125: 315 = 100: x ?
125
315
 = 
100
x
Applying cross muliplication, we get
Q u e s t i o n : 9
If the cost of 93 m of a certain kind of plastic sheet is Rs 1395, then what would it cost to buy 105 m of such plastic sheet?
S o l u t i o n :
Length of
plastic sheet 
inmetre
93 105
Cost inRs 1395 x
Let the cost of the plastic sheet per metre be Rs x.
If more sheets are bought, the cost will also be more. Therefore, it is a direct variation. We get: 93: 105 = 1395: x ?
93
105
 = 
1395
x
Applying cross muliplication, we get: x = 
105 × 1395
93
= 1575Thus
Q u e s t i o n : 1 0
Suneeta types 1080 words in one hour. What is her GWAM grosswordsaminuterate
?
S o l u t i o n :
Number of words 1080 x
Time inminute 60 1
Let x be her GWAM.
If the time taken is less, GWAM will also be less. Therefore, it is a direct variation. 1080: x = 60: 1 ?
1080
x
 = 
60
1
Applying cross muliplication, we get: x = 
1080×1
60
= 18Thus, her GWAM will
( )
Number of Boxes 68 x
Shelf-length inm 13.6 20.4
Number of copies 136 x
Length the shelf inm 3.4 5.1
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 12 minutes?
S o l u t i o n :
Distance inkm 50 x
Time inminute 60 12
Let the distance be x km.
If the time taken is less, the distance covered will also be less. Therefore, it is a direct variation. 50: x = 60: 12 ?
50
x
 = 
60
12
Applying cross muliplication, we get: x =
50×12
60
= 10Thus, the required
Q u e s t i o n : 1 2
68 boxes of a certain commodity require a shelf-length of 13.6 m. How many boxes of the same commodity would occupy a shelf length of 20.4 m?
S o l u t i o n :
Let x be the number of boxes that occupy a shelf-length of 20.4 m.
If the length of the shelf increases, the number of boxes will also increase. Therefore, it is a case of direct variation.
68
x
 = 
13.6
20.4
? 68 × 20. 4 = x × 13. 6 ? x = 
68×20.4
13.6
= 
1387.2
13.6
= 102Thus,
Q u e s t i o n : 1 3
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?
S o l u t i o n :
Let x be the number of copies that would occupy a shelf-length of 5.1 m.
Since the number of copies and the length of the shelf are in direct variation, we have:
136
x
 = 
3.4
5.1
? 136 × 5. 1 = x × 3. 4 ? x = 
136×5.1
3.4
= 204Thus, 204 copies will occupy a shelf of length
Q u e s t i o n : 1 4
The second class railway fare for 240 km of Journey is Rs 15.00. What would be the fare for a journey of 139.2 km?
S o l u t i o n :
Let Rs x be the fare for a journey of 139.2 km.
Distance inkm 240 139.2
Fare inRs. 15 x
Since the distance travelled and the fare are in direct variation, we have:
240
139.2
 = 
15
x
? 240 × x = 15 × 139. 2 ? x = 
15 × 139.2
240
= 
2088
240
= 8. 7Thus, the fare for a journey of 139. 2 km will be 
Q u e s t i o n : 1 5
If the thickness of a pile of 12 cardboards is 35 mm, find the thickness of a pile of 294 cardboards.
S o l u t i o n :
Let x cm be the thickness of a pile of 294 cardboards.
 
Thickness incm 3.5 x
Cardboard  12 294
Since the pile of the cardboards and its thickness are in direct variation, we have:
3.5
x
 = 
12
294
? 3. 5 × 294 = x × 12 ? x = 
3.5 × 294
12
= 
1029
12
= 85. 75 cmThus, the thickness of a pile of 294 
Q u e s t i o n : 1 6
The cost of 97 metre of cloth is Rs 242.50. What length of this can be purchased for Rs 302.50?
S o l u t i o n :
Let x metre be the length of the cloth that can be purchased for Rs 302.50.
 
Length inm 97 x
Cost inRs 242.50 302.50
Since the length of the cloth and its cost are in direct variation, we have:
97
x
 = 
242.50
302.50
? 97 × 302. 50 = x × 242. 50 ? x = 
97 × 302.50
242.50
= 
29342.50
242.50
= 121Thus, the required length will be 121 metre
Q u e s t i o n : 1 7
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 number of men required to dig a trench of 27 metre.
 
Number of men 11 x
Length inm
27
4
27
Since the length of the trench and the number of men are in direct variation, we have:
11
x
 = 
27/4
27
? 11 × 27 = x × 
27
4
? x = 
11 × 27 × 4
27
= 44Thus, 44 men will be required to dig a trench of 27
Q u e s t i o n : 1 8
A worker is paid Rs 210 for 6 days work. If his total income of the month is Rs 875, for how many days did he work?
S o l u t i o n :
Let x be the number of days for which the worker is paid Rs 875.
 
Income inRs. 210 875
Number of days 6 x
Since the income of the worker and the number of working days are in direct variation, we have:
210
875
 = 
6
x
? 210 × x = 875 × 6 ? x = 
875 × 6
210
= 
5250
210
= 25Thus, the required number of days
Q u e s t i o n : 1 9
A woker is paid Rs 200 for 8 days work. If he works for 20 days, how much will he get?
S o l u t i o n :
 Let Rs x be the income for 20 days of work.
 
Income inRs 200 x
Number of days 8 20
Since the income and the number of working days are in direct variation, we have:
200
x
 = 
8
20
? 200 × 20 = 8x ? x = 
200 × 20
8
= 
4000
8
= 500Thus, the worker will get Rs 500 for working 20 days
Q u e s t i o n : 2 0
The amount 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?
S o l u t i o n :
Let x gm be the weight that would produce an extension of 17.4 cm.
 
Weight ingm 150 x
Length incm 2.9 17.4
Since the amount of extension in an elastic string and the weight hung on it are in direct variation, we have:
150
x
 = 
2.9
17.4
? 17. 4 × 150 = 2. 9 × x ? x = 
17.4 × 150
2.9
= 
2610
2.9
= 900Thus, the required
Q u e s t i o n : 2 1
The amount 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 weight
of 700 gm.
S o l u t i o n :
Let x cm be the extension produced by the weight of 700 gm.
 
Weight ingm 250 700
Length incm 3.5 x
Since the amount of extension in an elastic spring varies and the weight hung on it is in direct variation, we have:
250
700
 = 
3.5
x
? x × 250 = 3. 5 × 700 ? x = 
3.5 × 700
250
= 
2450
250
= 9. 8Thus, the
Q u e s t i o n : 2 2
In 10 days, the earth picks up 2.6 × 10
8
 pounds of dust from the atmosphere. How much dust will it pick up in 45 days?
S o l u t i o n :
Let the amount of dust picked up by the earth in 45 days be x pounds.
Since the amount of dust picked up by the earth and the number of days are in direct variation, we have: Ratio of the dust picked up by the earth in pounds = ratio of the number of days 
Q u e s t i o n : 2 3
In 15 days, the earth picks up 1.2 × 10
8
 kg of dust from the atmosphere. In how many days it will pick up 4.8 × 10
8
 kg of dust?
S o l u t i o n :
Let x be the number of days taken by the earth to pick up 4. 8 ×10
8
kg of dust.
Since the amount of dust picked up by the earth and the number of days are in direct variation, we get:
15
x
 = 
1.2 × 10
8
4.8 × 10
8
? x = 15 × 
4.8
1.2
? x = 60
Thus, the required number of days will be 60.