Direct and Inverse Variation — Statement I-II Type Questions

Statement I: If on increasing a quantity $x$, another quantity $y$ also increases, then $x$ and $y$ vary inversely to each other.

Statement II: Two quantities $x$ and $y$ vary inversely if $x = \dfrac{k}{y}$, where $k$ is a constant.

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

If on increasing a quantity $x$, another quantity $y$ also increases, then $x$ and $y$ vary directly and not inversely to each other.

So, Statement I is false.

Two quantities $x$ and $y$ vary inversely if $xy = k$, i.e. if $x = \dfrac{k}{y}$, where $k$ is a constant.

So, Statement II is true.

Hence, Option 2 is the correct option.

Statement I: If 5 pipes can fill a tank in 10 hours, then 10 pipes are needed to fill the same tank in half the time.

Statement II: Time required to do a certain work = $\dfrac{\text{work to be done}}{\text{one day's work}}$

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

Less the time in which the tank is to be filled, more will be the number of pipes required, i.e. number of pipes and time are in inverse variation.

Half of the time = $\dfrac{10}{2} = 5$ hours

Let $x$ pipes be needed to fill the tank in 5 hours.

$\Rightarrow 5 \times 10 = x \times 5\\[1em] \Rightarrow 50 = 5x\\[1em] \Rightarrow x = \dfrac{50}{5}\\[1em] \Rightarrow x = 10$

So, Statement I is true.

Time required to do a certain work = $\dfrac{\text{work to be done}}{\text{one day's work}}$

So, Statement II is true.

Hence, Option 3 is the correct option.

Statement I: Three friends Rohit, Virat and Rishabh build a wall. Individually, they can do the task in 1, 2 and 4 days respectively. We can say that together they will take less than 1 day to build the wall.

Statement II: If they get ₹9000 for the job, they should divide it equally among themselves.

1. Statement I is true but Statement II is false.

2. Statement I is false but Statement II is true.

3. Both Statement I and Statement II are true.

4. Both Statement I and Statement II are false.

Answer

Rohit's one day's work = $\dfrac{1}{1} = 1$

Virat's one day's work = $\dfrac{1}{2}$

Rishabh's one day's work = $\dfrac{1}{4}$

$\text{(Rohit + Virat + Rishabh)'s one day's work} = 1 + \dfrac{1}{2} + \dfrac{1}{4}\\[1em] = \dfrac{4 + 2 + 1}{4}\\[1em] = \dfrac{7}{4}$

Number of days taken by them working together = $\dfrac{1}{\dfrac{7}{4}} = \dfrac{4}{7}$

As $\dfrac{4}{7}$ is less than 1, they will take less than 1 day to build the wall.

So, Statement I is true.

Remuneration is in proportion of the work done. Since Rohit, Virat and Rishabh do different amounts of work, they should divide the money in the ratio $1 : \dfrac{1}{2} : \dfrac{1}{4}$, i.e. 4 : 2 : 1 and not equally among themselves.

So, Statement II is false.

Hence, Option 1 is the correct option.