**2.13 Variation :**

Observe the below mentioned examples which we
encounter in our daily life:

Example 1: If 180 men in 6 days, working 10
hours daily, can dig a trench of 60m long, 1 m wide and 1 m deep, how many days
are needed for 100 men, working 8 hours a day, to dig a trench of 100m long,
1.5m wide and 1.2m deep?

Example 2: The weight of a body varies
inversely as the square of its distance from the centre of the earth. If the
radius of the earth is about 6380 KM, how much would a
80 KG man weigh 1600 KM above the surface of the earth?

How can we find solutions to such
problems?

Variation means change. We say things have changed
with time and we hear quite often that this was not the case with our time and
things have changed. We often hear that in our time the rice is used to cost
few Anas per KG, where is it now
costs in multiples of 10 Rs. Has the price of rice increased directly
with passage of time?. No. it has come down also. So raise in
the price of rice or rise in the price of gold, petrol or any item has not
always increased with passage of time. So price raise is not directly
proportional to time. Does
the height of a person
with increase in age? Though height grows continuously in the initial stage, after
certain age it stops. So height does is not directly vary with age.

**2.13.1** **Direct variation(Proportion)**

Is there some thing which always increase with
time? Yes, distance covered by moving train or bus always increases with the time
as per its speed. We also know that

Distance travelled = Speed*time or d=st. If
time travelled is less, then distance covered is less. In this case we say
distance is directly proportional to time and we write dt and read this relation
as: d is directly proportional to t

Note that d/t = k- a constant (speed). Here k is a constant which does not change and is called 'constant of proportionality' where as d and t are
variables. Interest paid by a bank to a
depositor or interest charged on a loan is directly proportional to the amount
when term and rate of interest is fixed.

We know circumference of a circle = 2pr. As the radius of a circle
increases or decreases, circumference increases or decreases. Also note that Cr because C/r = 2p which is a constant. Similarly Area of a circle = pr^{2}. Thus Ar and A/r^{2}= p. Here also area of a
circle increases or decreases with increase or decrease in the radius. To know
the distance between 2 places, we use maps. The distance between two places in
a map is multiplied scale mentioned in the map to get actual distances(Ex
1 cm = 10 Kms). Is weight directly proportional to age? –NO.

**2.13 Problem 1 :** The distance through which a body falls from rest varies
as square of time it takes to fall that distance. It is known that body falls
64 cm in 2 seconds. How far does that body fall in 6 seconds?

**Solution:**

Note that we can not
solve as in unitary method like

2sec >>> 64cm

_{} 6sec >>> (64/2)*6= 192

This is not the
solution.

It is given that dt^{2}^{}

_{} d/t^{2}=
k

_{}k = 64/4= 16

Now k = 16= d/6^{2}=d/36

_{}d= 16*36= 576

Thus the body falls in 576 cms in 6 seconds

**2.13.2** **Inverse variation(Proportion)**

Have you heard about growers throwing tomatoes on
the road when there is bumper crop? Why it is so? Note that:

·
When there is too much of supply of vegetables or paddy,
wheat etc, the price of them crashes.

·
When there is too much of supply of fuel in the
international market the price of Petrol
and Diesel drops

Examples:

1.
When more number of people are involved in a manual work,
time taken to complete the job decrease

2.
We reach a state of weightlessness when we move far away
from earth

In direct variation, when a value of a variable increases, the value of another dependant
variable also increases.

However, in the examples listed above when a value of a
variable increases, the value of another
dependant variable decreases. In such
cases, product dependent on dependent variables is constant.

In other words, If x and y are 2 variables then x1/y. We also say
x inversely
varies with y and accordingly xy=k a constant.

Like this we can also have x1/y^{2} , x1/y^{4}
, x1/_{} . . . . and then xy^{2}, xy^{4},
x_{} will be constant respectively.

**2.13 Problem 2 :** When a ball is thrown upwards, the time, T seconds during
which the ball remains in the air is directly proportional to the square root
of the height, h meters. reached. We know T=4.47sec
when h=25m.

(i)
Find the formula for T in terms of h

(ii)
Find T when h=50

(iii)
If the ball is thrown upwards and remains in the air for 5
seconds, find the height reached.

**Solution:**

T_{}

_{}T= k_{}

_{}4.47= 5k

_{}k = 0.894^{}

_{}T = 0.894 _{}

When h =50

T=
0.894*_{}= 0.894*7.07_{} 6.32

When
T =5

_{}= T/k = 5/0.894_{}5.60_{}31.36m

^{ }

**2.13.3 Joint Variation**

Can
a variation depend upon multiple variables? We know the formula for interest calculation which for
simple interest and compound interest is

SI
= PTR/100

and

CI
= P(1+R/100)^{T}-P

What do we observe? we observe that Simple Interest is directly proportional to
Principal P, term T and rate of interest R and Compound interest is dependent
on those three variables.

We also know that weight of a
person depends upon the distance he is
away from the centre of earth.

**2.13.4 Work, People,
Days, hours**

We know the work done is
directly proportional to number of men(M), number of days(D) and number of
hours(H).

In other words,

WM, WD, WH

_{} WM*D*H or M*D*H/W
= constant

**2.13 Problem 3 :** If 36 men can build a
wall of 140M long in 21 days, how many men are required to build a similar wall
of length 50M in 18 days?

**Solution:**

Here W_{1}=
140, M_{1}=36, D_{1}=21 and given W_{2}= 50, D_{2}=18
'H' is number of hours though not given is equal. we need to find M_{2}
such that

M*D*H/W = constant

Hence 36*21*H/140 = M_{2}*18*H/50

On solving we get M_{2}=
15

^{ }

**2.13 Problem 4 :** Tap A Can fill a cistern in 8 hours and tap B can empty
it in 12 hours. How long will it take to
fill the cistern if both of them are opened together

**Solution:**

Part filled in 1 hour =
(1/8-1/12)= (3-2)/24= 1/24.

Time taken to fill the
tank= 24 hours

**2.13 Problem 5 :** If 180 men in 6 days, working 10 hours daily, can dig a
trench of 60m long, 1 m wide and 1 m deep, how many days are needed for 100
men, working 8 hours a day, to dig a trench of 100m long, 1.5m wide and 1.2m
deep?

**Solution:**

Here W_{1}=
60*1*1, M_{1}=180, D_{1}=6 and H_{1} =10, and given W_{2}=
100*1.5*1.2, M_{2}=100, and H_{2} =8, we need to find D_{2} such
that

M*D*H/W = constant

M_{1}*D_{1}*H_{1}/W_{1
}= M_{2}*D_{2}*H_{2}/W_{2}

Hence 180*6*10/60 =
100*D_{2}*8*/(100*1.5*1.2)

On solving we get D_{2}
= 40.5

**2.13 Problem 6 :** The weight of a body varies inversely as the square of
its distance from the centre of the earth. If the radius of the earth is about
6380 KM, how much would a 80 KG man weigh 1600 KM
above the surface of the earth?

**Solution:**

Here W1/d^{2}

Weight of the person
when he is on earth = 80 KG. d_{1 }=Radius of earth=6380 KM

Let W_{2
} be his weight when he is
1600 KM above the earth's surface

_{} W_{1 }d_{1}^{2}=
W_{2 }d_{2}^{2}

80*6380^{2}= W_{2}*7980^{2}

On solving we get W_{2}=
51.14

**2.13 Problem 7 :** Suppose A alone can perform a work in 5 days more than A
and B working together. Suppose further, B working alone can complete
the same work in 20 days more than A and B working together. Find out
how much time it will take A and B both working
together?

**Solution:**

Let **x** be the time taken by A and B both working together to
perform the work.

It is given that A alone can complete the work in x+5 days

It is also given that B
alone can complete the work in x+20 days

From formula, The time
taken by both A and B working together is(**=x**)= (x+5)*(x+20)/{(x+5)+(x+20)}

_{} x= x^{2}+25x+100/2x+25

_{}2x^{2}+25x= x^{2}+25x+100

_{} x^{2}=100

_{} x=10

The time taken by both
A and B, when
they work together to complete the work is 10 days

**Verification:**

Time taken by A alone
to complete the work = x+5 = 15 days

Time taken by B
alone to complete the work = x+20 = 30
days

From formula, the time
taken by both A and B, when they work together= 30*15/45= 10

**2.13 Problem 8 :** It takes 1 hour to fill a tank by a pump A. 1 hour and 40
minutes using a pump B. A third pump C takes average time of pump A and B to
fill up the tank. Suppose the pumps A and B are started together and 2 pumps of
capacity C are used to drain
the water. Will the tank ever get filled? If so in how much time
it gets filled ?

**Solution:**

Time taken by pump A =
60 Minutes

Time taken by pump B =
100 Minutes

Time taken by pump C =
80 Minutes

Let t be the time taken to fill up
the tank

By formula

1/t = 1/60+1/100-
2(1/80)

= 1/60+1/100-1/40= (10+6-15)/600 = 1/600

Time taken to fill up the
tank = 600 minutes= 10 hours

Summary of learning

xy , x/y = k a
constant

x1/y. xy=k a constant

WM*D*H or M*D*H/W
= k a constant