myhelper: Exercise 10 B

Exercise 10 B

Question 1

1. State the relationship between the given variables as an equation, using $k$ for the constant of variation.

(i) The volume $V$ of a gas at a fixed temperature varies inversely as the pressure $P$.

(ii) The current $I$ in an electrical circuit of fixed voltage, varies inversely as the resistance $R$.

(iii) The height $h$ of a cylinder of fixed volume, varies inversely as the area $A$ of the base.

(iv) The frequency $f$ of an electromagnetic wave is inversely proportional to the length $l$ of the wave.

2. Fill in the blanks in the following tables by determining first whether x and y vary directly or inversely :

(a)

x	3	6	?	27	?
y	11	22	33	?	880

(b)

x	30	20	15	?	?
y	6	4	?	2	1

(c)

x	2	3	4	?	8
y	48	?	24	16	?

(d)

x	1	5	10	?	?
y	125	?	12.5	5	1

3. (i) If $y$ varies inversely as $x$, and if $y=9$ when $x=2$, find $y$ when $x=3$.

(ii) If $u$ is inversely proportional to $v$, and if $u=12$ when $v=3$, find $u$ when $v=9$.

(iii) If $c$ is inversely proportional to $d$, and if $c=18$ when $d=\frac{2}{3}$, find $d$ when $c=\frac{6}{7}$.

(iv) If $m$ is inversely proportional to $n$, and if $m=0.02$ when $n=5$, find $m$ when $n=0.2$.

4. Navin cycles to his school at an average speed of $12 \mathrm{~km} / \mathrm{hr}$. It takes him 20 minutes to reach the school. If he wants to reach his school in 15 minutes, what should be his average speed ?

5. The time needed to travel from one place to another is inversely proportional to the speed. A person travelling $72 \mathrm{~km} / \mathrm{hr}$ can go from Dehradun to Lucknow in 10 hours. How fast must the person travel to make the trip in 9 hours?

6. 28 pumps can empty a reservoir in 18 hours. In how many hours can 42 such pumps do the same work?

7. A stock of food grains is enough for 400 persons for 9 days. How long will the same stock last for 300 persons?

8. A contractor who had a work force of 630 persons, undertook to complete a portion of a stadium in 14 months. He was asked to complete the job in 9 months. How many extra persons had he to employ?

9. Working 4 hours a day, Savita can type a manuscript in 15 days. How many hours a day should she work so as to finish the work in 10 days ?

10. A train moving at a speed of $60 \mathrm{~km} / \mathrm{hr}$ covers a certain distance in $7.5$ hours. What should be the speed of the train to cover the same distance in 6 hours?

11. A garrison of 800 men had provisions for 39 days. However, a reinforcement of 500 men arrived. For how many days will the food last now ?

12. A besieged town has provisions to last for 3 weeks. Its population is 22400 . How many people must be sent away in order that the provisions may last for 7 weeks?

13. A hostel had rations for 60 days for 500 students. After 12 days, 300 more students join the hostel. How long will the remaining rations last?

Multiple Choice Questions (MCQS)

14. $x$ and $y$ vary inversely as each other. When $x=10, y=6$. Which of the following is not a possible pair of corresponding values of $x$ and $y$.

(a) 15 and 4

(b) 20 and 3

(d) $0.6$ and 100

15. If $a$ varies inversely as $b+2$ and if $a=8$ when $b=1.5$, find $a$ when $b=5$.

(a) $2 \frac{2}{5}$

(b) $5 \frac{3}{5}$

(d) $4 \frac{2}{5}$

16. The number of hours needed to clear the trees from some land is inversely proportional to the number of people who are working. If it would take 4 people 15 hours to do the job, how many people would be needed to complete the job in 6 hours?

(a) 22 people

(b) 8 people

(d) 15 people

17. A pulley revolves at a speed that is inversely proportional to its diameter. A pulley with a diameter of $12 \mathrm{~cm}$ is belted to a pulley with a diameter of $8 \mathrm{~cm}$. If the smaller pulley is revolving at a rate of $96 \mathrm{rpm}$ (revolutions per minute), how fast is the larger pulley revolving?

(a) $144 \mathrm{rpm}$

(b) $64 \mathrm{rpm}$

(d) $48 \mathrm{rpm}$

High Order Thinking Skills (HOTS)

18. A group of 210 men had provisions for 60 days. After 10 days, 60 men left. How long will the remainign food last ?

[Hint. The remaining food would have lasted 50 days for 210 men. Find out how long will the food last for 150 men].

EXERCISE 10 B

Q1 | Ex-10B | Class 8 | Direct and inverse variation | SChand Composite Maths | myhelper

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Question 1

The pressure P of an enclosed gas , held at a constant temperature , is inversely proportional to the volume V of the gas . Express P in terms of V and the constant of variations K . Calculate
(i) The value of K when P = 500 and V = 2

Sol :

Given : $\text{P}\propto \dfrac{1}{\text{V}}$ or $\text{P}= \dfrac{k}{\text{V}}$

⇒ PV = K

⇒ 500×2 = K

⇒K = 1000

(ii) The value of P when V = 5

Sol :

⇒$\text{P}= \dfrac{k}{\text{V}}$

⇒$\text{P}= \dfrac{1000}{5}$

⇒P = 200

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Question 2

How many days would it take 67 men to build a wall which 134 mens can build in 3 weeks ?

Sol :

Number of men (m)	134	67
Number of weeks(w)	3

As number of weeks (w) varies inversely with number of mens (m)

⇒$\text{No. of weeks}\propto \dfrac{1}{\text{No. of mens}}$ or

⇒$\text{No. of weeks}=\dfrac{k}{\text{No. of mens}}$

⇒k = w×m , where k is variable constant

⇒k = 3×134 = 402

Now , m = 67 , d = ? ⇒ $\text{No. of weeks}=\dfrac{k}{\text{No. of mens}}$

⇒$\text{w}=\dfrac{402}{67}$

⇒w = 6

= 6 weeks

ALTERNATE METHOD

134÷67=2

Simple explanation through logic:

If 2x the amount of people takes a certain time to complete a task, half the amount of people will take 2x the time to complete the task.

3 weeks × 2 = 6 weeks

6 weeks = 42 days

Some basic concept

Why we are not dividing 134 by 3 ?

Answer : Because No. of mens is inversely proportional to no . of weeks in other words on increasing first value(no. of mens) then other value decreases(no. of weeks)

Why we are multiplying 134 by 3 ?

Answer : To understand this firstly lets discuss why we use multiplication

Suppose we have to give 2 pencil to every five person , then whats the Total no. of pencils

2 + 2 + 2 + 2 + 2 = 10 or 2×5 = 10

In the same way 134 mens work every week , total 3 weeks to complete work

No . of mens in 1st week + No. of mens work in 2nd week + No. of mens work in 3rd week = 402 or No. of mens ×3 = 402

402/67= 6 weeks

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Question 3

A car can complete a certain journey in 12 hours if it travels at 65 km/h . How much time will it take if it travels at 78 km/h ?

Sol :

Speed(S)	65 km/p	78 km/h
Time (T)	12 hours

As Speed (S) varies inversely with Time (T)

⇒$\text{S}\propto \dfrac{1}{\text{T}}$ or

⇒$\text{S}=\dfrac{k}{\text{T}}$

⇒k = S×T , where k is variable constant

⇒k = 65×12 = 780

Now , S = 78 , T = ? ⇒ $\text{S}=\dfrac{k}{T}$

⇒$\text{S}=\dfrac{780}{78}$

⇒S = 10

= 10 hours

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Question 4

A workforce of 210 men with a contractor can finish a work in 16 months . How many more man men should be employed so that the work is completed in 14 months ?

Sol :

Number of men (m)	210
Time(T)	16 months	14 months

As Number of mens (m) varies inversely with Time (T)

⇒$\text{m}= \dfrac{1}{\text{T}}$ or

⇒$\text{m}=\dfrac{k}{T}$

⇒k = T×m , where k is variable constant

⇒k = 16×210 = 3360

Now , T = 14 , m = ? ⇒ $\text{m}=\dfrac{k}{\text{T}}$

⇒$\text{m}=\dfrac{3360}{14}$

⇒m = 240

To complete work in 14 months , we need

= 240 - 210

= 30 more men

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Question 5

It is found that a book will contains 540 pages if 28 lines are allowed in a page . How many lines should be allowed in a page if the book has to contain 360 pages ?

Sol :

Number of pages (P)	540	360
Lines (L)	28

As Number of pages (P) varies inversely with Lines (L) ,

⇒$\text{P}\propto \dfrac{1}{\text{L}}$ or

⇒$\text{P}= \dfrac{k}{\text{L}}$ or

⇒k = P×L

⇒k = 540×28

⇒k = 15120 , where k is variable constant

Now , P = 360 , L = ? ⇒ , $\text{L}= \dfrac{k}{\text{P}}$ or

⇒$\text{L}= \dfrac{15120}{360}$

⇒L = 42

= 42 lines per page

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Question 6

Sreyash has enough money to buy 54 machines worth 200 each . How many machines can he buys if he gets a discount of 20 on each machine ?

Sol :

Number of Machines (M)	54
Cost of each (C)	200	200 - 20 = 180

As Number of Machines (M) varies inversely with Cost (C) ,

⇒$\text{M}\propto \dfrac{1}{\text{C}}$ or

⇒$\text{M}= \dfrac{k}{\text{C}}$ or

⇒k = M×C

⇒k = 54×200

⇒k = 10800 , where k is variable constant

Now , C = 180 , M = ? ⇒ , $\text{M}= \dfrac{k}{\text{C}}$ or

⇒$\text{M}= \dfrac{10800}{180}$

⇒M = 60

= 60 machines

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Question 7

If a ball moves at 135 m/s , it will strike an object in 4 seconds . If it moves at 120 m/s , how long will it takes to strike the same objects ?

Sol :

Speed (S)	135 m/s	120 m/s
Time (T)	4 seconds

As Speed (S) varies inversely with Time (T) ,

⇒$\text{S}\propto \dfrac{1}{\text{T}}$ or

⇒$\text{S}= \dfrac{k}{\text{T}}$ or

⇒k = S×T

⇒k = 135×4

⇒k = 540 , where k is variable constant

Now , S = 120 , T = ? ⇒ , $\text{T}= \dfrac{k}{\text{S}}$ or

⇒$\text{T}= \dfrac{540}{120}$

⇒T = 4.5 seconds

= 4.5 seconds

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Question 8

A man eats 200 g of rice a day and he has enough rice to last him 35 days . How long would the stock of rice last him if he were to eat 250 g of rice a day ?

Sol :

Rice (R)	200 g	250 g
Number of Days (D)	35 days

As Rice (R) varies inversely with Number of Days (D) ,

⇒$\text{R}\propto \dfrac{1}{\text{D}}$ or

⇒$\text{R}= \dfrac{k}{\text{D}}$ or

⇒k = R×D

⇒k = 200×35

⇒k = 7000 , where k is variable constant

Now , R = 250 , D = ? ⇒ , $\text{D}= \dfrac{k}{\text{R}}$ or

⇒$\text{D}= \dfrac{7000}{250}$

⇒D = 28 days

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Question 9

A wheel of circumference 2.8 m revolve 765 times in traversing a certain distance . How many revolutions does a wheel of circumference 1.7 m make in traversing the same distance .

Sol :

Circumference of wheel (W)	2.8 m	1.7 m
Number of Revolution (R)	765

As Circumference of wheel (W) varies inversely with Number of Revolution (R)

⇒$\text{W}\propto \dfrac{1}{\text{R}}$ or

⇒$\text{W}= \dfrac{k}{\text{R}}$ or

⇒k = W×R

⇒k = 2.8×765

⇒k = 2142 , where k is variable constant

Now , W = 1.7 , R = ? ⇒ , $\text{R}= \dfrac{k}{\text{W}}$ or

⇒$\text{R}= \dfrac{2142}{1.7}$

⇒R = 1260 times

= 1260 times

S.chand Class 8 Maths Solution Chapter 10 Variation and Direct and Inverse Proportions Exercise 10 B

Exercise 10 B

Question 1

S.chand books class 8 maths solution chapter 10 Direct and Inverse variation exercise 10 B

EXERCISE 10 B

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