Chapter 10: Direct and Inverse Proportions

Ans: 45 persons complete a work in 20 days

1 person can complete work in = $45 \times  20 = 900$ days

Time taken by 75 persons$ =\dfrac{900}{75}$ = 12 days = $12\times 24 = 288$ hr

Ans: Distance traveled in 75 steps $= 50$ m

Distance covered in 1 step = $\dfrac{50}{75} $m

Distance covered in 375 steps $= \left( \dfrac{50}{75} \right)\times 375=$ $\dfrac{18750}{75}$= 250 m= $\dfrac{250}{1000}$  km= 0.25km

Ans: False, Two quantities x and y are said to vary directly with each other, if

$x\propto y$

$\Rightarrow  x=ky$

Ans: False, When the speed is kept fixed, time and distance vary directly with each other.

$\text{Speed}=\dfrac{\text{distance}}{\text{time}}$

Since, speed is fixed so,

$Const\propto \dfrac{d}{t} \Rightarrow t\propto d$

Ans: False, When the distance is kept fixed, speed and time vary indirectly/inversely with each other. Since, if we increase speed, then time taken will be less and vice-versa.

$\text{Speed}=\dfrac{\text{distance}}{\text{time}}$

Since, distance is fixed so,

$s\propto \dfrac{1}{t}$

Ans: False, Length of a side of a square and its area does not vary directly with  each other, e.g.

Let a be the length of each side of a square.$  Area={{a}^{2}}$

So, if we increase the length of the side of a square, then their area increases but not directly.

Ans: False, Length of a side of an equilateral triangle and its perimeter vary directly with each other, e.g. Let a be the side of an equilateral triangle

$\text{Perimeter}=3\times \text{Side} \Rightarrow \text{Perimeter} \propto \text{Side}$

So, if we increase the length of the side of the equilateral triangle, then their     perimeter will also increase.

Ans: False, If d varies inversely as $t^2$, then we can write $dt^2 = k$, where k is some constant.

Since, two quantities x and y are said to be in Inverse proportion, if an increase in x causes a proportional decrease in y and vice-versa, in such a manner that the product of their corresponding values remains constant.

Ans: True

Height of tree $= 24$ m

Length of shadow of tree $= 15$ m

Let height of pole $= x$

Length of shadow of pole $= 6$ m

Now,

$\dfrac{24}{15} =\dfrac{x}{6}$

$\Rightarrow  x =\dfrac{24\times 6}{15}$

$\Rightarrow x =9.6$ m

Ans: False

$x$ and $y$ are in direct proportion, then,

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

if $x = 3$ and $y = 5$, then

$\Rightarrow k =\dfrac{3}{5}$

Now, $x-1 = 2$ and $y-1=4$

$\Rightarrow $  $\dfrac{x-1}{y-1}=\dfrac{2}{4} =\dfrac{1}{2}$