Chapter 10: Direct and Inverse Proportions

Ans: Both $x $and $y$ are said to vary inversely with each other if for some positive number $k$, $xy = k.$

Ans: $x$ and $y$ are said to vary directly with each other if for some positive number $k,$ $ \dfrac{x}{y}$ = $k.$

Ans: Two quantities are said to vary directly with each other if they increase (decrease) together in such a manner that the ratio of their corresponding values remains constant.

Ans: Two quantities are said to vary inversely with each other if an increase in one causes a decrease in the other in such a manner that the product of their corresponding values remains constant.

Ans: Time taken by 12 pumps to empty a reservoir = 20 hr

Time taken by 1 pump to empty the reservoir $= 20 \times  12 = 240$ hr

Hence, time taken by 45 pumps to empty the reservoir

$=\dfrac{240}{45}=\dfrac{240\times 60}{45}=\dfrac{14400}{45} =$ 320 min$= 5\times 60+20$ min= $5$ hour $20$ min

Ans: If x varies inversely as y, then,

$     x\times y = k$

$\Rightarrow $ If$ x = 60$ and $y = 10$

Then,

$ x\times y = 60\times 10 = 600 = k$

Hence, $x\times 2 = 600$

$\Rightarrow $ $x      =\dfrac{600}{2}= 300$

Ans: If $x$ and $y $varies directly, then,

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

If $x=12$ and$ y = 48$, then

$\Rightarrow $  $k =\dfrac{12}{48}=\dfrac{1}{4}$

Now, if $x=6$, and $k =\dfrac{1}{4},$ then y will be,

$       \dfrac{6}{y} =\dfrac{1}{4}$

$\Rightarrow $  $y = 6\times 4=24$

Ans: When the speed remains constant, the distance traveled is directly proportional to the time.

Ans: On increasing a, b increases in such a manner that a/b remains constant and positive, then a and b are said to vary directly with each other.

Ans: If on increasing a, b decreases in such a manner that ab remains constant and positive, then a and b are said to vary inversely with each other.