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Inverse proportion - class-XII

Description: inverse proportion
Number of Questions: 24
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Tags: ratio and proportions proportion maths rule of three ratio direct proportion and inverse proportion ratio, rate and proportion direct and inverse proportion direct and inverse proportions comparing quantities
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A car takes 2 hours to reach a destination by travelling at the speed of 60 km/h. when the car travels at the speed of 80 km/h it takes 

maths rule of three inverse proportion direct proportion and inverse proportion types of proportions

  1. one and half hour

  2. one hour

  3. two hour

  4. None

Show answer
Correct Option: A
Explanation:

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

Speed (in km/hr) 60 80
Time (in hours) 2 x

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.

Which is an example of inverse proportion?

maths rule of three inverse proportion direct proportion and inverse proportion types of proportions

  1. More amount of sweets, more total cost

  2. More length of cloth, more cost

  3. More expenditure, less saving

  4. More height of object, more length of its shadow

Show answer
Correct Option: C
Explanation:

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 $y$.

If $x$ increases and hence  $y$ decreases proportionally, it is called Inverse proportion i.e $x\propto\dfrac{1}{y}$
For example, In option C, if our expenses are more, then we are left with lesser savings from a fixed salary or income.
hence, Expenditure and savings are in inverse proportion.
So the answer is C

Which is example of inverse proportion?

maths rule of three inverse proportion direct proportion and inverse proportion types of proportions

  1. Length of edge of square & its area

  2. Speed and time taken

  3. Length of edge of square & its perimeter

  4. Quantity of pencils & total cost

Show answer
Correct Option: B
Explanation:

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 $y$.


we know that,
$speed = \dfrac{distance}{time}$

i.e., Speed is directly proportion to distance and at a fixed distance  inverse proportion to time.

Hence at fixed distance, if speed increses, time taken will decrease and vice versa. which is Option B

A car travels $40\ kms$ in $30$ mins. If the speed of bus remains same, how far can it travel in $3$ hrs?

maths rule of three inverse proportion direct proportion and inverse proportion types of proportions

  1. $120\ km$

  2. $80\ km$

  3. $240\ km$

  4. None of these

Show answer
Correct Option: C
Explanation:
A car travels $40$ kms in $30$ mins
Let $x$ be the distance covered by a bus in $3$ hrs $=180$ mins
Then, $\dfrac {40}{30} = \dfrac {x}{180}$

$\Rightarrow x = \dfrac {40\times 180}{30} = 240\ km$.

The number of workers increased the days to complete the work decreased."is example of 

maths direct proportion and inverse proportion inverse proportion rule of three types of proportions

  1. Inverse ratio

  2. Direct ratio

  3. Both of A and B

  4. none of above

Show answer
Correct Option: A
Explanation:

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, $x \alpha \dfrac{1}{y}$ 

In the given example, if the no. of workers are increased then the time required to do the job will decrease, likewise if no. of workers decrease, then time to do the same job will increase i.e. no. of workers and time to do job are inversely proportional to each other.
hence the answer is inverse proportion
So, the answer is option A.

Four angles of a quadrilateral are in the ratio $3:5:7:9$. The greatest angle is _________.

maths direct proportion and inverse proportion inverse proportion rule of three types of proportions

  1. $125^o$

  2. $75^o$

  3. $135^o$

  4. $120^o$

Show answer
Correct Option: C
Explanation:
Let the angles of quadrilateral be $x,y,z,w.$
 
Given ,$x:y:z:w=3:5:7:9$

$x=3k$
$y=5k$
$z=7k$
$w=9k$

we k.n.t sum of angles of a quadrilateral is $360^{\circ }$
$x+y+z+w=3k+5k+7k+9k=360^{\circ }$
$\Rightarrow 24k=360^{\circ }$
$\Rightarrow k=\dfrac{360^{\circ }}{24}=15^{\circ }$

Greatest angle is $w=9k=9(15^{\circ })=135^{\circ }$

A stone is dropped into a well and the report of the stone striking the bottom is heard $7.7$ seconds after it is dropped. Assume that the stone falls $16t^2m$ in t seconds and that the velocity of sound is $1,120$  per second. The depth of the well is:

maths direct proportion and inverse proportion inverse proportion rule of three types of proportions

  1. $784$m

  2. $342$m

  3. $1568$m

  4. $156.8$m

  5. none of these

Show answer
Correct Option: A
Explanation:

Since the distance d travelled by the stone and the sound is the same,

 $d = r _1t _1 = r _2t _2$.
 $\therefore$ The times of travel are inversely proportional to the rates of fail: 
$r _1/r _2 = t _2/t _1$,
$\therefore \dfrac{16t^2/t}{1120} = \dfrac{7.7-t}{t}; $
 $\therefore t = 7 (seconds);$
$ \therefore 16t^2 = 16\times 7^2 = 784 (metre)$.

A farmer has enough food to feed $28$ animals in his cattle for 9 days. How long would the food last if there were $8$ more animals in his cattle ?

maths direct proportion and inverse proportion inverse proportion rule of three types of proportions

  1. $5$

  2. $7$

  3. $10$

  4. None of these

Show answer
Correct Option: B
Explanation:
let say each animal eat $x$ amount of food per day
$\rightarrow$ Amount of food eaten by $28$ animal for $1$ day
$=28 \times x$
$\rightarrow$ Amount of food required for $28$ animal for $9$ days
$=28 x \times 9$
$=(28 \times 9)x \ 252x$
$\therefore$ The farmer has $252x$ amount of food.
now, the number if animal $=28+8=36$.
$\boxed{No. of\ days\ the\ food\ will\ last = \dfrac{amount\ of\ food}{food\ required\ for\ 1\ day}}........1$
food required for $1$ day $=(no.of\ animals) \times x$
food requires for $1$ day $=36x $ (put\ this\ in\ $1$)
No. o days food will last $=\dfrac{252x}{36x}= \dfrac{252}{36}=7$
$\boxed{\because The\ food\ last\ for\ 7\ days\ if\ 8\ animals\ are\ added}$  

































Mark the correct alternative of the following.
If six men can do a piece of work in $6$ days, then $3$ men can do same work in?

maths direct proportion and inverse proportion inverse proportion rule of three types of proportions

  1. $10$ days

  2. $12$ days

  3. $15$ days

  4. $18$ days

Show answer
Correct Option: B
Explanation:

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


$6 \times 6 = 3 \times x$


$x = \dfrac{36}{3}$

$x = 12 $ days

Ten men, working for 6 days of 10 hours each, finish $ \dfrac {5}{21} $ of a piece of work. How many men working at the same rate and for the same number of hours each day, will be required to complete the remaining work in 8 days?

maths direct proportion and inverse proportion inverse proportion rule of three types of proportions

  1. 24

  2. 26

  3. 25

  4. 21

Show answer
Correct Option: A
Explanation:

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

Present ages of $X\;and\;Y$ are in the ratio $5\,\colon\,6$ respectively. Seven years hence this ratio will become $6\,\colon\,7$ respectively. What is $X's$ present age ?

maths direct proportion and inverse proportion inverse proportion rule of three types of proportions

  1. $35$ years

  2. $42$ years

  3. $49$ years

  4. Can't be determined

Show answer
Correct Option: A
Explanation:

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


 $x$ $ 2$  $4$ $ 20$  $15$
 $y$  $3$  $7$ $ 10$  $12$
 $z$  $10$ $ 5$ $ 1$  $4/3$

Observe the table and find the quantities which are in inverse proportion

maths direct proportion and inverse proportion inverse proportion rule of three types of proportions

  1. $x \ and \  y$

  2. $x \ and \ z$

  3. $y \ and \ z$

  4. None of the above

Show answer
Correct Option: B
Explanation:

By definition of inverse proportion,

$a \  \alpha \  \dfrac{1}{b}$ i.e. $a = K \dfrac{1}{b}$                $K=$ constant of proportionality
$\therefore a\times b = K$  
From given example, consider
$x \times y =2\times 3 \neq 4\times 7 \neq 20\times 10\neq 15\times 12$
$x\times z = 2\times 10 = 4\times 5 = 20\times 1 = 15\times \dfrac{4}{3} = 20$
$\therefore$ $x$ and $z$ obey the equation $1$
Hence $x$ and $z$ are in inverse proportion.
Answer is B

$ x$  $2$ $ 5$ $ 25$
$ y$  $25$  $10$  $m$

If $x$ & $y$ are in inverse proportion, find m

maths direct proportion and inverse proportion inverse proportion rule of three types of proportions

  1. $1$

  2. $2$

  3. $3$

  4. $4$

Show answer
Correct Option: B
Explanation:

The given example is of inverse proportion.

by definition, 
$x\propto \dfrac{1}{y}$ i.e. $x = K\dfrac{1}{y}$ , where $K$ is constant of proportionality
$\therefore xy = K$
$x\times y = 2\times 25 = 50 = K$
now, $25\times m = 50$
$\therefore m = 2$
Answer is option B

Which of the following $x$ & $y$ are in inverse proportion?

maths direct proportion and inverse proportion inverse proportion rule of three types of proportions

  1. $\dfrac{x}{y}=K$

  2. $x+y=K$

  3. $x-y=K$

  4. $xy=K$

Show answer
Correct Option: D
Explanation:
 $\dfrac{x}{y}=k$$\implies x=ky$$\implies x\alpha y$  $x+y=k$$\implies x=k-y$  $x-y=k$$\implies x=k+y$ $xy=k$$\implies x=\dfrac{k}{y}$$\implies x\alpha \dfrac{1}{y}$ 

Answer $(D)$

 $x$  $2$ $ 3$ $ 5$ $ 6$ $10$
$ y$  $15$  $10$ $ b$  $5$ $ 3$

Identify the inverse proportional quantities.

maths direct proportion and inverse proportion inverse proportion rule of three types of proportions

  1. $4$

  2. $5$

  3. $6$

  4. $10$

Show answer
Correct Option: C
Explanation:

The given example is of inverse proportion.

by definition, 
$x\propto \dfrac{1}{y}$ i.e. $x = K\dfrac{1}{y}$ , where $K$ is constant of proportionality
$\therefore xy = K$
$x\times y = 2\times 15 = 30 = K$
now, $5\times b = 30$
$\therefore b = 6$
Answer is option C

Find inverse proportionaly constant if $x$ and $y$ are in inverse proportion.

$x$ $9$ $6$ $3$ $18$
$y$ $2$ $3$ $6$ $1$
maths direct proportion and inverse proportion inverse proportion rule of three types of proportions

  1. $9$

  2. $18$

  3. $27$

  4. $30$

Show answer
Correct Option: B
Explanation:

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

A work is done by $10$ workers in $6$ hours? How many workers will be required to do the same work in $4$ hrs?

maths direct proportion and inverse proportion inverse proportion rule of three types of proportions

  1. $10$

  2. $15$

  3. $20$

  4. None of these

Show answer
Correct Option: B
Explanation:

It is inverse variation.
$6\times 10 = 4\times x$
$\Rightarrow x = \dfrac {6\times 10}{4} = 15\ workers$

$35$ workers can build a house in $16$ days. How many days will $28$ workers working at the same rate take to build the same house?

maths direct proportion and inverse proportion inverse proportion rule of three types of proportions

  1. $16\ days$

  2. $28\ days$

  3. $20\ days$

  4. $10\ days$

Show answer
Correct Option: C
Explanation:

By inverse proportion,

$28\times x = 35\times 16$

$x = \dfrac {35\times 16}{28} = 20\ days$

Four pipes can fill a tank in $70$ min. How long will it take to fill the tank by $7$ pipes?

maths direct proportion and inverse proportion inverse proportion rule of three types of proportions

  1. $20\ min$

  2. $35\ min$

  3. $40\ min$

  4. None of these

Show answer
Correct Option: C
Explanation:

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

If $\displaystyle \frac { a }{ b } -\frac { c }{ d } =0$ and bc=7, then determine the true statement among the following.

maths direct proportion and inverse proportion inverse proportion rule of three types of proportions

  1. a and b are directly proportional.

  2. a and c are inversely proportional.

  3. a and d are inversely proportional.

  4. b and c are directly proportional.

  5. c and d are inversely proportional.

Show answer
Correct Option: C
Explanation:

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

$\Rightarrow \dfrac{a}{b}=\dfrac{c}{d}$
$\Rightarrow ad=bc$
According to the question $bc=7$
$\therefore ad=7$
Then $a$ and $d$ are inversely proportional to eachother.

Which of the following are in inverse proportion?

maths direct proportion and inverse proportion inverse proportion rule of three types of proportions

  1. The number of workers on a job and the time to complete the job.

  2. The time taken for a journey and the distance travelled in a uniform speed.

  3. Area of cultivated land and the crop harvested.

  4. The time taken for a fixed journey and the speed of the vehicle.

  5. The population of a country and the area of land per person.

Show answer
Correct Option: A,D,E
Explanation:

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, $x \alpha \dfrac{1}{y}$ 


In option A, if the no. of workers are increased then the time required to do the job will decrease, likewise if no. of workers decrease, then time to do the same job will increase i.e. no. of workers and time to do the job are inversely proportional to each other.


Similarly is the option D and E where the time taken to complete the journey is inversely proportional to the speed of vehicle and population is inversely proportional to area of the land respectively, are both examples of inverse proportion. 

So, the answer is A, D, E.

The length of a pendulum varies inversely as the square of the number of beats it makes per minute. If a pendulum, $65$ cm long, makes $27$ beats per minute, then the length of the pendulum that makes $24$ beats per minutes is 

maths direct proportion and inverse proportion inverse proportion rule of three types of proportions

  1. $91$ cm

  2. $85$ cm

  3. $81$ cm

  4. $71$ cm

Show answer
Correct Option: C
Explanation:
Length $\alpha \cfrac{1}{(\text {No. of beats} )^2}$
$ \Rightarrow L = \cfrac{k}{\text {(beat)}^2}$, where $k$ is constant.
$\Rightarrow 65 = \cfrac{k}{(27)^2}$
$\Rightarrow k = 65 \times (27)^2$ 
Also, $k=L\times(24)^2$
$\Rightarrow 65 \times (27)^2= L \times(24)^2$
$\Rightarrow L = 81 cm $(approx)

Which of the following is an example of indirect proportion?

maths direct proportion and inverse proportion inverse proportion rule of three types of proportions

  1. Number of mangoes in a bag and weight of the bag.

  2. Earning and hours worked.

  3. More the speed, lesser the time taken.

  4. None of these.

Show answer
Correct Option: C
Explanation:

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.

A fort is provisioned for $42$ days. After $10$ days, a reinforcement of $200$ men arrives and the food will now last only $24$ days. How many men were there in the fort?

maths direct proportion and inverse proportion inverse proportion rule of three types of proportions

  1. $200$

  2. $600$

  3. $500$

  4. $300$

Show answer
Correct Option: B
Explanation:

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

After $10$ days: $x$ soldiers will remain with $32$ days of provision
therefore for $(x+200)$ soldiers provision will decrease to $24$ days
therefore By inverse proportion,
$x\times32= (200+x)\times 24$
$\therefore \dfrac{32}{24} = \dfrac{200 + x}{x}$
$\therefore \dfrac{32}{24} = \dfrac{200}{x} + 1$
$\therefore \dfrac{32}{24} - 1 = \dfrac{200}{x}$
$\therefore \dfrac{8}{24} = \dfrac{200}{x}$
$\therefore x = \dfrac{200\times 24}{8}$
$\therefore x = 600$
Answer is $600$ days

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