Direct and Inverse Variation — Multiple Choice Questions

Two quantities $x$ and $y$ are said to be in inverse variation if

1. $xy = k$

2. $x \propto \dfrac{1}{y}$

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

4. all of these

Answer

Two quantities $x$ and $y$ are said to be in inverse variation if their product is constant.

$\Rightarrow xy = k\\[1em] \Rightarrow x = \dfrac{k}{y}\\[1em] \Rightarrow x \propto \dfrac{1}{y}$

So, all the three given relations represent inverse variation.

Hence, Option 4 is the correct option.

If 12 metre of wire costs ₹240, then cost of 8 metre wire is

1. ₹160

2. ₹200

3. ₹120

4. ₹180

Answer

More the length of the wire, more will be its cost, i.e. length and cost are in direct variation.

Let the cost of 8 metre wire be ₹ $x$.

$\Rightarrow \dfrac{12}{240} = \dfrac{8}{x}\\[1em] \Rightarrow x = \dfrac{240 \times 8}{12}\\[1em] \Rightarrow x = \dfrac{1{,}920}{12}\\[1em] \Rightarrow x = 160$

Hence, Option 1 is the correct option.

If 5 kg wheat cost ₹120, then cost of 20 kg wheat is

1. ₹400

2. ₹420

3. ₹440

4. ₹480

Answer

More the quantity of wheat, more will be its cost, i.e. quantity and cost are in direct variation.

Let the cost of 20 kg wheat be ₹ $x$.

$\Rightarrow \dfrac{5}{120} = \dfrac{20}{x}\\[1em] \Rightarrow x = \dfrac{120 \times 20}{5}\\[1em] \Rightarrow x = \dfrac{2{,}400}{5}\\[1em] \Rightarrow x = 480$

Hence, Option 4 is the correct option.

If 10 men can complete a work in 6 days, then 30 men can complete the same work in

1. 2 days

2. 3 days

3. 4 days

4. 5 days

Answer

More the number of men at work, less will be the number of days taken, i.e. number of men and number of days are in inverse variation.

Let 30 men complete the work in $x$ days.

$\Rightarrow 10 \times 6 = 30 \times x\\[1em] \Rightarrow 60 = 30x\\[1em] \Rightarrow x = \dfrac{60}{30}\\[1em] \Rightarrow x = 2$

Hence, Option 1 is the correct option.

If a car travels 80 km in 5 litres of petrol, then distance covered by it in 15 litres of petrol is

1. 400 km

2. 240 km

3. 200 km

4. 100 km

Answer

More the quantity of petrol, more will be the distance covered, i.e. petrol and distance are in direct variation.

Let the distance covered in 15 litres of petrol be $x$ km.

$\Rightarrow \dfrac{80}{5} = \dfrac{x}{15}\\[1em] \Rightarrow x = \dfrac{80 \times 15}{5}\\[1em] \Rightarrow x = \dfrac{1200}{5}\\[1em] \Rightarrow x = 240$

Hence, Option 2 is the correct option.

In a college mess there was enough food for 200 students for 20 days. If 50 new students joined them, then the food will last for

1. 15 days

2. 16 days

3. 17 days

4. 18 days

Answer

Number of students after 50 new students joined = 200 + 50 = 250

More the number of students, less will be the number of days for which the food lasts, i.e. number of students and number of days are in inverse variation.

Let the food last for $x$ days for 250 students.

$\Rightarrow 200 \times 20 = 250 \times x\\[1em] \Rightarrow 4000 = 250x\\[1em] \Rightarrow x = \dfrac{4000}{250}\\[1em] \Rightarrow x = 16$

Hence, Option 2 is the correct option.

3 persons can paint a house in 8 days, then 4 persons can paint the same house in

1. 5 days

2. 6 days

3. 7 days

4. none of these

Answer

More the number of persons at work, less will be the number of days taken, i.e. number of persons and number of days are in inverse variation.

Let 4 persons paint the house in $x$ days.

$\Rightarrow 3 \times 8 = 4 \times x\\[1em] \Rightarrow 24 = 4x\\[1em] \Rightarrow x = \dfrac{24}{4}\\[1em] \Rightarrow x = 6$

Hence, Option 2 is the correct option.

A photograph of bacteria is enlarged 100000 times attains a length of 5 cm, then actual length of the bacteria is

1. 0.00005 cm

2. 5 × 10^-5 cm

3. 5 × 10^-7 m

4. all of these

Answer

The photograph is enlarged 100000 times, so the actual length of the bacteria is $\dfrac{1}{100000}$ times the length in the photograph.

Actual length of the bacteria = $\dfrac{5}{100000}$ cm

= 0.00005 cm

= 5 × 10^-5 cm

Converting the length into metres,

5 × 10^-5 cm = 5 × 10^-5 × $\dfrac{1}{100}$ m = 5 × 10^-7 m

So, all the three given values represent the actual length of the bacteria.

Hence, Option 4 is the correct option.

A tree 12 metre high casts a shadow of length 8 metre. Height of the tree, whose shadow is 6 metre in length is

1. 6 m

2. 9 m

3. 15 m

4. none of these

Answer

At the same time, more the height of an object, more will be the length of its shadow, i.e. height and length of shadow are in direct variation.

Let the height of the tree be $x$ m.

$\Rightarrow \dfrac{12}{8} = \dfrac{x}{6}\\[1em] \Rightarrow x = \dfrac{12 \times 6}{8}\\[1em] \Rightarrow x = \dfrac{72}{8}\\[1em] \Rightarrow x = 9$

Hence, Option 2 is the correct option.

If 5 pipes can fill the tank in 1 hour, then 4 pipes will fill the tank in

1. 75 minutes

2. 70 minutes

3. 65 minutes

4. none of these

Answer

Converting the time into minutes,

1 hour = 60 minutes

Less the number of pipes, more will be the time taken to fill the tank, i.e. number of pipes and time are in inverse variation.

Let 4 pipes fill the tank in $x$ minutes.

$\Rightarrow 5 \times 60 = 4 \times x\\[1em] \Rightarrow 300 = 4x\\[1em] \Rightarrow x = \dfrac{300}{4}\\[1em] \Rightarrow x = 75$

Hence, Option 1 is the correct option.

A tap fills a tank in 8 hours and another tap at the bottom empties it in 10 hours. If both work together, the tank will be filled in

1. 18 hours

2. 24 hours

3. 36 hours

4. 40 hours

Answer

Portion of the tank filled by the first tap in one hour = $\dfrac{1}{8}$

Portion of the tank emptied by the second tap in one hour = $\dfrac{1}{10}$

$\text{Portion of the tank filled in one hour when both taps work together} = \dfrac{1}{8} - \dfrac{1}{10}\\[1em] = \dfrac{5 - 4}{40}\\[1em] = \dfrac{1}{40}$

Time taken to fill the tank = $\dfrac{1}{\dfrac{1}{40}} = 40$ hours

Hence, Option 4 is the correct option.