Direct and Inverse Proportions

CBSE Worksheet-1
Class 08 - Mathematics (Direct and Inverse Proportions)

General Instructions: All questions are compulsory. Q.1 to Q.2 carries one mark each. Q.3 to Q.7 carries two marks each. Q.8 and Q.9 carries three marks each. Q.10 to Q.12 carries four marks each.

In the table state whether x and y vary directly.

x 1 3 5 6

y 6 4 4 1
Is the population of a country and the area of land per person are in inverse proportion ?
State true or false:
1. A varies jointly as B and C means A ∝ B/C.
2. Journey (x km) undertaken by a car and the petrol (y litres) consumed by it is an example of inverse variation.
3. More number of pipes, lesser time required to fill the tank.
4. Distance x and time y, speed remaining the same is an example of direct variation.
Fill Up the following:
1. A farmer has enough food to feed 28 animals in his cattle for 9 days, if there were 8 more animals in his cattle, Number of days would the food last be ________.
2. y is equal to product of k and x. The term k is ________.
3. Area of cultivated land and the crop harvested. The above statement are in ________ proportion.
4. If 18 dolls cost Rs. 630, number of dolls can bought for Rs. 455 be ________.

x	1	3	5	6
y	6	4	4	1

Match the following:

Column A	Column B
1. In x/y, y increases when x	(a) positive
2. The inversely proportional generally means that one variable is bigger and another one is	(b) opposite way
3. If one quantity is inversely proportional to another, it changes in the	(c) smaller
4. Constant of variation is always	(d) increases

A car takes 2 hours to reach a destination by travelling at the speed of 60 km/h. How long will it take when the car travels at the speed of 80 km/h ?
How long will a boy take to run around a square field of side 35m, if he runs at a speed of 9 km/hr?
A factory requires 42 machines to produce a given number of articles in 63 days. How many machines would required to produce the same number of articles in 54 days ?
If y varies directly as x, and y = 25 when x = 15, find the constant of variation. Then find x when y = 40.
A photograph of a bacteria enlarged 50000 times attains a length of 5 cm as shown in the diagram. What is the actual length of the bacteria ? If the photograph is enlarged 20000 times only, what be its enlarged length ?
A machine in a soft drink factory fills 840 bottles in six hours. How many bottles will it fill in five hours ?
Bacteria enlarged 60000 times attains a length of 3 cm. what is the length of bacteria if it is enlarged 10000 time only.

CBSE Worksheet-1
Class 08 - Mathematics (Direct and Inverse Proportions)
Solution

No
The population of a country and the area of land per person are in inverse proportion.
1. False; A varies jointly as B and C means A ∝ BC
2. False
3. True
4. False
1. 7 days
2. proportionally constant
3. direct
4. 13
1. (d),
2. (c),
3. (b),
4. (a)
Let it take x hours. We have the following table.

Speed (in km/h) 60 80

Number of hours 2 x

Lesser the speed, more the number of hours to reach the destination.
Hence, 60 × 2 = 80 × x
∴ $x = \frac{60 \times 2}{80}$
∴ $x = \frac{3}{2}$
∴ $x = 1 \frac{1}{2}$
Thus, $1 \frac{1}{2}$ hours would be taken.
Speed = 9 km/hr $= 9 \times (5 / 18)$ m/sec = 5/2 m/sec
Side of square is 35m
Perimeter= Distance travelled $= (35 \times 4) m = 140 m$ .
Time taken $= 140 \times (2 / 5) \sec = 56 \sec$
Suppose that x machines would be required. We have the following table.

Number of machines 42 x

Number of days 63 54

Lesser the number of machines, more will be the number of days to produce the same number of articles.
So, this is a case of inverse proportion.
Hence, 42 × 63 = x × 54
∴ $x = \frac{42 \times 63}{54}$
∴ x = 49
Hence, 49 machines would be required.
y ∝ x
y = kx
25 = k (15)
$k = \frac{25}{15} = \frac{5}{3}$
Now $k = \frac{5}{3}$ , y = 40 then x = ?
$y = \frac{5}{3} x$
$40 = \frac{5}{3} x$
$x = 40 \times \frac{3}{5} = 24$
Actual length of the bacteria
$\frac{5}{50000} c m$
$= \frac{1}{10000} c m .$
= 10–4 cm
More the number of times a photograph of a bacteria is enlarged, more the length attained. So, the number of times a photograph of a bacteria is enlarged and the length attained are directly proportional to each other.
So, $\frac{x_{1}}{x_{2}} = \frac{x_{2}}{y_{2}}$
∴ $\frac{50000}{5} = \frac{20000}{y_{2}}$
∴ 50000 y2 = 5 × 20000
∴ $y_{2} = \frac{5 \times 20000}{50000}$
∴ y2 = 2
Hence, its enlarged length would be 2 cm.
Let it fill x bottles in five hours. We put the given information in the form of a table as shown below :

Number of bottles filled 840 x

Number of hours 6 5

More the numbers of hours, more the number of bottles filled would be. So, the number of bottles filled and the number of hours are directly proportional to each other.
So, $\frac{x_{1}}{x_{2}} = \frac{y_{1}}{y_{2}}$
∴ $\frac{840}{x_{2}} = \frac{6}{5}$
∴ 6x2 = 840 × 5
∴ x2 = $\frac{840 \times 5}{6}$
∴ x2 = 700
Hence, 700 bottles will be filled.
Length of bacteria 3 x

Enlargement of bacteria 60000 10000

$\frac{3}{60000} = \frac{x}{10000}$
or $x = \frac{3 \times 10000}{60000} = 1 / 2 c m$
Length of bacteria = 0.5 cm.

Speed (in km/h)	60	80
Number of hours	2	x

Number of machines	42	x
Number of days	63	54

Number of bottles filled	840	x
Number of hours	6	5

Length of bacteria	3	x
Enlargement of bacteria	60000	10000

Direct and Inverse Proportions - Worksheets