S.chand publication New Learning Composite mathematics solution of class 8 Chapter 9 Variation Exercise 9B

 Exercise 9B

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Q1 | Ex-9B | Class 8 | S.Chand | New Learning Composite maths | Variation | myhelper

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Question 1 

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

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

Sol :

V=volume of gas

P=pressure

K=temperature(constant)

$V\propto \frac{1}{P}$

∴$V=\frac{K}{P}$

(b) The current l in an electrical circuit of fixed voltage varies inversely as the resistance R.

Sol :

I=current

R=ressistance

K=voltage(constant)

$I \propto \frac{1}{R}$

∴$I=\frac{K}{R}$

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

Sol :

h=height

A=Area

K=volume(constant)

$h\propto \frac{1}{A}$

∴$h=\frac{K}{A}$

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

Sol :

f=frequency

l=length of wave

k=constant

$f\propto \frac{1}{l}$

∴$f=\frac{K}{l}$

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Q2 | Ex-9B | Class 8 | S.Chand | New Learning Composite maths | Variation | myhelper

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Question 2

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

(a)

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

Sol :

$\frac{y}{x}=\frac{11}{3}=k$

∴$\frac{y_4}{x_4}=k$

$x_4=33\times \frac{3}{11}$=9

$\frac{y_5}{x_5}=k$

$y_5=\frac{11}{5}\times 27$=99

$\frac{y_6}{x_6}=k$

$x_6=880\times \frac{3}{11}=240$

Ans 9,99,240  (Directly)

(b)

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

Sol :

y ∝ x

∴$\frac{y}{x}=k=\frac{6}{30}=\frac{1}{5}$

$\frac{y_4}{x_4}=k$

$y_4=15\times \frac{1}{5}$

=3

$\frac{y_5}{x_5}=k$

$x_5=\frac{y_5}{k}$

x₅=2×5=10

$\frac{y_6}{x_6}=k$

$x_6=\frac{y_6}{k}$

x₆=1×5=5

Ans 3,10,5 (Directly)

(c)

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

Sol :

$y \propto \frac{1}{x}$

xy=k=48×2=96

y₂x₂=k

$y_2=\frac{x}{x_2}$

$=\frac{96}{3}=32$

y₄x₄=k

$x_4=\frac{k}{y_1}$

$x_4=\frac{96}{16}=6$

y₅x₅=k

$y_5=\frac{k}{x_5}$

$x_4=\frac{96}{8}=12$

Ans 32, 6, 12 (Inversly)

(d)

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

Sol :

$y\propto \frac{1}{x}$

∴xy=k=125×1=125

x₂y₂=k

$y=\frac{125}{5}=25$

x₄y₄=k

$x_4=\frac{125}{5}=25$

x₅y₅=k

$x_5=\frac{125}{1}=125$

Ans 25, 25, 125

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Q3 | Ex-9B | Class 8 | S.Chand | New Learning Composite maths | Variation | myhelper

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Question 3

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

Sol :

y	9	?	?
x	2	12	3

$y \propto \frac{1}{x}$ 

∴yx=k

Now , y=9 , x=2

∴k=yx

or k=9×2=18

Again, 

x=12

∴$y=\frac{18}{12}=\frac{3}{2}$

x=3

∴$y=\frac{k}{x}=\frac{18}{3}=6$

Ans $\frac{3}{2}$ ,3

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

Sol :

$u \propto \frac{1}{v}$  

∴uv=k

Now u=12 ,v=3 ∴k=12×3=36

Agan , v=9  ∴$u=\frac{k}{v}=\frac{36}{9}=4$

(c) If c is inversely proportional to d, and if c = 18 when d = 2/3, find d when c = 6/7

Sol :

$c\propto \frac{1}{d}$ ∴cd=k

Now , 

c=18 ,$d=\frac{2}{3}$  ∴$k=18 \times \frac{2}{3}=12$

Again $c=\frac{6}{7}$  ∴$d=\frac{k}{c}12\times \frac{7}{6}=14$

(d) If m is inversely propotional to n, and if m = 0.02 when n = 5, find m when n = 0.2

Sol :

$m \propto \frac{1}{n}$ , ∴mn=k

Now ,m=0.02 , n=5  ∴k=0.02×5=0.10

mn=k

m×0.2=0.10

m=0.5

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Question 4

Navin cycles to his school at an average speed of 12 km/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?

Sol :

Speed=s=12 km/hr , time=t=20min

∴$s\propto \frac{1}{t}$  ∴st=k

s₁=12 , t₁=20 , k=12×20=240

t₂=15min  ∴$S_2=\frac{k}{t_2}=\frac{240}{15}=16$km/hr

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Q5 | Ex-9B | Class 8 | S.Chand | New Learning Composite maths | Variation | myhelper

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Question 5

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

Sol :

$T \propto \frac{1}{S}$  ∴TS=K

S₁=72km/hr  ,T₁=10 h  ∴K=S₁T₁=72×10=720

∴T₁=9 h  , $S_1=\frac{K}{T_1}=\frac{720}{9}=80$km/h

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Question 6

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

Sol :

$P\propto \frac{1}{T}$  ∴PT=K

P₁=28  ,T₁=18 h  ∴K=P₁T₁=18×28=504

∴P₂=42  , $T_2=\frac{K}{P_2}=\frac{504}{42}=12$h

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Question 7

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

Sol :

$M\propto \frac{1}{D}$  MD=K

∴M₁=400 per  D₁=9  ∴K=M₁D₁=400×9=3600      

∴M₂=300 per  $D_2=\frac{K}{M_2}=\frac{3600}{300}=12$days

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Question 8

A contractor, who had a workforce 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?

Sol :

$M\propto \frac{1}{D}$  ∴MD=K

∴M₁=630 per  D₁=14 months  ∴K=M₁D₁=630×14=8820      

∴D₂=9 months  $M_2=\frac{K}{D_2}=\frac{8820}{9}=980$days

∴Extra person=980-630=350

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Q9 | Ex-9B | Class 8 | S.Chand | New Learning Composite maths | Variation | myhelper

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Question 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?

Sol :

$T\propto \frac{1}{D}$   ∴TD=K

∴T₁=4 hour  D₁=15 days  ∴K=T₁D₁=4×15=60      

D₂=10 days  $T_2=\frac{k}{D_2}=\frac{60}{10}$=6 hour  

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Q10 | Ex-9B | Class 8 | S.Chand | New Learning Composite maths | Variation | myhelper

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Question 10

A train moving at a speed of 60 km/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?

Sol :

$S \propto \frac{1}{T}$   ∴ST=K

∴S₁=60 km/hour  T₁=7.5 hour  ∴K=S₁T₁=60×7.5=450      

T₂=6 hour  $S_2=\frac{450}{6}$=75 km/hour  

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Question 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?

Sol :

$M \propto \frac{1}{D}$  ∴DM=K

∴M₁=800 min  D₁=39 days  ∴K=M₁D₁=800×39=31200      

M₂=500 min  $D_2=\frac{31200}{500}$=62.4 days

∴The food last now=(62.4-39)=23.4≈24days

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Question 12

A beseieged 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?

Sol :

$P \propto \frac{1}{W}$  ∴PW=K

∴P₁=22400 min  W₁=3 days  ∴K=P₁W₁=22400×3=67200      

W₂=7 $P_2=\frac{67200}{7}$=9600 

∴Number of provisions=22400-9600

=12800

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Question 13

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

Sol :

(60-12)=48 days for (500+300)=800

students	500	800
days	48	x

∴$x=\frac{48\times 500}{800}$

=30 days