Inverse proportion - class-XII

Let the car takes x hours to reach a destination by travelling at the speed of 80 km/h. Then,

Clearly, more the speed, less will be the time taken. So, it is a case of inverse proportion.
$ \therefore ,,, 60 \times 2 = 80 \times x  \Rightarrow x = \displaystyle \frac{60 \times 2}{80} = \frac{3}{2}$
Hence, the time taken will be $\displaystyle 1\frac{1}{2}$ hours.

2 quantities say it be $x , y$ are said to be in proportion when the change in value of $x$ , leads to the equal change in value of $y$.

2 quantities say it be $x , y$ are said to be in inverse proportion when the increse in value of $x$ , leads to the decrease in value of $y$.

Inverse proportion is a relation between two quantities such that one increases in proportion as the other decreases i.e. If $x$ increases as $y$ decreases, $x \alpha \dfrac{1}{y}$ 

Work done by $6 = $ work done by $3$

Lets first find the number of working hours required to complete $\dfrac { 5 }{ 21 }$ of the work.
M = number of men
D = number of days
H = number of hours worked per day
N = work done
MDH = N
10$\times $6$\times $10 = $\dfrac { 5 }{ 21 }$
600 hours to complete $\dfrac { 5 }{ 21 }$ of work
Let the total number of hours required to complete the work be $x$ then
$x$ = $\dfrac { 5 }{ 21 }$ $\div $600
   = 600$\times $$\dfrac { 21 }{ 5
 }$
$x$   = 2520 hours
so the remaining work is $x$$-$  $\dfrac { 5 }{ 21 }$
               = $\dfrac {16 }{ 21 }$ $x$

               =$\dfrac {16 }{ 21 }$$\times $2520
               = 1920 hours required to complete remaining $\dfrac {16 }{ 21 }$ of the work
 now the equation we get is
MDH = N
MDH = 1920
M $\times $ 8$\times $10 = 1920
M =  $\dfrac { 1920 }{ 80 } $
M = 24
 Ans. 24 men

As per question,
$\frac { x }{ y } =\frac { 5 }{ 6 } $
$6x=5y$
$6x-5y=0$  multiply eq by 6.
$36x-30y=0$  ...eq 1
$\frac { x+7 }{ y+7 } =\frac { 6 }{ 7 } $
$7x+49=6y+42$  
$7x-6y=-7$  ...multiply eq by 5.
$35x-30y=-35$.... eq 2
eq1-eq2
$x=35$
Answer (A) 35 Years

By definition of inverse proportion,

The given example is of inverse proportion.

Answer $(D)$

The given example is of inverse proportion.

$\because x$ and $y$ are inverse porportion
$\therefore x \times y = k \Rightarrow 9\times 2 = 6\times 3 = 3\times 6 = 18\times 1 = 18 = k$

By inverse proportion,

By the principle of inverse proportion
$4\times 70 = 7\times x$
$\Rightarrow x = \dfrac {4\times 70}{7} = 40\ min$

Given, $\dfrac{a}{b}-\dfrac{c}{d}=0$

Inverse proportion is a relation between two quantities such that one increases in proportion as the other decreases
i.e. If $x$ increases as $y$ decreases, $x \alpha \dfrac{1}{y}$ 

Indirect proportion: When one quantity increases another quantity decreases.
In option C, if the speed of the vehicle increases then the time taken by it to cover a distance, decreases.

Let $x$ soldiers had provision for $42$ days in a fort.