Chapter 10: Direct and Inverse Proportions

Ans: Number of students in a hostel and consumption of food are directly proportional to each other. For example: let 200 students in a hostel consume 100 kg of rice in a month. Then, 400 students in a hostel can consume 200 kg of rice in a month.

Ans: Area of the walls of a room and the cost of white washing the balls are directly proportional to each other.

For example: let ₹10000 required for white washing a room with $\left( 12\times 8 \right)m$ size.

Then, ₹20000 required for white washing a room with $\left( 12\times 16 \right)m$ size.

Ans: The number of people working, and the quantity of work are directly proportional to each other.

For example: let 200 workers complete 20% of our work.

Then, 400 workers can complete 40% of the same work.

Ans: Simple interest on a given some under rate of interest are directly proportional to each other.

For example: Let P = ₹1000, R = 10% and T = 1yr

$SI=\dfrac{P\times R\times T}{100}=\dfrac{1000\times 10\times 1}{100}=$ ₹ 100

But if P = ₹1000, R = 20% and T = 1yr

$SI=\dfrac{P\times R\times T}{100}=\dfrac{1000\times 20\times 1}{100}=$ ₹ 200

Ans: Compound interest on some under some investors are neither dependent directly or inversely.

Ans: The quantity of rice and its cost are directly proportional to each other, e.g. 

Let 1 kg of rice price = Rs 40

Then, 2 kg of rice price $= Rs 2\times 40 = Rs 80$

Ans: The height of a tree and the number of years is neither directly nor inversely proportional to each other.

Ans: Increase in cost and number of shirts that can be purchased, if the budget remains the same are inversely proportional to each other.

E.g.: Let 2 shirts price = Rs 800

After increasing the price of each shirt,1 shirt price became Rs 800 where budget = Rs 800.

Ans: Area of land and its cost are directly proportional to each other.

E.g., Let $200{{m}^{2}}$ land cost = Rs. 1000

Then, $400{{m}^{2}}$ land cost = Rs. 2000

Ans: Sales tax and the amount of the bill are directly proportional to each other.

e.g. Let bill amount = Rs. 1000

Sales tax = 10%

Then, sales tax $=\dfrac{10}{100}\times 1000=Rs. 100$

But if, bill amount = 2000

Sales tax = 10%

Then, sales tax = $\dfrac{10}{100}\times 2000=10\times 20=Rs. 200$

Ans: If $x$ varies inversely with $y$,

$xy = k$ (constant) .... (i)

If$ x = 20$ and $y = 600 $

$xy = 20\times 600      $

$ =12000  $

$k = 12000  $

When $x = 400$ then from equation-(i),

$ \Rightarrow $   $y\times 400=k $

$\Rightarrow $   $y\times 400=12000 $

$\Rightarrow $  $y = 30  $

Ans: If x varies directly as y

$ \dfrac{x}{y}=k$

If $x = 80 $and $y = 160$

$\Rightarrow $   $\dfrac{x}{y}=\dfrac{80}{160}=\dfrac{1}{2}$

When $x = 64$,

$ \dfrac{64}{y}=\dfrac{1}{2}$

$\Rightarrow   y=128$

Ans: If l varies directly as m,

$\dfrac{l}{m}=\dfrac{5}{\dfrac{2}{3}}\ldots \left( i \right)$

$\dfrac{l}{m} =5\times \dfrac{3}{2}=\dfrac{15}{2}$

$\Rightarrow k=\dfrac{15}{2}$

When $m=\dfrac{16}{3}, $then from (i),

$\dfrac{l}{\dfrac{16}{3}}=\dfrac{15}{2}$

$\Rightarrow $   $l=\dfrac{15}{2}\times \dfrac{16}{3}=40$

Answer: If $x$ varies inversely as y

$xy=k$ (constant) …(i)

If $x=1.5$ and $y=60$

$\therefore xy=1.5\times 60=90$

$\Rightarrow $ $k=90$

When $y=4.5,$then from Eq. (i)

$4.5\times y=k$

$\Rightarrow 4.5\times y=90$

$\Rightarrow  y=\dfrac{90}{4.5}=20$

Ans: Since, for 300 people, flour is enough for 42 days.

Therefore, for 1 person flour enough$=300\times 42=12600$ days

Now, 20 more people have joined the camp.

So, total persons$=300+20=320$

Therefore, for 320 persons flour enough $=\dfrac{12600}{320}=\dfrac{3}{8}$ days

Ans: 560 people complete it partially in 9 months

$\therefore $1 person completes it in 9×560= 5040

$\therefore $ In 5 months the work can be completed =$\dfrac{5040}{5}$=1008 persons

Now, the number of extra persons should he employ to complete the work in 5 months

=1008-560=448 persons

Ans: 1 hour = 60 minutes $\Rightarrow $Half hour = 30 minutes

In 6 minutes Sobi can type = 108 words

In 1 minute she can type = $\dfrac{108}{6}$

In 30 minutes she can type = $\dfrac{108}{6}\times $ 30 = 540 words

Ans: car covers a distance in 40 min with an average speed =60km/h = $\dfrac{60 \times 1000}{60}$m/min

In 1 min, the same distance can be cover with speed =$\dfrac{60 \times 1000\times 40 }{60}$= 40000 m/min

In 25 min, the same distance can be cover with speed = $\dfrac{40000 }{25}$= 1600 m/min

= $\dfrac{1600 }{1000}$ $\times  60$ = 16 $\times  6$ = 96  km/h